Maths formulae booklet (iit jee)

MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E D U C A T I O N S
QUADRATIC EQUATION & EXPRESSION
1. Quadratic expression :
A polynomial of degree two of the form ax2
+ bx + c, a ≠ 0 is
called a quadratic expression in x.
2. Quadratic equation :
An equation ax2
+ bx + c = 0, a ≠ 0, a, b, c ∈ R has two and
only two roots, given by
α =
− + −b b ac
a
2
4
2
and β =
− − −b b ac
a
2
4
2
3. Nature of roots :
Nature of the roots of the given equation depends upon the
nature of its discriminant D i.e. b2
– 4ac.
Suppose a, b, c ∈ R, a ≠ 0 then
(i) If D > 0 ⇒ roots are real and distinct (unequal)
(ii) If D = 0 ⇒ roots are real and equal (Coincident)
(iii) If D < 0 ⇒ roots are imaginary and unequal i.e.
non real complex numbers.
Suppose a, b, c ∈ Q a ≠ 0 then
(i) If D > 0 and D is a perfect square ⇒ roots are rational
& unequal
(ii) If D > 0 and D is not a perfect square ⇒ roots are
irrational and unequal.
For a quadratic equation their will exist exactly 2 roots real
or imaginary. If the equation ax2
+ bx + c = 0 is satisfied for
more than 2 distinct values of x, then it will be an identity &
will be satisfied by all x. Also in this case a = b = c = 0.
PAGE # 2PAGE # 1
4. Conjugate roots :
Irrational roots and complex roots occur in conjugate pairs
i.e.
if one root α + iβ, then other root α – iβ
if one root α + β , then other root α – β
5. Sum of roots :
S = α + β =
−b
a
=
−Coefficient of x
Coefficient of x2
Product of roots :
P = αβ =
c
a
=
cons t term
Coefficient of x
tan
2
6. Formation of an equation with given roots :
x2
– Sx + P = 0
⇒ x2
– (Sum of roots) x + Product of roots = 0
7. Roots under particular cases :
For the equation ax2
+ bx + c = 0, a ≠ 0
(i) If b = 0 ⇒ roots are of equal magnitude but of opposite
sign.
(ii) If c = 0 ⇒ one root is zero and other is –b/a
(iii) If b = c = 0 ⇒ both roots are zero
(iv) If a = c ⇒ roots are reciprocal to each other.

E D U C A T I O N S
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 4PAGE # 3
(v) If a > 0, c < 0 or a < 0, c > 0 ⇒ roots are of opposite
signs
(vi) If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 ⇒ both
roots are –ve
(vii) If a > 0, b < 0 , c > 0 or a < 0, b > 0, c < 0 ⇒ both
roots are +ve.
8. Symmetric function of the roots :
If roots of quadratic equation ax2
+ bx + c, a ≠ 0 are α and β,
then
(i) (α – β) = ( )α β αβ+ −2
4 = ±
b ac
a
2
4−
(ii) α2
+ β2
= (α + β)2
– 2αβ =
b ac
a
2
2
2−
(iii) α2
– β2
= (α + β) ( )α β αβ+ −2
4 =
− −b b ac
a
2
2
4
(iv) α3
+ β3
= (α + β)3
– 3(α + β) αβ =
− −b b ac
a
( )2
3
3
(v) α3
– β3
= (α – β) [α2
+ β2
– αβ]
= ( )α β αβ+ −2
4 [α2
+ β2
– αβ]
=
( )b ac b ac
a
2 2
3
4− −
(vi) α4
+ β4
= (α2
+ β2
)2
– 2α2
β2
={(α + β)2
–2αβ}2
– 2α2
β2
=
b ac
a
2
2
2
2−F
HG
I
KJ –
2 2
2
c
a
(vii) α4
– β4
=(α2
+ β2
) (α2
– β2
) =
− − −b b ac b ac
a
( )2 2
4
2 4
(viii) α2
+ αβ + β2
=(α + β)2
– αβ =
b ac
a
2
2
+
(ix)
α
β
+
β
α
=
α β
αβ
2 2
+
=
( )α β αβ
αβ
+ −2
2
(x)
α
β
F
HG I
KJ
2
+
β
α
F
HG I
KJ
2
=
α β
α β
4 4
2 2
+
=
[( ) ]b ac a c
a c
2 2 2 2
2 2
2 2− −
9. Condition for common roots :
The equations a1
x2
+ b1
x + c1
= 0 and a2
x2
+ b2
x + c2
= 0
have
(i) One common root if
b c b c
c a c a
1 2 2 1
1 2 2 1
−
− =
c a c a
a b a b
1 2 2 1
1 2 2 1
−
−
(ii) Both roots common if
a
a
1
2
=
b
b
1
2
=
c
c
1
2

E D U C A T I O N S
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 6PAGE # 5
10. Maximum and Minimum value of quadratic expression :
In a quadratic expression ax2
+ bx + c = a x
b
a
D
a
+
F
HG I
KJ −
L
N
MM
O
Q
PP2 4
2
2 ,
Where D = b2
– 4ac
(i) If a > 0, quadratic expression has minimum value
4
4
2
ac b
a
−
at x =
−b
a2
and there is no maximum value.
(ii) If a < 0, quadratic expression has maximum value
4
4
2
ac b
a
−
at x =
−b
a2
and there is no minimum value.
11. Location of roots :
Let f(x) = ax2
+ bx + c, a ≠ 0 then w.r.to f(x) = 0
(i) If k lies between the roots then a.f(k) < 0
(necessary & sufficient)
(ii) If between k1
& k2
their is exactly one root of k1
, k2
themselves are not roots
f(k1
) . f(k2
) < 0 (necessary & sufficient)
(iii) If both the roots are less than a number k
D ≥ 0, a.f(k) > 0,
−b
a2
< k (necessary & sufficient)
(iv) If both the roots are greater than k
D ≥ 0, a.f(k) > 0,
−b
a2
> k (necessary & sufficient)
(v) If both the roots lies in the interval (k1
, k2
)
D ≥ 0, a.f(k1
) > 0, a.f(k2
) > 0, k1
<
−b
a2
< k2
(vi) If k1
, k2
lies between the roots
a.f(k1
) < 0, a.f(k2
) < 0
(vii) λ will be the repeated root of f(x) = 0 if
f(λ) = 0 and f'(λ) = 0
12. For cubic equation ax3
+ bx2
+ cx + d = 0 :
We have α + β + γ =
−b
a
, αβ + βγ + γα =
c
a
and αβγ =
−d
a
where α, β, γ are its roots.
13. For biquadratic equation ax4
+ bx3
+ cx2
+ dx + e = 0 :
We have α + β + γ + δ = –
b
a
, αβγ + βγδ + γδα + γδβ =
−d
a
αβ + αγ + αδ + βγ + βδ + γδ=
c
a
and αβγδ =
e
a

E D U C A T I O N S
E D U C A T I O N S
COMPLEX NUMBER
1. Complex Number :
A number of the form z = x + iy (x, y ∈ R, i = −1 ) is called
a complex number, where x is called a real part i.e. x = Re(z)
and y is called an imaginary part i.e. y = Im(z).
Modulus |z| = x y2 2
+ ,
amplitude or amp(z) = arg(z) = θ = tan–1
y
x
.
(i) Polar representation :
x = r cosθ, y = r sinθ, r = |z| = x y2 2
+
(ii) Exponential form :
z = reiθ
, where r = |z|, θ = amp.(z)
(iii) Vector representation :
P(x, y) then its vector representation is z = OP
→
2. Integral Power of lota :
i = −1 , i2
= –1, i3
= –i , i4
= 1
Hence i4n+1
= i, i4n+2
= –1, i4n+3
= –i, i4n
or i4(n+1)
= 1
3. Complex conjugate of z :
If z = x + iy, then z = x – iy is called complex conjugate
of z
* z is the mirror image of z in the real axis.
* |z| = | z |
* z + z = 2Re(z) = purely real
* z – z = 2i Im
(z) = purely imaginary
* z z = |z|2
* z z zn1 2+ + +.... = z 1
+ z 2
+ .......... + z n
* z z1 2− = z 1
– z 2
* z z1 2
= z 1 z 2
*
z
z
1
2
F
HG
I
KJ =
z
z
1
2
F
HG I
KJ (provided z2 ≠ 0)
* zn
e j = ( z )n
* zc h = z
* If α = f(z), then α = f( z )
Where α = f(z) is a function in a complex variable
with real coefficients.
* z + z = 0 or z = – z ⇒ z = 0 or z is purely imaginary
* z = z ⇒ z is purely real
4. Modulus of a complex number :
Magnitude of a complex number z is denoted as |z| and is
defined as
|z| = (Re( )) (Im( ))z z2 2
+ , |z| ≥ 0
(i) z z = |z|2
= | z |2
(ii) z–1
=
z
z| |2
(iii) |z1
± z2
|2
= |z1
|2
+ |z2
|2
± 2 Re (z1
z2 )
PAGE # 8PAGE # 7

E D U C A T I O N S
E D U C A T I O N S
(iv) |z1
+ z2
|2
+ |z1
– z2
|2
= 2 [|z1
|2
+ |z2
|2
]
(v) |z1
± z2
| ≤ |z1
| + |z2
|
(vi) |z1
± z2
| ≥ |z1
| – |z2
|
5. Argument of a complex number :
Argument of a complex number z is the ∠ made by its radius
vector with +ve direction of real axis.
arg z = θ , z ∈ 1st
quad.
= π – θ , z ∈ 2nd
quad.
= – θ , z ∈ 3rd
quad.
= θ – π , z ∈ 4th
quad.
(i) arg (any real + ve no.) = 0
(ii) arg (any real – ve no.) = π
(iii) arg (z – z ) = ± π/2
(iv) arg (z1
.z2
) = arg z1
+ arg z2
+ 2 k π
(v) arg
z
z
1
2
F
HG I
KJ = arg z1
– arg z2
+ 2 k π
(vi) arg ( z ) = –arg z = arg
1
z
F
HG I
KJ , if z is non real
= arg z, if z is real
(vii) arg (– z) = arg z + π, arg z ∈ (– π , 0]
= arg z – π, arg z ∈ (0, π ]
(viii) arg (zn
) = n arg z + 2 k π
(ix) arg z + arg z = 0
argument function behaves like log function.
6. Square root of a complex no.
a ib+ = ±
| | | |z a
i
z a+
+
−L
N
MM
O
Q
PP2 2 , for b > 0
= ±
| | | |z a
i
z a+
−
−L
N
MM
O
Q
PP2 2 , for b < 0
7. De-Moiver's Theorem :
It states that if n is rational number, then
(cosθ + isinθ)n
= cosθ + isin nθ
and (cosθ + isinθ)–n
= cos nθ – i sin nθ
8. Euler's formulae as z = reiθ
, where
eiθ
= cosθ + isinθ and e–iθ
= cosθ – i sinθ
∴ eiθ
+ e–iθ
= 2cosθ and eiθ
– e–iθ
= 2 isinθ
9. nth
roots of complex number z1/n
= r1/n cos sin
2 2m
n
i
m
n
π θ π θ+F
HG I
KJ +
+F
HG I
KJL
NM O
QP,
where m = 0, 1, 2, ......(n – 1)
(i) Sum of all roots of z1/n
is always equal to zero
(ii) Product of all roots of z1/n
= (–1)n–1
z
10. Cube root of unity :
cube roots of unity are 1, ω, ω2
where
ω =
− +1 3
2
i
and 1 + ω + ω2
= 0, ω3
= 1
PAGE # 10PAGE # 9

E D U C A T I O N S
E D U C A T I O N S
11. Some important result :
If z = cosθ + isinθ
(i) z +
1
z
= 2cosθ
(ii) z –
1
z
= 2 isinθ
(iii) zn +
1
zn = 2cosnθ
(iv) If x = cosα + isinα , y = cos β + i sin β & z = cosγ + isinγ
and given x + y + z = 0, then
(a)
1
x
+
1
y
+
1
z
= 0 (b) yz + zx + xy = 0
(c) x2
+ y2
+ z2
= 0 (d) x3
+ y3
+ z3
= 3xyz
12. Equation of Circle :
* |z – z1
| = r represents a circle with centre z1
and
radius r.
* |z| = r represents circle with centre at origin.
* |z – z1
| < r and |z – z1
| > r represents interior and
exterior of circle |z – z1
| = r.
* z z + a z + a z + b = 0 represents a general circle
where a ∈ c and b ∈ R.
* Let |z| = r be the given circle, then equation of
tangent at the point z1
is z z 1
+ z z1
= 2r2
* diametric form of circle :
arg
z z
z z
−
−
F
HG I
KJ1
2
= ±
π
2
,
or
z z
z z
−
−
1
2
+
z z
z z
−
−
1
2
= 0
or z
z z
−
+1 2
2
=
| |z z1 2
2
−
or |z – z1
|2
+ |z – z2
|2
= |z1
– z2
|2
Where z1
, z2
are end points of diameter and z is any
point on circle.
13. Some important points :
(i) Distance formula PQ = |z2
– z1
|
(ii) Section formula
For internal division =
m z m z
m m
1 2 2 1
1 2
+
+
MATHS FORMULA - POCKET BOOK MATHS FORMULA - P
For external division =
m z m z
m m
1 2 2 1
1 2
−
−
(iii) Equation of straight line.
* Parametric form z = tz1
+ (1 – t)z2
where t ∈ R
* Non parametric form
z z
z z
z z
1
1
1
1 1
2 2
= 0.
* Three points z1
, z2
, z3
are collinear if
z z
z z
z z
1 1
2 2
3 3
1
1
1
= 0
or slope of AB = slope of BC = slope of AC.
PAGE # 12PAGE # 11

E D U C A T I O N S
E D U C A T I O N S
(iv) The complex equation
z z
z z
−
−
1
2
= k represents a circle
if k ≠ 1 and a straight line if k = 1.
(v) The triangle whose vertices are the points represented
by complex numbers z1
, z2
, z3
is equilateral if
1
2 3z z− +
1
3 1z z− +
1
1 2z z− = 0
i.e. if z1
2
+ z2
2
+ z3
2
= z1
z2
+ z2
z3
+ z1
z3
.
(vi) |z – z1
| = |z – z2
| = λ , represents an ellipse if
|z1
– z2
| < λ , having the points z1
and z2
as its foci
and if |z1
– z2
| = λ , then z lies on a line segment
connecting z1
& z2
(vii) |z – z1
| ~ |z – z2
| = λ represents a hyperbola if
|z1
– z2
| > λ , having the points z1
and z2
as its foci,
and if |z1
– z2
| = λ , then z lies on the line passing
through z1
and z2
excluding the points between z1
& z2
.
(viii) If four points z1
, z2
, z3
, z4
are concyclic,
then
z z
z z
1 2
1 4
−
−
F
HG I
KJ z z
z z
3 4
3 2
−
−
F
HG I
KJ is purely real.
(ix) If three complex numbers are in A.P., then they lie on
a straight line in the complex plane.
(x) If z1
, z2
, z3
be the vertices of an equilateral triangle
and z0
be the circumcentre,
then z1
2
+ z2
2
+ z3
2
= 3z0
2
.
(xi) If z1
, z3
, z3
....... zn
be the vertices of a regular
polygon of n sides & z0
be its centroid, then
z1
2
+ z2
2
+ ......... + zn
2
= nz0
2
.
PAGE # 14PAGE # 13
(xii) If z1
, z2
, z3
be the vertices of a triangle, then the
triangle is equilateral
iff (z1
– z2
)2
+ (z2
– z3
)2
+ (z3
– z1
)2
= 0.
(xiii) If z1
, z2
, z3
are the vertices of an isosceles triangle,
right angled at z2
,
then z1
2
+ z2
2
+ z3
2
= 2z2
(z1
+ z3
).
(xiv) z1
, z2
, z3
. z4
are vertices of a parallelogram then
z1
+ z3
= z2
+ z4

E D U C A T I O N S
E D U C A T I O N S
PERMUTATION & COMBINATION
1. Factorial notation -
The continuous product of first n natural numbers is called
factorial
i.e. n or n! = 1. 2. 3........(n – 1).n
n! = n(n – 1)! = n(n – 1)(n – 2)! & so on
or n (n – 1)......... (n – r + 1) =
n
n r
!
( )!−
Here 0! = 1 and (–n)! = meaningless.
2. Fundamental principle of counting -
(i) Addition rule : If there are two operations such that
they can be done independently in m and n ways
respectively, then either (any one) of these two
operations can be done by (m + n) ways.
Addition ⇒ OR (or) Option
(ii) Multiplication rule : Let there are two tasks of an
operation and if these two tasks can be performed in m
and n different number of ways respectively, then the
two tasks together can be done in m × n ways.
Multiplication ⇒ And (or) Condition
(iii) Bijection Rule : Number of favourable cases
= Total number of cases
– Unfavourable number of cases.
3. Permutations (Arrangement of objects) -
(i) The number of permutations of n different things taken
r at a time is n
pr
=
n
n r
!
( )!−
(ii) The number of permutations of n dissimilar things taken
all at a time is n
pn
= n!
(iii) The number of permutations of n distinct objects taken
r at a time, when repetition of objects is allowed is nr
.
(iv) If out of n objects, 'a' are alike of one kind, 'b' are alike
of second kind and 'c' are alike of third kind and the
rest distinct, then the number of ways of permuting
the n objects is
n
a b c
!
! ! !
4. Restricted Permutations -
(i) The number of permutations of n dissimilar things taken
r at a time, when m particular things always occupy
definite places = n–m
pr–m
(ii) The number of permutations of n different things taken
r at a time, when m particular things are always to be
excluded (included)
= n–m
Pr
(n–m
Cr–m
× r!)
5. Circular Permutations -
When clockwise & anticlockwise orders are treated as
different.
(i) The number of circular permutations of n different things
taken r at a time
n
rP
r
(ii) The number of circular permutations of n different things
taken altogether
n
nP
n
= (n – 1)!
When clockwise & anticlockwise orders are treated
as same.
(i) The number of circular permutations of n different things
taken r at a time
n
rP
r2
(ii) The number of circular permutations of n different things
taken all together
n
nP
n2
=
1
2
(n – 1)!
PAGE # 16PAGE # 15

E D U C A T I O N S
E D U C A T I O N S
6. Combination (selection of objects) -
The number of combinations of n different things taken r at
a time is denoted by n
Cr
or C (n, r)
n
Cr
=
n
r n r
!
!( )!−
=
n
rP
r!
(i) n
Cr
= n
Cn–r
(ii) n
Cr
+ n
Cr–1
= n+1
Cr
(iii) n
Cr
= n
Cs ⇒ r = s or r + s = n
(iv) n
C0
= n
Cn
= 1
(v) n
C1
= n
Cn–1
= n
(vi) n
Cr
=
n
r
n–1
Cr–1
(vii) n
Cr
=
1
r
(n – r + 1) n
Cr–1
7. Restricted combinations -
The number of combinations of n distinct objects taken r at
a time, when k particular objects are always to be
(i) included is n–k
Cr–k
(ii) excluded is n–k
Cr
(iii) included and s particular things are to be excluded is
n–k–s
Cr–k
8. Total number of combinations in different cases -
(i) The number of selections of n identical objects, taken
at least one = n
(ii) The number of selections from n different objects, taken
at least one
= n
C1
+ n
C2
+ n
C3
+ ....... + n
Cn
= 2n
– 1
(iii) The number of selections of r objects out of n iden-
tical objects is 1.
(iv) Total number of selections of zero or more objects
from n identical objects is n + 1.
(v) Total number of selections of zero or more objects
out of n different objects
= n
C0
+ n
C1
+ n
C2
+ n
C3
+ ....... + n
Cn
= 2n
(vi) The total number of selections of at least one out of
a1
+ a2
+ ...... + an
objects where a1
are alike (of
one kind), a2
are alike (of second kind), ......... an
are
alike (of nth
kind) is
[(a1
+ 1) (a2
+ 1) (a3
+ 1) + ...... + (an
+ 1)] – 1
(vii) The number of selections taking atleast one out of
a1
+ a2
+ a3
+ ....... + an
+ k objects when a1
are
alike (of one kind), a2
are alike (of second kind),
........ an
are alike (of kth
kind) and k are distinct is
[(a1
+ 1) (a2
+ 1) (a3
+ 1) .......... (an
+ 1)] 2k
– 1
9. Division and distribution -
(i) The number of ways in which (m + n + p) different
objects can be divided into there groups containing m,
n, & p different objects respectively is
( )!
! ! !
m n p
m n p
+ +
(ii) The total number of ways in which n different objects
are to be divided into r groups of group sizes n1
, n2
, n3
,
............. nr
respectively such that size of no two groups
is same is
n
n n nr
!
! !............ !1 2
.
(iii) The total number of ways in which n different objects
are to be divided into groups such that k1
groups have
group size n1
, k2
groups have group size n2
and so on,
kr
groups have group size nr
, is given as
n
n n n k k kk k
r
k
r
r
!
( !) ( !) .............( !) ! !............ !1 2 1 2
1 2
.
PAGE # 18PAGE # 17

E D U C A T I O N S
E D U C A T I O N S
(iv) The total number of ways in which n different objects
are divided into k groups of fixed group size and are
distributed among k persons (one group to each) is
given as
(number of ways of group formation) × k!
10. Selection of light objects and multinomial theorem -
(i) The coefficient of xn
in the expansion of (1 – x–r
) is
equal to n + r – 1
Cr – 1
(ii) The number of solution of the equation x1
+ x2
+ ..........
+ xr
= n, n ∈ N under the condition n1 ≤ x1 ≤ n'1
,
n2 ≤ x2 ≤ n'2
, ................ nr ≤ xr ≤ n'r
where all x'i
s are integers is given as
Coefficient of xn
is
x x x x x x x x x
n n n n n n n n nr r r1 1 1 2 2 21 1 1
+ + + + + + + + +L
NM O
QP
+ + +
... ... ... ...
' ' '
e j e j e j
11. Derangement Theorem -
(i) If n things are arranged in a row, then the number of
ways in which they can be rearranged so that no one
of them occupies the place assigned to it is
= n! 1
1
1
1
2!
1
3!
1
4!
1
1
− + − + − + −
L
NM O
QP!
.... ( )
!
n
n
(ii) If n things are arranged at n places then the number of
ways to rearrange exactly r things at right places is
=
n
r
! 1
1
1
1
2
1
3
1
4
1
1
− + − + + + −
−
L
NM O
QP−
! ! ! !
.... ( )
( )!
n r
n r
12. Some Important results -
(a) Number of total different straight lines formed by joining
the n points on a plane of which m(<n) are collinear is
n
C2
– m
C2
+ 1.
(b) Number of total triangles formed by joining the n
points on a plane of which m(< n) are collinear is
n
C3
– m
C3
.
(c) Number of diagonals in a polygon of n sides is
n
C2
– n.
(d) If m parallel lines in a plane are intersected by a
family of other n parallel lines. Then total number of
parallelogram so formed is m
C2
× n
C2
.
(e) Given n points on the circumference of a circle, then
number of straight lines n
C2
number of triangles n
C3
number of quadrilaterals n
C4
(f) If n straight lines are drawn in the plane such that
no two lines are parallel and no three lines are
concurrent. Then the number of part into which these
lines divide the plane is = 1 + Σn
(g) Number of rectangles of any size in a square of n × n
is r
r
n
3
1=
∑ and number of squares of any size is r
r
n
2
1=
∑ .
(h) Number of rectangles of any size in a rectangle of
n × p is
np
4
(n + 1) (p + 1) and number of squares
of any size is
r
n
=
∑
1
(n + 1 – r) (p + 1 – r).
PAGE # 20PAGE # 19

E D U C A T I O N S
E D U C A T I O N S
PROBABILITY
1. Mathematical definition of probability :
Probability of an event
=
No of favourable cases to event A
Total no of cases
.
.
Note : (i) 0 ≤ P (A) ≤ 1
(ii) Probability of an impossible event is zero
(iii) Probability of a sure event is one.
(iv) P(A) + P(Not A) = 1 i.e. P(A) + P( A ) = 1
2. Odds for an event :
If P(A) =
m
n
and P( A ) =
n m
n
−
Then odds in favour of A =
P A
P A
( )
( )
=
m
n m−
and odds in against of A =
P A
P
( )
(A)
=
n m
m
−
3. Set theoretical notation of probability and some impor-
tant results :
(i) P(A + B) = 1 – P( A B )
(ii) P(A/B) =
P AB
P B
( )
( )
(iii) P(A + B) = P(AB) + P( A B) + P(A B )
(iv) A ⊂ B ⇒ P(A) ≤ P(B)
(v) P( AB ) = P(B) – P(AB)
(vi) P(AB) ≤ P(A) P(B) ≤ P(A + B) ≤ P(A) + P(B)
(vii) P(Exactly one event) = P(A B ) + P( A B)
(viii) P( A + B ) = 1 – P(AB) = P(A) + P(B) – 2P(AB)
= P(A + B) – P(AB)
(ix) P(neither A nor B) = P ( A B ) = 1 – P(A + B)
(x) When a coin is tossed n times or n coins are tossed
once, the probability of each simple event is
1
2n .
(xi) When a dice is rolled n times or n dice are rolled once,
the probability of each simple event is
1
6n .
(xii) When n cards are drawn (1 ≤ n ≤ 52) from well shuffled
deck of 52 cards, the probability of each simple event
is
1
52
Cn
.
(xiii) If n cards are drawn one after the other with replace-
ment, the probability of each simple event is
1
52( )n .
(xiv) P(none) = 1 – P (atleast one)
(xv) Playing cards :
(a) Total cards : 52 (26 red, 26 black)
(b) Four suits : Heart, diamond, spade, club (13 cards
each)
(c) Court (face) cards : 12 (4 kings, 4 queens, 4
jacks)
(d) Honour cards : 16 (4 Aces, 4 kings, 4 queens, 4
Jacks)
PAGE # 22PAGE # 21

E D U C A T I O N S
E D U C A T I O N S
(xvi) Probability regarding n letters and their envelopes :
If n letters corresponding to n envelopes are placed in
the envelopes at random, then
(a) Probability that all the letters are in right enve-
lopes =
1
n!
(b) Probability that all letters are not in right enve-
lopes = 1 –
1
n!
(c) Probability that no letters are in right envelope
=
1
2!
–
1
3!
+
1
4!
.... + (–1)n
1
n!
(d) Probability that exactly r letters are in right
envelopes =
1
r!
1
2
1
3
1
4
1
1
! ! !
..... ( )
( )!
− + + + −
−
L
NM O
QP−n r
n r
4. Addition Theorem of Probability :
(i) When events are mutually exclusive
i.e. n (A ∩ B) = 0 ⇒ P(A ∩ B) = 0
∴ P(A ∪ B) = P(A) + P(B)
(ii) When events are not mutually exclusive i.e.
P(A ∩ B) ≠ 0
∴ P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
or P(A + B) = P(A) + P(B) – P(AB)
(iii) When events are independent i.e. P(A ∩ B) = P(A) P (B)
∴ P(A + B) = P(A) + P(B) – P(A) P(B)
5. Conditional probability :
P(A/B) = Probability of occurrence of A, given that B has
already happened =
P A B
P B
( )
( )
∩
P(B/A) = Probability of occurrence of B, given that A has
already happened =
P A B
P A
( )
( )
∩
Note : If the outcomes of the experiment are equally
likely, then P(A/B) =
No of sample pts in A B
No of pts in B
. .
. .
∩
.
(i) If A and B are independent event, then P(A/B) = P(A)
and P(B/A) = P(B)
(ii) Multiplication Theorem :
P(A ∩ B) = P(A/B). P(B), P(B) ≠ 0
or P(A ∩ B) = P(B/A) P(A), P(A) ≠ 0
Generalized :
P(E1 ∩ E2 ∩ E3 ∩ ............... ∩ En
)
= P(E1
) P(E2
/E1
) P(E3
/E1 ∩ E2
) P(E4
/E1 ∩ E2 ∩ E3
) .........
If events are independent, then
P(E1 ∩ E2 ∩ E3 ∩ ....... ∩ En
) = P(E1
) P(E2
) ....... P(En
)
6. Probability of at least one of the n Independent events :
If P1
, P2
, ....... Pn
are the probabilities of n independent
events A, A2
, .... An
then the probability of happening of at
least one of these event is.
1 – [(1 – P1
) (1 – P2
)......(1 – Pn
)]
or P(A1
+ A2
+ ... + An
) = 1 – P ( A1
) P ( A2
) .... P( An
)
PAGE # 24PAGE # 23

E D U C A T I O N S
E D U C A T I O N S
7. Total Probability :
Let A1
, A2
, ............. An
are n mutually exclusive & set of
exhaustive events and event A can occur through any one
of these events, then probability of occurence of A
P(A) = P(A ∩ A1
) + P(A ∩ A2
) + ............. + P(A ∩ An
)
=
r
n
=
∑
1
P(Ar
) P(A/Ar
)
8. Baye's Rule :
Let A1
, A2
, A3
be any three mutually exclusive & exhaustive
events (i.e. A1
∪ A2
∪ A3
= sample space & A1 ∩ A2 ∩ A3
= φ)
an sample space S and B is any other event on sample space
then,
P(Ai
/B) =
P B A P A
P B A P A P B A P A P B A P A
i i( / ) ( )
( / ) ( ) ( / ) ( ) ( / ) ( )1 1 2 2 3 3+ + ,
i = 1, 2, 3
9. Probability distribution :
(i) If a random variable x assumes values x1
, x2
, ......xn
with probabilities P1
, P2
, ..... Pn
respectively then
(a) P1
+ P2
+ P3
+ ..... + Pn
= 1
(b) mean E(x) =Σ Pi
xi
(c) Variance = Σx2
Pi
– (mean)2
= Σ (x2
) – (E(x))2
(ii) Binomial distribution : If an experiment is repeated n
times, the successive trials being independent of one
another, then the probability of -
r success is n
Cr
Pr
qn–r
atleast r success is
k r
n
=
∑ n
Ck
Pk
qn–k
where p is probability of success in a single trial, q = 1 – p
(a) mean E(x) = np
(b) E (x2
) = npq + n2
p2
(c) Variance E(x2
) – (E(x))2
= npq
(d) Standard deviation = npq
10. Truth of the statement :
(i) If two persons A and B speaks truth with the probabil-
ity p1
& p2
respectively and if they agree on a state-
ment, then the probability that they are speaking truth
will be given by
p p
p p p p
1 2
1 2 1 21 1+ − −( ) ( ) .
(ii) If A and B both assert that an event has occurred,
probability of occurrence of which is α then the prob-
ability that event has occurred.
Given that the probability of A & B speaking truth is p1
, p2
.
α
α α
p p
p p p p
1 2
1 2 1 21 1 1+ − − −( ) ( ) ( )
(iii) If in the second part the probability that their lies
(jhuth) coincides is β then from above case required
probability will be
α
α α β
p p
p p p p
1 2
1 2 1 21 1 1+ − − −( ) ( ) ( ) .
PAGE # 26PAGE # 25

E D U C A T I O N S
E D U C A T I O N S
PROGRESSION AND SERIES
1. Arithmetic Progression (A.P.) :
(a) General A.P. — a, a + d, a + 2d, ...... , a + (n – 1) d
where a is the first term and d is the common difference
(b) General (nth
) term of an A.P. —
Tn
= a + (n – 1)d [nth
term from the beginning]
If an A.P. having m terms, then nth
term from end
= a + (m – n)d
(c) Sum of n terms of an A.P. —
Sn
=
n
2
[2a + (n – 1)d] =
n
2
[a + Tn
]
Note : If sum of n terms i.e. Sn
is given then Tn
= Sn
– Sn–1
where Sn–1
is sum of (n – 1) terms.
(d) Supposition of terms in A.P. —
(i) Three terms as a - d, a, a + d
(ii) Four terms as a – 3d, a – d, a + d, a + 3d
(iii) Five terms as a – 2d, a – d, a, a + d, a + 2d
(e) Arithmetic mean (A.M.) :
(i) A.M. of n numbers A1
, A2
, ................ An
is defined
as
A.M. =
A A A
n
n1 2+ + +.........
=
ΣA
n
i
=
Sum of numbers
n
(ii) For an A.P., A.M. of the terms taken symmetrically
from the beginning and from the end will always
be constant and will be equal to middle term or
A.M. of middle term.
(iii) If A is the A.M. between two given nos. a and b,
then
A =
a b+
2
i.e. 2A = a + b
PAGE # 27 PAGE # 28
(iv) If A1
, A2
,...... An
are n A.M's between a and b,
then A1
= a + d, A2
= a + 2d,...... An
= a + nd,
where d =
b a
n
−
+ 1
(v) Sum of n A.M's inserted between a and b is
n
2
(a + b)
(vi) Any term of an A.P. (except first term) is equal to
the half of the sum of term equidistant from the
term i.e. an
=
1
2
(an–r
+ an+r
), r < n
2. Geometric Progression (G.P.)
(a) General G.P. — a, ar, ar2
, ......
where a is the first term and r is the common ratio
(b) General (nth
) term of a G.P. — Tn
= arn–1
If a G.P. having m terms then nth
term from end = arm–n
(c) Sum of n terms of a G.P. —
Sn
=
a r
r
n
( )1
1
−
−
=
a T r
r
n−
−1
, r < 1
=
a r
r
n
( )−
−
1
1
=
T r a
r
n −
− 1
, r > 1
(d) Sum of an infinite G.P. — S∞
=
a
r1 −
, |r|<1
(e) Supposition of terms in G.P. —
(i) Three terms as
a
r
, a, ar
(ii) Four terms as
a
r3 ,
a
r
, ar, ar3
(iii) Five terms as
a
r2 ,
a
r
, a, ar, ar2

E D U C A T I O N S
E D U C A T I O N S
(f) Geometric Mean (G.M.) —
(i) Geometrical mean of n numbers x1
, x2
, .......... xn
is defined as
G.M. = (x1
x2
............... xn
)1/n
.
(i) If G is the G.M. between two given numbers a
and b, then
G2
= ab ⇒ G = ab
(ii) If G1
, G2
, .............. Gn
are n G.M's between a
and b, then
G1
= ar, G2
= ar2
,..... Gn
= arn
, where r =
b
a
F
HG I
KJ
+1 1/n
(iii) Product of the n G.M.'s inserted between a & b is
(ab)n/2
3. Arithmetico - Geometric Progression (A.G.P.) :
(a) General form — a, (a + d)r, (a + 2d) r2
, .............
(b) General (nth
) term — Tn
= [a + (n – 1) d] rn–1
(c) Sum of n terms of an A.G.P — Sn
=
a
r1 −
+ r.
d r
r
n
( )
( )
1
1
1
2
−
−
−
(d) Sum of infinite terms of an A.G.P.
S∞ =
a
r1 −
+
dr
r( )1 2
−
4. Sum standard results :
(a) Σn = 1 + 2 + 3 + ..... + n =
n n( )+ 1
2
(b) Σn2
= 12
+ 22
+ 32
+ ..... + n2
=
n n n( )( )+ +1 2 1
6
(c) Σn3
= 13
+ 23
+ 33
+ .... + n3
=
n n( )+L
NM O
QP1
2
2
(d) Σa = a + a + .... + (n times) = na
(e) Σ(2n – 1) = 1 + 3 + 5 + .... (2n – 1) = n2
(f) Σ2n = 2 + 4 + 6 + .... + 2n = n (n + 1)
5. Harmonic Progression (H.P)
(a) General H.P. —
1
a
,
1
a d+
,
1
2a d+
+......
(b) General (nth
term) of a H.P. — Tn
=
1
1a n d+ −( )
=
1
n term coresponding to A Pth
. .
(c) Harmonic Mean (H.M.)
(i) If H is the H.M. between a and b, then H =
2ab
a b+
(ii) If H1
, H2
,......,Hn
are n H.M's between a and b,
then H1
=
ab n
bn a
( )+
+
1
, ....., Hn
=
ab n
na b
( )+
+
1
or first find n A.M.'s between
1
a
&
1
b
, then their
reciprocal will be required H.M's.
6. Relation Between A.M., G.M. and H.M.
(i) AH = G2
(ii) A ≥ G ≥ H
(iii) If A and G are A.M. and G.M. respectively between two
+ve numbers, then these numbers are
A ± A G2 2
−
PAGE # 29 PAGE # 30

E D U C A T I O N S
E D U C A T I O N S
BINOMIAL THEOREM
1. Binomial Theorem for any +ve integral index :
(x + a)n
= n
C0
xn
+ n
C1
xn–1
a + n
C2
xn–2
a2
+ .......
+ n
Cr
xn–r
ar
+ .... + n
Cn
an
=
r
n
=
∑0
n
Cr
xn–r
ar
(i) General term - Tr+1
= n
Cr
xn–r
ar
is the (r + 1)th
term from
beginning.
(ii) (m + 1)th
term from the end = (n – m + 1)th
from begin-
ning = Tn–m+1
(iii) middle term
(a) If n is even then middle term =
n
th
2
1+
F
HG I
KJ term
(b) If n is odd then middle term =
n
th
+F
HG I
KJ1
2
and
n
th
+F
HG I
KJ3
2
term
Binomial coefficient of middle term is the greatest bino-
mial coefficient.
2. To determine a particular term in the given expasion :
Let the given expansion be x
x
n
α
β
±
F
HG I
KJ1
, if xn
occurs in
Tr+1
(r + 1)th
term then r is given by n α – r (α + β) = m
and for x0
, n α – r (α + β) = 0
3. Properties of Binomial coefficients :
For the sake of convenience the coefficients n
C0
, n
C1
,
n
C2
..... n
rC ..... n
nC are usually denoted by C0
, C1
,.....
Cr
.......... Cn
respectively.
* C0
+ C1
+ C2
+ ..... + Cn
= 2n
* C0
– C1
+ C2
– C3
+ ..... + Cn
= 0
* C0
+ C2
+ C4
+ ..... = C1
+ C3
+ C5
+ .... = 2n–1
* n
rC =
n
r
Cn
r
−
−
1
1 =
n
r
n
r
−
−
1
1
n
rC−
−
2
2
and so on ...
* 2n
n rC + =
2n
n r n r
!
! !− +c h c h
* n
rC + n
rC −1 = n
rC+1
* C1
+ 2C2
+ 3C3
+ ... + nCn
= n.2n–1
* C1
– 2C2
+ 3C3
......... = 0
* C0
+ 2C1
+ 3C2
+ ......+ (n + 1)Cn
= (n + 2)2n–1
* C0
2
+ C1
2
+ C2
2
+ ..... + Cn
2
=
2
2
n
n
c h
c h
!
!
= 2n
Cn
* C0
2
– C1
2
+ C2
2
– C3
2
+ .....
=
0
1
,
,
/2
/2
if n is odd
C if n is even
n n
n−
RS|
T|c h
PAGE # 31 PAGE # 32

E D U C A T I O N S
E D U C A T I O N S
Note : 2 1
0
n
C+
+ 2 1
1
n
C+
+ .... + 2 1n
nC+
= 2 1
1
n
nC+
+ +
2 1
2
n
nC+
+ + ..... 2 1
2 1
n
nC+
+ = 22n
* C0
+
C1
2
+
C2
3
+ ..... +
C
n
n
+ 1
=
2 1
1
1n
n
+
−
+
* C0
–
C1
2
+
C2
3
–
C3
4
.... +
( )−
+
1
1
n
nC
n
=
1
1n +
4. Greatest term :
(i) If
( )n a
x a
+
+
1
∈ Z (integer) then the expansion has two
greatest terms. These are kth
and (k + 1)th
where x & a
are +ve real nos.
(ii) If
( )n a
x a
+
+
1
∉ Z then the expansion has only one great-
est term. This is (k + 1)th
term k =
( )n a
x a
+
+
L
NM O
QP1
,
{[.] denotes greatest integer less than or equal to x}
5. Multinomial Theorem :
(i) (x + a)n
=
r
n
=
∑
0
n
Cr
xn–r
ar
, n ∈N
=
r
n
=
∑
0
n
n r) r
!
( ! !−
xn–r
ar
= r s n+ =
∑ n
s r
!
! !
xs
ar
,
where s = n – r
(ii) (x + y + z)n
=
r s t n+ + =
∑
n
s r t
!
! ! !
xr
ys
zt
Generalized (x1
+ x2
+..... xk
)n
= r r r nk1 2+ + =
∑
...
n
r r rk
!
! !.... !1 2
x x x
r r
k
rk
1 2
1 2
.....
6. Total no. of terms in the expansion (x1
+ x2
+... xn
)m
is
m+n–1
Cn–1
PAGE # 33 PAGE # 34

E D U C A T I O N S
E D U C A T I O N S
TRIGONOMETRIC RATIO AND IDENTITIES
1. Some important results :
(i) Arc length AB = r θ
Area of circular sector =
1
2
r2
θ
(ii) For a regular polygon of side a and number of sides n
(a) Internal angle of polygon = (n – 2)
π
n
(b) Sum of all internal angles = (n – 2) π
(c) Radius of incircle of this polygon r =
a
2
cot
π
n
(d) Radius of circumcircle of this polygon R
=
a
2
cosec
π
n
(e) Area of the polygon =
1
4
na2
cot
π
n
F
HG I
KJ
(f) Area of triangle =
1
4
a2
cos
π
n
(g) Area of incircle = π
a
n2
2
cot
πF
HG I
KJ
(h) Area of circumcircle = π
a
ec
n2
2
cos
πF
HG I
KJ
2. Relation between system of measurement of angles :
D
90
=
G
100
=
2C
π
& π radian = 1800
3. Trigonometric identities :
(i) sin2
θ + cos2
θ = 1
(ii) cosec2
θ – cot2
θ = 1
(iii) sec2
θ – tan2
θ = 1
4. Sign convention :
y
II quadrant I quadrant
sin & cosec All +ve
are +ve
x' O x
III quadrant IV quadrant
tan & cot cos & sec
are +ve are +ve
y'
5. T-ratios of allied angles : The signs of trigonometrical ratio
in different quadrant.
Allied∠ of (–θ) 900
± θ 1800
± θ 2700
± θ 3600
± θ
T-ratios
sinθ –sinθ cosθ m sinθ –cosθ ±sinθ
cosθ cosθ m sinθ –cosθ ±sinθ cosθ
tanθ –tanθ m cotθ ±tanθ m cotθ ±tanθ
cotθ –cotθ m tanθ ±cotθ m tanθ ±cotθ
secθ secθ m cosecθ –secθ ±cosecθ secθ
cosecθ –cosecθ secθ m cosecθ –sec θ ±cosecθ
6. Sum & differences of angles of t-ratios :
(i) sin(A ± B) = sinA cosB ± cosA sinB
(ii) cos(A ± B) = cosA cosB ± sinA sinB
(iii) tan (A ± B) =
tan tan
tan tan
A B
A B
±
1 m
PAGE # 35 PAGE # 36

E D U C A T I O N S
E D U C A T I O N S
(iv) cot (A ± B) =
cot cot
cot cot
A B
B A
m 1
±
(v) sin(A + B) sin(A – B) = sin2
A – sin2
B = cos2
B – cos2
A
(vi) cos(A + B) cos (A – B) = cos2
A – sin2
B = cos2
B – sin2
A
(vii) tan(A + B + C)
=
tan tan tan tan tan tan
tan tan tan tan tan tan
A B C A B C
A B B C C A
+ + −
− − −1
=
S S
S
1 3
21
−
−
Generalized tan (A + B + C + ......... )
=
S S S S
S S S S
1 3 5 7
2 4 6 81
− + − +
− + − + −
......
......
Where S1
= Σ tan A
S2
= Σ tan A tan B,
S3
= Σ tan A tan B tan C & so on
(viii) sin (A + B + C) = Σ sin A cos B cos C – Π sin A
= Π cos A (Numerator of tan (A + B + C))
(ix) cos (A + B + C) = Π cos A – Σ sin A sin B cos C
= Π cos A (Denominator of tan (A + B + C))
for a triangle A + B + C = π
Σ tan A = Π tan A
Σ sin A = Σ sin A cos B cos C
1 + Π cos A = Σ sin A sin B cos C
(viii) sin750
=
3 1
2 2
+
= cos150
(ix) cos750
=
3 1
2 2
−
= sin150
(x) tan750
= 2 + 3 = cot150
(xi) cot750
= 2 – 3 = tan150
7. Formulaes for product into sum or difference and vice-
versa :
(i) 2sinA cosB = sin(A + B) + sin(A – B)
(ii) 2cosA sinB = sin(A + B) – sin(A – B)
(iii) 2cosA cosB = cos(A + B) + cos(A – B)
(iv) 2sinA sinB = cos(A – B) – cos(A + B)
(v) sinC + sinD = 2sin
C D+F
HG I
KJ2
cos
C D−F
HG I
KJ2
(vi) sinC – sinD = 2cos
C D+F
HG I
KJ2 sin
C D−F
HG I
KJ2
(vii) cosC + cosD = 2cos
C D+F
HG I
KJ2
cos
C D−F
HG I
KJ2
(viii) cosC – cosD = 2sin
C D+F
HG I
KJ2
sin
D C−F
HG I
KJ2
(ix) tanA + tanB =
sin( )
cos cos
A B
A B
+
8. T-ratios of multiple and submultiple angles :
(i) sin2A = 2sinA cosA =
2
1 2
tan
tan
A
A+
= (sin A + cos A)2
– 1 = 1 – (sin A – cos A)2
⇒ sinA = 2sinA/2 cosA/2 =
2 2
1 22
tan /
tan /
A
A+
(ii) cos2A = cos2
A – sin2
A = 2cos2
A – 1
= 1 – 2sin2
A =
1
1
2
2
−
+
tan
tan
A
A
PAGE # 37 PAGE # 38

E D U C A T I O N S
E D U C A T I O N S
(iii) tan2A =
2
1 2
tan
tan
A
A−
⇒ tanA =
2 2
1 22
tan /
tan /
A
A−
(iv) sin3θ = 3sinθ – 4sin3
θ = 4sin(600
– θ ) sin(600
+ θ ) sin θ
= sin θ (2 cos θ – 1) (2 cos θ + 1)
(v) cos3θ = 4cos3
θ – 3cosθ
= 4cos(600
– θ ) cos(600
+ θ ) cos θ
= cos θ (1 – 2 sin θ ) (1 + 2 sin θ )
(vi) tan3A =
3
1 3
3
2
tan tan
tan
A A
A
−
−
= tan(600
– A) tan(600
+ A)tanA
(vii) sinA/2 =
1
2
− cos A
(viii) cosA/2 =
1
2
+ cos A
(ix) tanA/2 =
1
1
−
+
cos
cos
A
A
=
1 − cos
sin
A
A
, A ≠ (2n + 1)π
9. Maximum and minimum value of the expression :
acosθ + bsinθ
Maximum (greatest) Value = a b2 2
+
Minimum (Least) value = – a b2 2
+
10. Conditional trigonometric identities :
If A, B, C are angles of triangle i.e. A + B + C = π, then
(i) sin2A + sin2B + sin2C = 4sinA sinB sinC
i.e. Σ sin 2A = 4 Π (sin A)
(ii) cos2A + cos2B + cos2C = –1– 4cosA cosB cosC
(iii) sinA + sinB + sinC = 4cosA/2 cosB/2 cosC/2
(iv) cosA + cosB + cosC = 1 + 4 sinA/2 sinB/2 sinC/2
(v) sin2
A + sin2
B + sin2
C = 1 – 2sinA sinB cosC
(vi) cos2
A + cos2
B + cos2
C = 1 – 2cosA cosB cosC
(vii) tanA + tanB + tanC = tanA tanB tanC
(viii) cotB cotC + cotC cotA + cotA cotB = 1
(ix) Σ tan A/2 tan B/2 = 1
(x) Σ cot A cot B = 1
(xi) Σ cot A/2 = Π cot A/2
11. Some useful series :
(i) sinα + sin(α + β) + sin(α + 2β) + .... + to nterms
=
sin sin
sin
α β
β
β
+
−F
HG I
KJL
NM O
QP L
NM O
QPn n1
2 2
2
, β ≠ 2nπ
(ii) cosα + cos(α + β) + cos(α + 2β) + .... + to nterms
=
cos sin
sin
α β
β
β
+
−F
HG I
KJL
NM O
QPn n1
2 2
2
β ≠ 2nπ
(iii) cosα .cos2α . cos22
α ....cos(2n–1
α) =
sin
sin
2
2
n
n
α
α
, α ≠ nπ
= 1 , α = 2kπ
= –1 , α = (2k+1)π
PAGE # 39 PAGE # 40

E D U C A T I O N S
E D U C A T I O N S
TRIGONOMETRIC EQUATIONS
1. General solution of the equations of the form
(i) sinθ = 0 ⇒ θ = nπ, n ∈ I
(ii) cosθ = 0 ⇒ θ = (2n + 1)
π
2
, n ∈ I
(iii) tanθ = 0 ⇒ θ = nπ, n ∈ I
(iv) sinθ = 1 ⇒ θ = 2nπ +
π
2
(v) cosθ = 1 ⇒ θ = 2πn
(vi) sinθ = –1 ⇒ θ = 2nπ –
π
2
or 2nπ +
3
2
π
(vii) cosθ = –1 ⇒ θ = (2n + 1)π
(viii) sinθ = sinα ⇒ θ = nπ + (–1)n
α
(ix) cosθ = cosα ⇒ θ = 2nπ ± α
(x) tanθ = tanα ⇒ θ = nπ + α
(xi) sin2
θ = sin2
α ⇒ θ = nπ ± α
(xii) cos2
θ = cos2
α ⇒ θ = nπ ± α
(xiii) tan2
θ = tan2
α ⇒ θ = nπ ± α
2. For general solution of the equation of the form
a cosθ + bsinθ = c, where c ≤ a b2 2
+ , divide both side by
a b2 2
+
and put
a
a b2 2
+
= cosα,
b
a b2 2
+
= sinα.
Thus the equation reduces to form
cos(θ – α) =
c
a b2 2
+
= cosβ(say)
now solve using above formula
(i) If while solving an equation, we have to square it, then
the roots found after squaring must be checked
wheather they satisfy the original equation or not.
(ii) If two equations are given then find the common val-
ues of θ between 0 & 2π and then add 2nπ to this
common solution (value).
PAGE # 41 PAGE # 42

E D U C A T I O N S
E D U C A T I O N S
INVERSE TRIGONOMETRIC FUNCTIONS
1. If y = sin x, then x = sin–1
y, similarly for other inverse T-
functions.
2. Domain and Range of Inverse T-functions :
Function Domain (D) Range (R)
sin–1
x – 1 ≤ x ≤ 1 –
π
2
≤ θ ≤
π
2
cos–1
x – 1 ≤ x ≤ 1 0 ≤ θ ≤ π
tan–1
x – ∞ < x < ∞ –
π
2
< θ <
π
2
cot–1
x – ∞ < x < ∞ 0 < θ < π
sec–1
x x ≤ – 1, x ≥ 1 0 ≤ θ ≤ π , θ ≠
π
2
cosec–1
x x ≤ – 1, x ≥ 1 –
π
2
≤ θ ≤
π
2
, θ ≠ 0
3. Properties of Inverse T-functions :
(i) sin–1
(sin θ ) = θ provided –
π
2
≤ θ ≤
π
2
cos–1
(cos θ ) = θ provided θ ≤ θ ≤ π
tan–1
(tan θ ) = θ provided –
π
2
< θ <
π
2
cot–1
(cot θ ) = θ provided 0 < θ < π
sec–1
(sec θ) = θ provided 0 ≤ θ <
π
2
or
π
2
< θ ≤ π
cosec–1
(cosec θ ) = θ provided –
π
2
≤ θ < 0
or 0 < θ ≤
π
2
(ii) sin (sin–1
x) = x provided – 1 ≤ x ≤ 1
cos (cos–1
x) = x provided – 1 ≤ x ≤ 1
tan (tan–1
x) = x provided – ∞ < x < ∞
cot (cot–1
x) = x provided – ∞ < x < ∞
sec (sec–1
x) = x provided – ∞ < x ≤ – 1 or 1 ≤ x < ∞
cosec (cosec–1
x) = x provided – ∞ < x ≤ – 1
or 1 ≤ x < ∞
(iii) sin–1
(– x) = – sin–1
x,
cos–1
(– x) = π – cos–1
x
tan–1
(– x) = – tan–1
x
cot–1
(– x) = π – cot–1
x
cosec–1
(– x) = – cosec–1
x
sec–1
(– x) = π – sec–1
x
(iv) sin–1
x + cos–1
x =
π
2
, ∀ x ∈ [– 1, 1]
tan–1
x + cot–1
x =
π
2
, ∀ x ∈ R
sec–1
x + cosec–1
x =
π
2
, ∀ x ∈ (– ∞, – 1] ∪ [1, ∞)
PAGE # 43 PAGE # 44

E D U C A T I O N S
E D U C A T I O N S
4. Value of one inverse function in terms of another
inverse function :
(i) sin–1
x = cos–1
1 2
− x = tan–1
x
x1 2
−
= cot–1
1 2
− x
x
= sec–1
1
1 2
− x
= cosec–1
1
x
, 0 ≤ x ≤ 1
(ii) cos–1
x = sin–1
1 2
− x = tan–1 1 2
− x
x
= cot–1
x
x1 2
−
= sec–1
1
x
= cosec–1
1
1 2
− x
, 0 ≤ x ≤ 1
(iii) tan–1
x = sin–1
x
x1 2
+
= cos–1
1
1 2
+ x
= cot–1
1
x
= sec–1
1 2
+ x = cosec–1 1 2
+ x
x
, x ≥ 0
(iv) sin–1
1
x
F
HG I
KJ = cosec–1
x, ∀ x ∈ (– ∞ , 1] ∪ [1, ∞ )
(v) cos–1
1
x
F
HG I
KJ = sec–1
x, ∀ x ∈ (– ∞ , 1] ∪ [1, ∞ )
(vi) tan–1
1
x
F
HG I
KJ =
cot
cot
−
−
>
− + <
RST|
1
1
0
0
x for x
x for xπ
5. Formulae for sum and difference of inverse trigonomet-
ric function :
(i) tan–1
x + tan–1
y = tan–1
x y
xy
+
−
F
HG I
KJ1
; if x > 0, y > 0, xy < 1
(ii) tan–1
x + tan–1
y = π + tan–1
x y
xy
+
−
F
HG I
KJ1
; if x > 0, y > 0, xy > 1
(iii) tan–1
x – tan–1
y = tan–1
x y
xy
−
+
F
HG I
KJ1 ; if xy > –1
(iv) tan–1
x – tan–1
y = π + tan–1
x y
xy
−
+
F
HG I
KJ1
; if x > 0, y < 0, xy < –1
(v) tan–1
x + tan–1
y + tan–1
z = tan–1
x y z xyz
xy yz zx
+ + −
− − −
F
HG I
KJ1
(vi) sin–1
x ± sin–1
y = sin–1 x y y x1 12 2
− ± −L
NM O
QP;
if x,y ≥ 0 & x2
+ y2
≤ 1
(vii) sin–1
x ± sin–1
y = π – sin–1 x y y x1 12 2
− ± −L
NM O
QP;
if x,y ≥ 0 & x2
+ y2
> 1
(viii) cos–1
x ± cos–1
y = cos–1 xy x ym 1 12 2
− −L
NM O
QP;
if x,y > 0 & x2
+ y2
≤ 1
(ix) cos–1
x ± cos–1
y = π – cos–1 xy x ym 1 12 2
− −L
NM O
QP;
if x,y > 0 & x2
+ y2
> 1
PAGE # 45 PAGE # 46

E D U C A T I O N S
E D U C A T I O N S
6. Inverse trigonometric ratios of multiple angles
(i) 2sin–1
x = sin–1
(2x 1 2
− x ), if –1 ≤ x ≤ 1
(ii) 2cos–1
x = cos–1
(2x2
–1), if –1 ≤ x ≤ 1
(iii) 2tan–1
x = tan–1
2
1 2
x
x−
F
HG I
KJ = sin–1
2
1 2
x
x+
F
HG I
KJ = cos–1
1
1
2
2
−
+
F
HG
I
KJx
x
(iv) 3sin–1
x = sin–1
(3x – 4x3
)
(v) 3cos–1
x = cos–1
(4x3
– 3x)
(vi) 3tan–1
x = tan–1
3
1 3
3
2
x x
x
−
−
F
HG
I
KJ
PROPERTIES & SOLUTION OF TRIANGLE
Properties of triangle :
1. A triangle has three sides and three angles.
In any ∆ABC, we write BC = a, AB = c, AC = b
B
B
C
C
A
A
a
bc
and ∠BAC = ∠A, ∠ABC = ∠B, ∠ACB = ∠C
2. In ∆∆∆∆∆ABC :
(i) A + B + C = π
(ii) a + b > c, b +c > a, c + a > b
(iii) a > 0, b > 0, c > 0
3. Sine formula :
a
Asin
=
b
Bsin
=
c
Csin
= k(say)
or
sinA
a
=
sinB
b
=
sinC
c
= k (say)
4. Cosine formula :
cos A =
b c a
bc
2 2 2
2
+ −
cos B =
c a b
ac
2 2 2
2
+ −
cos C =
a b c
ab
2 2 2
2
+ −
PAGE # 47 PAGE # 48

E D U C A T I O N S
E D U C A T I O N S
5. Projection formula :
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A
6. Napier's Analogies :
tan
A B−
2
=
a b
a b
−
+
cot
C
2
tan
B C−
2
=
b c
b c
−
+
cot
A
2
tan
C A−
2
=
c a
c a
−
+
cot
B
2
7. Half angled formula - In any ∆∆∆∆∆ABC :
(a) sin
A
2
=
( ) ( )s b s c
bc
− −
sin
B
2
=
( ) ( )s c s a
ca
− −
sin
C
2
=
( ) ( )s a s b
ab
− −
where 2s = a + b + c
(b) cos
A
2
=
s s a
bc
( )−
cos
B
2
=
s s b
ca
( )−
cos
C
2
=
s s c
ab
( )−
(c) tan
A
2
=
( ) ( )
( )
s b s c
s s a
− −
−
tan
B
2
=
( ) ( )
( )
s c s a
s s b
− −
−
tan
C
2
=
( ) ( )
( )
s b s a
s s c
− −
−
8. ∆∆∆∆∆, Area of triangle :
(i) ∆ =
1
2
ab sin C =
1
2
bc sin A =
1
2
ca sin B
(ii) ∆ = s s a s b s c( ) ( ) ( )− − −
9. tan
A
2
tan
B
2
=
s c
s
−
tan
B
2
tan
C
2
=
s a
s
−
tan
C
2
tan
A
2
=
s b
s
−
10. Circumcircle of triangle and its radius :
(i) R =
a
A2sin
=
b
B2sin
=
c
C2sin
(ii) R =
abc
4∆
Where R is circumradius
PAGE # 49 PAGE # 50

E D U C A T I O N S
E D U C A T I O N S
11. Incircle of a triangle and its radius :
(iii) r =
∆
s
(iv) r = (s – a) tan
A
2
= (s – b) tan
B
2
= (s – c) tan
C
2
(v) r = 4R sin
A
2
sin
B
2
sin
C
2
(vi) cos A + cos B + cos C = 1 +
r
R
(vii) r =
a
B C
A
sin sin
cos
2 2
2
=
b
A C
B
sin sin
cos
2 2
2
=
c
B A
C
sin sin
cos
2 2
2
12. The radii of the escribed circles are given by :
(i) r1
=
∆
s a−
, r2
=
∆
s b−
, r3
=
∆
s c−
(ii) r1
= s tan
A
2
, r2
= s tan
B
2
, r3
= s tan
C
2
(iii) r1
= 4R sin
A
2
cos
B
2
cos
C
2
,
r2
= 4R cos
A
2
sin
B
2
cos
C
2
,
r3
= 4R cos
A
2
cos
B
2
sin
C
2
(iv) r1
+ r2
+ r3
– r = 4R
(v)
1
1r +
1
2r +
1
3r =
1
r
(vi)
1
1
2
r
+
1
2
2
r
+
1
3
2
r
+
1
2
r
=
a b c2 2 2
2
+ +
∆
(vii)
1
bc
+
1
ca
+
1
ab
=
1
2Rr
(viii) r1
r2
+ r2
r3
+ r3
r1
= s2
(ix) ∆ = 2R2
sin A sin B sin C = 4Rr cos
A
2
cos
B
2
cos
C
2
(x) r1
=
a
B C
A
cos cos
cos
2 2
2
, r2
=
b
C A
B
cos cos
cos
2 2
2
,
r3
=
c
A B
C
cos cos
cos
2 2
2
PAGE # 51 PAGE # 52

E D U C A T I O N S
E D U C A T I O N S
(ii) d = h (cotα – cotβ)
α β
d
h
HEIGHT AND DISTANCE
1. Angle of elevation and depression :
If an observer is at O and object is at P then ∠ XOP is
called angle of elevation of P as seen from O.
If an observer is at P and object is at O, then ∠ QPO is
called angle of depression of O as seen from P.
2. Some useful result :
(i) In any triangle ABC if AD : DB = m : n
∠ ACD = α , ∠ BCD = β & ∠ BDC = θ
then (m + n) cotθ = m cotα – ncot β
C
B
B
A
A
Dm n
α β
θ
= ncotA – mcotB [m – n Theorem]
PAGE # 53 PAGE # 54

E D U C A T I O N S
E D U C A T I O N S
POINT
1. Distance formula :
Distance between two points P(x1
, y1
) and Q(x2
, y2
) is
given by d(P, Q) = PQ
= ( ) ( )x x y y2 1
2
2 1
2
− + −
= ( ) ( )Difference of x coordinate Difference of ycoordinate2 2
+
Note : (i) d(P, Q) ≥ 0
(ii) d(P, Q) = 0 ⇔ P = Q
(iii) d(P, Q) = d(Q, P)
(iv) Distance of a point (x, y) from origin
(0, 0) = x y2 2
+
2. Use of Distance Formula :
(a) In Triangle :
Calculate AB, BC, CA
(i) If AB = BC = CA, then ∆ is equilateral.
(ii) If any two sides are equal then ∆ is isosceles.
(iii) If sum of square of any two sides is equal to
the third, then ∆ is right triangle.
(iv) Sum of any two equal to left third they do not
form a triangle
i.e. AB = BC + CA or BC = AC + AB
or AC = AB + BC. Here points are collinear.
(b) In Parallelogram :
Calculate AB, BC, CD and AD.
(i) If AB = CD, AD = BC, then ABCD is a parallelo-
gram.
(ii) If AB = CD, AD = BC and AC = BD, then ABCD
is a rectangle.
(iii) If AB = BC = CD = AD, then ABCD is a rhombus.
(iv) If AB = BC = CD = AD and AC = BD, then ABCD
is a square.
(C) For circumcentre of a triangle :
Circumcentre of a triangle is equidistant from vertices
i.e. PA = PB = PC.
Here P is circumcentre and PA is radius.
(i) Circumcentre of an acute angled triangle is in-
side the triangle.
(ii) Circumcentre of a right triangle is mid point of
the hypotenuse.
(iii) Circumcentre of an obtuse angled triangle is
outside the triangle.
3. Section formula :
(i) Internally :
AP
BP
=
m
n
= λ , Here λ > 0
A(x , y )1 1
B(x , y )2 2P
m n
P
mx nx
m n
my ny
m n
2 1 2 1+
+
+
+
F
HG I
KJ,
(ii) Externally :
AP
BP
=
m
n
= λ
A(x , y )1 1
B(x , y )2 2
P
n
m
P
mx nx
m n
my ny
m n
2 1 2 1−
−
−
−
F
HG I
KJ,
(iii) Coordinates of mid point of PQ are
x x y y1 2 1 2
2 2
+ +F
HG I
KJ,
(iv) The line ax + by + c = 0 divides the line joining the
points
(x1
, y1
) & (x2
, y2
) in the ratio = –
( )
( )
ax by c
ax by c
1
2 2
+ +
+ +
PAGE # 55 PAGE # 56

E D U C A T I O N S
E D U C A T I O N S
(v) For parallelogram – midpoint of diagonal AC = mid point
of diagonal BD
(vi) Coordinates of centroid G
x x x y y y1 2 3 1 2 3
3 3
+ + + +F
HG I
KJ,
(vii) Coordinates of incentre I
ax bx cx
a b c
ay by cy
a b c
1 2 3 1 2 3+ +
+ +
+ +
+ +
F
HG I
KJ,
(viii) Coordinates of orthocentre are obtained by solving the
equation of any two altitudes.
4. Area of Triangle :
The area of triangle ABC with vertices A(x1
, y1
), B(x2
, y2
)
and C(x3
, y3
).
∆ =
1
2
x y
x y
x y
1 1
2 2
3 3
1
1
1
(Determinant method)
=
1
2
x y
x y
x y
x y
1 1
2 2
3 3
1 1
=
1
2
[x1
y2
+ x2
y3
+ x3
y1
– x2
y1
– x3
y2
– x1
y3
]
[Stair method]
Note :
(i) Three points A, B, C are collinear if area of triangle
is zero.
(ii) If in a triangle point arrange in anticlockwise then
value of ∆ be +ve and if in clockwise then ∆ will be
–ve.
5. Area of Polygon :
Area of polygon having vertices (x1
, y1
), (x2
, y2
), (x3
, y3
)
........ (xn
, yn
) is given by area
=
1
2
x y
x y
x y
x y
x y
n n
1 1
2 2
3 3
1 1
M M
. Points must be taken in order.
6. Rotational Transformation :
If coordinates of any point P(x, y) with reference to new
axis will be (x', y') then
xB yB
x'→ cosθ sinθ
y'→ –sinθ cosθ
(i) Three pts. A, B, C are collinear, if area of triangle is
zero
(ii) Centroid G of ∆ABC divides the median AD or BE or CF in
the ratio 2 : 1
(iii) In an equilateral triangle, orthocentre, centroid,
circumcentre, incentre coincide.
(iv) Orthocentre, centroid and circumcentre are always
collinear and centroid divides the line joining orthocentre
and circumcentre in the ratio 2 : 1
(v) Area of triangle formed by coordinate axes & the line
ax + by + c = 0 is
c
ab
2
2
.
PAGE # 57 PAGE # 58

E D U C A T I O N S
E D U C A T I O N S
STRAIGHT LINE
1. Slope of a Line : The tangent of the angle that a line
makes with +ve direction of the x-axis in the anticlockwise
sense is called slope or gradient of the line and is generally
denoted by m. Thus m = tan θ .
(i) Slope of line || to x-axis is m = 0
(ii) Slope of line || to y-axis is m = ∞ (not defined)
(iii) Slope of the line equally inclined with the axes is 1
or – 1
(iv) Slope of the line through the points A(x1
, y1
) and
B(x2
, y2
) is
y y
x x
2 1
2 1
−
− .
(v) Slope of the line ax + by + c = 0, b ≠ 0 is –
a
b
(vi) Slope of two parallel lines are equal.
(vii) If m1
& m2
are slopes of two ⊥ lines then m1
m2
= – 1.
2. Standard form of the equation of a line :
(i) Equation of x-axis is y = 0
(ii) Equation of y-axis is x = 0
(iii) Equation of a straight line || to x-axis at a distance
b from it is y = b
(iv) Equation of a straight line || to y-axis at a distance
a from it is x = a
(v) Slope form : Equation of a line through the origin and
having slope m is y = mx.
(vi) Slope Intercept form : Equation of a line with slope m
and making an intercept c on the y-axis is y = mx + c.
(vii) Point slope form : Equation of a line with slope m
and passing through the point (x1
, y1
) is
y – y1
= m(x – x1
)
(viii) Two point form : Equation of a line passing through
the points (x1
, y1
) & (x2
, y2
) is
y y
y y
−
−
1
2 1
=
x x
x x
−
−
1
2 1
(ix) Intercept form : Equation of a line making intercepts a
and b respectively on x-axis and y-axis is
x
a
+
y
b
= 1.
(x) Parametric or distance or symmetrical form of the
line : Equation of a line passing through (x1
, y1
) and
making an angle θ , 0 ≤ θ ≤ π , θ ≠
π
2
with the
+ve direction of x-axis is
x x− 1
cos θ
=
y y− 1
sinθ
= r
⇒ x = x1
+ r cos θ , y = y1
+ r sin θ
Where r is the distance of any point P(x, y) on the
line from the point (x1
, y1
)
(xi) Normal or perpendicular form : Equation of a line
such that the length of the perpendicular from the
origin on it is p and the angle which the perpendicular
makes with the +ve direction of x-axis is α , is
x cos α + y sin α = p.
3. Angle between two lines :
(i) Two lines a1
x + b1
y + c1
= 0 & a2
x + b2
y + c2
= 0 are
(a) Parallel if
a
a
1
2
=
b
b
1
2
≠
c
c
1
2
(b) Perpendicular if a1
a2
+ b1
b2
= 0
(c) Identical or coincident if
a
a
1
2
=
b
b
1
2
=
c
c
1
2
(d) If not above three, then θ = tan–1
a b a b
a a b b
2 1 1 2
1 2 1 2
−
−
(ii) Two lines y = m1
x + c and y = m2
x + c are
(a) Parallel if m1
= m2
(b) Perpendicular if m1
m2
= –1
(c) If not above two, then θ = tan–1
m m
m m
1 2
1 21
−
+
PAGE # 59 PAGE # 60

E D U C A T I O N S
E D U C A T I O N S
4. Position of a point with respect to a straight line :
The line L(xi
, yi
) i = 1, 2 will be of same sign or of opposite
sign according to the point A(x1
, y1
) & B (x2
, y2
) lie on same
side or on opposite side of L (x, y) respectively.
5. Equation of a line parallel (or perpendicular) to the line
ax + by + c = 0 is ax + by + c' = 0 (or bx – ay + λ = 0)
6. Equation of st. lines through (x1
,y1
) making an angle ααααα
with y = mx + c is
y – y1
=
m
m
± tan
tan
α
α1 m
(x – x1
)
7. length of perpendicular from (x1
, y1
) on ax + by + c = 0
is
| |ax by c
a b
1 1
2 2
+ +
+
8. Distance between two parallel lines ax + by + ci
= 0,
i = 1, 2 is
| |c c
a b
1 2
2 2
−
+
9. Condition of concurrency for three straight lines
Li
≡≡≡≡≡ ai
x + bi
y + ci
= 0, i = 1, 2, 3 is
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= 0
10. Equation of bisectors of angles between two lines :
a x b y c
a b
1 1 1
1
2
1
2
+ +
+ = ±
a x b y c
a b
2 2 2
2
2
2
2
+ +
+
11. Family of straight lines :
The general equation of family of straight line will be written
in one parameter
The equation of straight line which passes through point of
intersection of two given lines L1
and L2
can be taken as
L1
+ λ L2
= 0
12. Homogeneous equation : If y = m1
x and y = m2
x be the
two equations
represented by ax2
+ 2hxy + by2
= 0 , then m1
+ m2
= –2h/b
and m1
m2
= a/b
13. General equation of second degree :
ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0 represent a pair of
straight line if ∆ ≡
a h g
h b f
g f c
= 0
If y = m1
x + c & y = m2
x + c represents two straight lines
then m1
+ m2
=
−2h
b
, m1
m2
=
a
b
.
14. Angle between pair of straight lines :
The angle between the lines represented by
ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0 or ax2
+ 2hxy + by2
= 0
is tanθ =
2 2
h ab
a b
−
+( )
(i) The two lines given by ax2
+ 2hxy + by2
= 0 are
(a) Parallel and coincident iff h2
– ab = 0
(b) Perpendicular iff a + b = 0
(ii) The two line given by ax2
+ 2hxy + by2
+ 2gx + 2fy + c
= 0 are
(a) Parallel if h2
– ab = 0 & af2
= bg2
(b) Perpendicular iff a + b = 0
(c) Coincident iff g2
– ac = 0
13. Combined equation of angle bisector of the angle between
the lines ax2
+ 2hxy + by2
= 0 is
x y
a b
2 2
−
−
=
xy
h
PAGE # 61 PAGE # 62

E D U C A T I O N S
E D U C A T I O N S
CIRCLE
1. General equation of a circle : x2
+ y2
+ 2gx + 2fy + c = 0
where g, f and c are constants
(i) Centre of the cirle is (–g, –f)
i.e. −
−F
HG I
KJ1
2
1
2
coeff of x coeff of y. , .
(ii) Radius is g f c2 2
+ −
2. Central (Centre radius) form of a circle :
(i) (x – h)2
+ (y – k)2
= r2
, where (h, k) is circle centre and
r is the radius.
(ii) x2
+ y2
= r2
, where (0, 0) origin is circle centre and r is
the radius.
3. Diameter form : If (x1
, y1
) and (x2
, y2
) are end pts. of a
diameter of a circle, then its equation is
(x – x1
) (x – x2
) + (y – y1
) (y – y2
) = 0
4. Parametric equations :
(i) The parametric equations of the circle x2
+ y2
= r2
are
x = rcosθ, y = r sinθ ,
where point θ ≡ (r cos θ , r sin θ )
(ii) The parametric equations of the circle
(x – h)2
+ (y – k)2
= r2
are x = h + rcosθ, y = k + rsinθ
(iii) The parametric equations of the circle
x2
+ y2
+ 2gx + 2fy + c = 0 are
x = –g + g f c2 2
+ − cosθ, y = –f + g f c2 2
+ − sinθ
(iv) For circle x2
+ y2
= a2
, equation of chord joining θ 1
& θ 2
is
xcos
θ θ1 2
2
+
+ ysin
θ θ1 2
2
+
= r cos
θ θ1 2
2
−
.
5. Concentric circles : Two circles having same centre C(h, k)
but different radii r1
& r2
respectively are called concentric
circles.
6. Position of a point w.r.t. a circle : A point (x1
, y1
) lies
outside, on or inside a circle
S ≡ x2
+ y2
+ 2gx + 2fy + c = 0 according as
S1 ≡ x1
2
+ y1
2
+ 2gx1
+ 2fy1
+ c is +ve, zero or –ve
7. Chord length (length of intercept) = 2 r p2 2
−
8. Intercepts made on coordinate axes by the circle :
(i) x axis = 2 g c2
−
(ii) y axis = 2 f c2
−
9. Length of tangent = S1
10. Length of the intercept made by line : y = mx + c with the
circle x2
+ y2
= a2
is
2
a m c
m
2 2 2
2
1
1
( )+ −
+
or (1 + m2
) |x1
– x2
|
where |x1
– x2
| = difference of roots i.e.
D
a
11. Condition of Tangency : Circle x2
+ y2
= a2
will touch the
line y = mx + c if c = ±a 1 2
+ m
PAGE # 63 PAGE # 64

E D U C A T I O N S
E D U C A T I O N S
12. Equation of tangent, T = 0 :
(i) Equation of tangent to the circle
x2
+ y2
+ 2gx + 2fy + c = 0 at any point (x1
, y1
) is
xx1
+ yy1
+ g(x + x1
) + f(y + y1
) + c = 0
(ii) Equation of tangent to the circle x2
+ y2
= a2
at any
point (x1
, y1
) is xx1
+ yy1
= a2
(iii) In slope form : From the condition of tangency for
every value of m.
The line y = mx ± a 1 2
+ m is a tangent to the circle
x2
+ y2
= a2
and its point of contact is
±
+
±
+
F
HG
I
KJam
m
a
m1 12 2
,
(iv) Equation of tangent at (a cos θ , a sin θ ) to the
circle x2
+ y2
= a2
is x cos θ + y sin θ = a.
13. Equation of normal :
(i) Equation of normal to the circle
x2
+ y2
+ 2gx + 2fy + c = 0 at any point P(x1
, y1
) is
y – y1
=
y f
x g
1
1
+
+ (x – x1
)
(ii) Equation of normal to the circle x2
+ y2
= a2
at any
point (x1
, y1
) is xy1
– x1
y = 0
14. Equation of pair of tangents SS1
= T2
15. The point of intersection of tangents drawn to the circle x2
+ y2
= r2
at point θ 1
& θ 2
is given as
r rcos
cos
,
sin
cos
θ θ
θ θ
θ θ
θ θ
1 2
1 2
1 2
1 2
2
2
2
2
+
−
+
−
F
H
GGG
I
K
JJJ
16. Equation of the chord of contact of the tangents drawn
from point P outside the circle is T = 0
17. Equation of a chord whose middle pt. is given by T = S1
18. Director circle : Equation of director circle for x2
+ y2
= a2
is
x2
+ y2
= 2a2
. Director circle is a concentric circle whose
radius is 2 times the radius of the given circle.
19. Equation of polar of point(x1
, y1
) w.r.t. the circle S = 0 is T = 0
20. Coordinates of pole : Coordinates of pole of the line
lx + my + n = 0 w.r.t the circle x2
+ y2
= a2
are
− −F
HG
I
KJa l
n
a m
n
2 2
,
21. Family of Circles :
(i) S + λS' = 0 represents a family of circles passing through
the pts. of intersection of
S = 0 & S' = 0 if λ ≠ –1
(ii) S + λ L = 0 represent a family of circles passing through
the point of intersection of S = 0 & L = 0
(iii) Equation of circle which touches the given straight line
L = 0 at the given point (x1
, y1
) is given as
(x – x1
)2
+ (y – y1
)2
+ λL = 0.
PAGE # 65 PAGE # 66

E D U C A T I O N S
E D U C A T I O N S
(iv) Equation of circle passing through two points A(x1
, y1
)
& B(x2
, y2
) is given as
(x – x1
) (x – x2
) + (y – y1
) (y – y2
) + λ
x y
x y
x y
1
1
1
1 1
2 2
= 0.
22. Equation of Common Chord is S – S1
= 0.
23. The angleθθθθθ of intersection of two circles with centres C1
& C2
and radii r1
& r2
is given by
cosθ =
r r d
r r
1
2
1
2 2
1 22
+ −
, where d = C1
C2
24. Position of two circles : Let two circles with centres C1
, C2
and radii r1
, r2
.
Then following cases arise as
(i) C1
C2
> r1
+ r2 ⇒ do not intersect or one outside the
other, 4 common tangents.
(ii) C1
C2
= r1
+ r2 ⇒ Circles touch externally, 3 common
tangents.
(iii) |r1
– r2
| < C1
C2
< r1
+ r2 ⇒ Intersection at two real
points, 2 common tangents.
(iv) C1
C2
= |r1
– r2
| ⇒ internal touch, 1 common tangent.
(v) C1
C2
< |r1
+ r2
| ⇒ one inside the other, no tangent.
Note : Point of contact divides C1
C2
in the ratio r1
: r2
internally or externally as the case may be
PAGE # 67 PAGE # 68
25. Equation of tangent at point of contact of circle is
S1
– S2
= 0
26. Radical axis and radical centre :
(i) Equation of radical axis is S – S1
= 0
(ii) The point of concurrency of the three radical axis of
three circles taken in pairs is called radical centre of
three circles.
27. Orthogonality condition :
If two circles S ≡ x2
+ y2
+ 2gx + 2fy + c = 0
and S' = x2
+ y2
+ 2g'x + 2f'y + c' = 0 intersect each other
orthogonally, then 2gg' + 2ff' = c + c'.

E D U C A T I O N S
E D U C A T I O N S
PARABOLA
1. Standard Parabola :
Imp. Terms y2
= 4ax y2
= – 4ax x2
= 4ay x2
= – 4ay
Vertex (v) (0, 0) (0, 0) (0, 0) (0, 0)
Focus (f) (a, 0) (–a, 0) (0, a) (0, –a)
Directrix (D) x = –a x = a y = –a y = a
Axis y = 0 y = 0 x = 0 x = 0
L.R. 4a 4a 4a 4a
Focal x + a a – x y + a a – y
distance
Parametric (at2
, 2at) (– at2
, 2at) (2at, at2
) (2at, – at2
)
Coordinates
Parametric x = at2
x = – at2
x = 2at x = 2at
Equations y = 2at y = 2at y = 2at2
y = – at2
y2
= 4ax
y2
= – 4ax
x2
= 4ay
x2
= – 4ay
PAGE # 69 PAGE # 70

E D U C A T I O N S
E D U C A T I O N S
2. Special Form of Parabola
* Parabola which has vertex at (h, k), latus rectum l
and axis parallel to x-axis is
(y – k)2
= l (x – h)
⇒ axis is y = k and focus at h k+
F
HG I
KJl
4
,
* Parabola which has vertex at (h, k), latus rectum l and
axis parallel to y-axis is
(x – h)2
= l (y – k)
⇒ axis is x = h and focus at h k, +
F
HG I
KJl
4
* Equation of the form ax2
+ bx + c = y represents
parabola.
i.e. y –
4
4
2
ac b
a
−
= a x
b
a
+
F
HG I
KJ2
2
,with vertex
−
−F
HG
I
KJb
a
ac b
a2
4
4
2
, and axes parallel to y-axis
Note : Parametric equation of parabola (y – k)2
= 4a(x – h) are x = h + at2
, y = k + 2at
3. Position of a point (x1
, y1
) and a line w.r.t. parabola
y2
= 4ax.
* The point (x1
, y1
) lies outside, on or inside the
parabola y2
= 4ax
according as y1
2
– 4ax1
>, = or < 0
* The line y = mx + c does not intersect, touches,
intersect a parabola y2
= 4ax according as
c > = < a/m
Note : Condition of tangency for parabola y2
= 4ax, we
have c = a/m and for other parabolas check disc. D = 0.
4. Equations of tangent in different forms :
(i) Point Form / Parametric form
Equations of tangent of all other standard parabolas
at (x1
, y1
) / at t (parameter)
Equation Tangent at Parametric Tangent of 't'
of parabola (x1
, y1
) coordinates't'
y2
=4ax yy1
=2a(x+x1
) (at2
, 2at) ty=x+at2
y2
=–4ax yy1
=–2a(x+x1
) (–at2
, 2at) ty=–x+at2
x2
=4ay xx1
=2a(y+y1
) (2at, at2
) tx=y + at2
x2
=–4ay xx1
=–2a(y+y1
) (2at, –at2
) tx =–y+at2
(ii) Slope form
Equations of tangent of all other parabolas in slope
form
Equation Point of Equations Condition of
of contact in of tangent Tangency
parabolas terms of in terms of
slope(m) slope (m)
y2
= 4ax
a
m
a
m2
2
,
F
HG I
KJ y = mx +
a
m
c =
a
m
y2
= – 4ax −
F
HG I
KJa
m
a
m2
2
, y = mx –
a
m
c = –
a
m
x2
= 4ay (2am, am2
) y = mx – am2
c = –am2
x2
= – 4ay (–2am, –am2
) y = mx + am2
c = am2
5. Point of intersection of tangents at any two points
P(at1
2
, 2at1
) and Q(at2
2
, 2at2
) on the parabola y2
= 4ax
is (at1
t2
, a(t1
+ t2
)) i.e. (a(G.M.)2
, a(2A.M.))
6. Combined equation of the pair of tangents drawn from a
point to a parabola is SS' = T2
, where S = y2
– 4ax,
S' = y1
2
– 4ax1
and T = yy1
– 2a(x + x1
)
PAGE # 71 PAGE # 72

E D U C A T I O N S
E D U C A T I O N S
7. Equations of normal in different forms
(i) Point Form / Parametric form
Equations of normals of all other standard parabolas
at (x1
, y1
) / at t (parameter)
Eqn
. of Normal Point Normals
parabola at (x1
, y1
) 't' at 't'
y2
= 4ax y–y1
=
−y
a
1
2
(x–x1
) (at2
, 2at) y+tx = 2at+at3
y2
= –4ax y–y1
=
y
a
1
2
(x–x1
) (–at2
, 2at) y–tx = 2at+at3
x2
= 4ay y–y1
= –
2
1
a
x (x–x1
) (2at, at2
) x+ty = 2at+at3
x2
= –4ay y–y1
=
2
1
a
x (x–x1
) (2at, –at2
) x–ty = 2at+at3
(ii) Slope form
Equations of normal, point of contact, and condition
of normality in terms of slope (m)
Eqn
. of Point of Equations Condition of
parabola contact of normal Normality
y2
= 4ax (am2
, –2am) y = mx–2am–am3
c = –2am–am3
y2
= – 4ax (–am2
, 2am) y = mx+2am+am3
c = am+am3
x2
= 4ay −
F
HG I
KJ2
2
a
m
a
m
, y = mx+2a+
a
m2 c = 2a+
a
m2
x2
= –4ay
2
2
a
m
a
m
, −
F
HG I
KJ y = mx–2a–
a
m2
c = –2a–
a
m2
Note :
(i) In circle normal is radius itself.
(ii) Sum of ordinates (y coordinate) of foot of normals
through a point is zero.
(iii) The centroid of the triangle formed by taking the foot
of normals as a vertices of concurrent normals of
y2
= 4ax lies on x-axis.
8. Condition for three normals from a point (h, 0) on x-axis
to parabola y2
= 4ax
(i) We get 3 normals if h > 2a
(ii) We get one normal if h ≤ 2a.
(iii) If point lies on x-axis, then one normal will be x-axis
itself.
9. (i) If normal of y2
= 4ax at t1
meet the parabola again
at t2
then t2
= – t1
–
2
1t
(ii) The normals to y2
= 4ax at t1
and t2
intersect each
other at the same parabola at t3
, then
t1
t2
= 2 and t3
= – t1
– t2
10. (i) Equation of focal chord of parabola y2
= 4ax at t1
is
y =
2
1
1
1
2
t
t −
(x – a)
If focal chord of y2
= 4ax cut (intersect) at t1
and
t2
then t1
t2
= – 1 (t1
must not be zero)
(ii) Angle formed by focal chord at vertex of parabola is
tan θ =
2
3
|t2
– t1
|
(iii) Intersecting point of normals at t1
and t2
on the
parabola y2
= 4ax is
(2a + a(t1
2
+ t2
2
+ t1
t2
), – at1
t2
(t1
+ t2
))
PAGE # 73 PAGE # 74

E D U C A T I O N S
E D U C A T I O N S
11. Equation of chord of parabola y2
= 4ax which is bisected
at (x1
, y1
) is given by T = S1
12. The locus of the mid point of a system of parallel chords
of a parabola is called its diameter. Its equation is y =
2a
m
.
13. Equation of polar at the point (x1
, y1
) with respect to
parabola y2
= 4ax is same as chord of contact and is given
by
T = 0 i.e. yy1
= 2a(x + x1
)
Coordinates of pole of the line l x + my + n = 0 w.r.t. the
parabola y2
= 4ax is
n am
l l
,
−F
HG I
KJ2
14. Diameter : It is locus of mid point of set of parallel chords
and equation is given by T = S1
15. Important results for Tangent :
(i) Angle made by focal radius of a point will be twice
the angle made by tangent of the point with axis of
parabola
(ii) The locus of foot of perpendicular drop from focus to
any tangent will be tangent at vertex.
(iii) If tangents drawn at ends point of a focal chord are
mutually perpendicular then their point of intersection
will lie on directrix.
(iv) Any light ray travelling parallel to axis of the parabola
will pass through focus after reflection through
parabola.
PAGE # 75 PAGE # 76
(v) Angle included between focal radius of a point and
perpendicular from a point to directrix will be bisected
of tangent at that point also the external angle will
be bisected by normal.
(vi) Intercepted portion of a tangent between the point
of tangency and directrix will make right angle at
focus.
(vii) Circle drawn on any focal radius as diameter will
touch tangent at vertex.
(viii) Circle drawn on any focal chord as diameter will touch
directrix.

E D U C A T I O N S
E D U C A T I O N S
ELLIPSE
1. Standard Ellipse (e < 1)
x
a
y
b
2
2
2
2
1+ =
RS|
T|
UV|
W|
For a > b For b > a
Centre (0, 0) (0, 0)
Vertices (±a, 0) (0, ±b)
Length of major axis 2a 2b
Length of minor axis 2b 2a
Foci (±ae, 0) (0, ±be)
Equation of directrices x = ±a/e y = ±b/e
Relation in a, b and e b2
= a2
(1 – e2
) a2
= b2
(1 – e2
)
Length of latus rectum 2b2
/a 2a2
/b
Ends of latus rectum ± ±
F
HG
I
KJae
b
a
,
2
± ±
F
HG
I
KJa
b
be
2
,
Parametric coordinates (a cos φ , b sin φ ) (a cos φ , b sin φ )
0 ≤ φ < 2 π
Focal radii SP = a – ex1
SP = b – ey1
S'P = a + ex1
S'P = b + ey1
Sum of focal radii SP + S'P = 2a 2b
Distance btn
foci 2ae 2be
Distance btn
directrices 2a/e 2b/e
Tangents at the vertices x = –a, x = a y = b, y = – b
Note : If P is any point on ellipse and length of perpendiculars
from to minor axis and major axis are p1
& p2
, then |xp
|
= p1
, |yp
| = p2
⇒
p
a
p
b
1
2
2
2
2
2
+ = 1
a > b
b > a
PAGE # 77 PAGE # 78
Imp. terms
Ellipse

E D U C A T I O N S
E D U C A T I O N S
2. Special form of ellipse :
If the centre of an ellipse is at point (h, k) and the
directions of the axes are parallel to the coordinate axes,
then its equation is
x h
a
−c h2
2
+
( )y k
b
− 2
2 = 1.
3. Auxillary Circle : The circle described by taking centre of
an ellipse as centre and major axis as a diameter is called
an auxillary circle of the ellipse.
If
x
a
2
2 +
y
b
2
2 = 1 is an ellipse then its auxillary circle is
x2
+ y2
= a2
.
Note : Ellipse is locus of a point which moves in such a
way that it divides the normal of a point on diameter of
a point of circle in fixed ratio.
4. Position of a point and a line w.r.t. an ellipse :
* The point lies outside, on or inside the ellipse if
S1
=
x
a
1
2
2 +
y
b
1
2
2 – 1 > , = or < 0
* The line y = mx + c does not intersect, touches,
intersect, the ellipse if
a2
m2
+ b2
< = > c2
5. Equation of tangent in different forms :
(i) Point form : The equation of the tangent to the
ellipse
x
a
2
2 +
y
b
2
2 = 1 at the point (x1
, y1
) is
xx
a
1
2 +
yy
b
1
2 = 1.
(ii) Slope form : If the line y = mx + c touches the
ellipse
x
a
2
2 +
y
b
2
2 = 1, then c2
= a2
m2
+ b2
. Hence, the
straight line y = mx ± a m b2 2 2
+ always represents
the tangents to the ellipse.
Point of contact :
Line y = mx ± a m b2 2 2
+ touches the ellipse
x
a
2
2 +
y
b
2
2 = 1 at
±
+
±
+
F
HG
I
KJa m
a m b
b
a m b
2
2 2 2
2
2 2 2
, .
(iii) Parametric form : The equation of tangent at any
point (a cos φ , b sin φ ) is
x
a
cos φ +
y
b
sin φ = 1.
6. Equation of pair of tangents from (x1
, y1
) to an ellipse
x
a
2
2 +
y
b
2
2 = 1 is given by SS1
= T2
7. Equation of normal in different forms :
(i) Point form : The equation of the normal at (x1
, y1
)
to the ellipse
x
a
2
2 +
y
b
2
2 = 1 is
a x
x
2
1
–
b x
y
2
1
= a2
– b2
.
PAGE # 79 PAGE # 80

E D U C A T I O N S
E D U C A T I O N S
(ii) Parametric form : The equation of the normal to the
ellipse
x
a
2
2 +
y
b
2
2 = 1 at (a cos φ , b sin φ ) is
ax sec φ – by cosec φ = a2
– b2
.
(iii) Slope form : If m is the slope of the normal to the
ellipse
x
a
2
2 +
y
b
2
2 = 1, then the equation of normal
is y = mx ±
m a b
a b m
( )2 2
2 2 2
−
+
.
The co-ordinates of the point of contact are
±
+
±
+
F
HG
I
KJa
a b m
mb
a b m
2
2 2 2
2
2 2 2
, .
Note : In general three normals can be drawn from a point
(x1
, y1
) to an ellipse
x
a
2
2 +
y
b
2
2 = 1.
8. Properties of tangents & normals :
(i) Product of length of perpendicular from either focii to
any tangent to the ellipse will be equal to square of
semi minor axis.
(ii) The locus of foot of perpendicular drawn from either
focii to any tangent lies on auxillary circle.
(iii) The circle drawn on any focal radius as diameter will
touch auxillary circle.
(iv) The protion of the tangent intercepted between the
point and directrix makes right angle at corresponding
focus.
(v) Sum of square of intercept made by auxillary circle on
any two perpendicular tangents of an ellipse will be
constant.
(vi) If a light ray originates from one of focii, then it will
pass through the other focus after reflection from
ellipse.
9. Equation of chord of contact of the tangents drawn from
the external point (x1
, y1
) to an ellipse is given by
xx
a
1
2 +
yy
b
1
2 = 0 i.e. T = 0.
10. The equation of a chord of an ellipse
x
a
2
2 +
y
b
2
2 = 1 whose
mid point is (x1
, y1
) is T = S1
.
11. Equation of chord joining the points (a cos θ , b sin θ ) and
(a cos φ , b sin φ ) on the ellipse
x
a
2
2 +
y
b
2
2 = 1 is
x
a
cos
θ φ+
2
+
y
b
sin
θ φ+
2
= cos
θ φ−
2
(i) Relation between eccentric angles of focal chord
⇒ tan
θ1
2
, tan
θ2
2
=
± −
±
e
e
1
1
(ii) Sum of feet of eccentric angles is odd π.
i.e. θ 1
+ θ 2
+ θ 3
+ θ 4
= (2n + 1) π .
PAGE # 81 PAGE # 82

E D U C A T I O N S
E D U C A T I O N S
12. Equation of polar of the point (x1
, y1
) w.r.t. the ellipse
x
a
2
2 +
y
b
2
2 = 1 is given by
xx
a
1
2 +
yy
b
1
2 = 0 i.e. T = 0.
The pole of the line l x + my + n = 0 w.r.t. the ellipse
x
a
2
2 +
y
b
2
2 = 1 is
− −F
HG
I
KJa
n
b n
n
2 2
l
, .
13. Eccentric angles of the extremities of latus rectum of the
ellipse
x
a
2
2 +
y
b
2
2 = 1 are tan–1 ±
F
HG I
KJb
ae
.
14. (i) Equation of the diameter bisecting the chords of
slope in the ellipse
x
a
2
2 +
y
b
2
2 = 1 is
y = –
b
a m
2
2 x
(ii) Conjugate Diameters : The straight lines y = m1
x,
y = m2
x are conjugate diameters of the ellipse
x
a
2
2 +
y
b
2
2 = 1 if m1
m2
= –
b
a
2
2 .
(iii) Properties of conjugate diameters :
(a) If CP and CQ be two conjugate semi-diameters
of the ellipse
x
a
2
2 +
y
b
2
2 = 1, then
CP2
+ CQ2
= a2
+ b2
(b) If θ and φ are the eccentric angles of the
extremities of two conjugate diameters, then
θ – φ = ±
π
2
(c) If CP, CQ be two conjugate semi-diameters of
the ellipse
x
a
2
2 +
y
b
2
2 = 1 and S, S' be two foci
of the ellipse, then SP.S'P = CQ2
(d) The tangents at the ends of a pair of conjugate
diameters of an ellipse form a parallelogram.
15. The area of the parallelogram formed by the tangents at
the ends of conjugate diameters of an ellipse is constant
and is equal to the product of the axis i.e. 4ab.
16. Length of subtangent and subnormal at p(x1
, y1
) to the
ellipse
x
a
2
2 +
y
b
2
2 = 1 is
a
x
2
1
– x1
& (1 – e2
) x1
PAGE # 83 PAGE # 84

E D U C A T I O N S
E D U C A T I O N S
HYPERBOLA
1. Standard Hyperbola :
x
a
2
2 –
y
b
2
2 = 1 –
x
a
2
2 +
y
b
2
2 = 1
or
x
a
2
2 –
y
b
2
2 = – 1
Centre (0, 0) (0, 0)
Length of
transverse axis 2a 2b
Length of
conjugate axis 2b 2a
Foci (±ae, 0) (0, ±be)
Equation of
directrices x = ±a/e y = ± b/e
Eccentricity e =
a b
a
2 2
2
+F
HG
I
KJ e =
a b
b
2 2
2
+F
HG
I
KJ
Length of L.R. 2b2
/a 2a2
/b
Parametric
co-ordinates (a sec φ , b tan φ ) (b sec φ , a tanφ )
0 ≤ φ < 2 π 0 ≤ φ < 2 π
Focal radii SP = ex1
– a SP = ey1
– b
S'P = ex1
+ a S'P = ey1
+ b
S'P – SP 2a 2b
Tangents at
the vertices x = – a, x = a y = – b, y = b
Equation of the y = 0 x = 0
transverse axis
Equation of the x = 0 y = 0
conjugate axis
Hyperbola
Conjugate Hyperbola
PAGE # 85 PAGE # 86
Imp. terms
Hyperbola

E D U C A T I O N S
E D U C A T I O N S
2. Special form of hyperbola :
If the centre of hyperbola is (h, k) and axes are parallel
to the co-ordinate axes, then its equation is
( )x h
a
− 2
2 –
( )y k
b
− 2
2 = 1.
3. Parametric equations of hyperbola :
The equations x = a sec φ and y = b tan φ are known
as the parametric equations of hyperbola
x
a
2
2 –
y
b
2
2 = 1
4. Position of a point and a line w.r.t. a hyperbola :
The point (x1
, y1
) lies inside, on or outside the hyperbola
x
a
2
2 –
y
b
2
2 = 1
according as
x
a
1
2
2 –
y
b
1
2
2 – 1 is +ve, zero or –ve.
The line y = mx + c does not intersect, touches, intersect
the hyperbola
according as c2
<, =, > a2
m2
– b2
.
5. Equations of tangents in different forms :
(a) Point form : The equation of the tangent to the
hyperbola
x
a
2
2 –
y
b
2
2 = 1
at (x1
, y1
) is
xx
a
1
2 –
yy
b
1
2 = 1.
(b) Parametric form : The equation of tangent to the
hyperbola
x
a
2
2 –
y
b
2
2 = 1 at (a sec φ , b tan φ ) is
x
a
sec φ –
y
b
tan φ = 1.
(c) Slope form : The equations of tangents of slope m
to the hyperbola
x
a
2
2 –
y
b
2
2 = 1 are y = mx ± a m b2 2 2
− and the
co-ordinates of points of contacts are
±
−
±
−
F
HG
I
KJa m
a m b
b
a m b
2
2 2 2
2
2 2 2
, .
6. Equation of pair of tangents from (x1
, y1
) to the hyperbola
x
a
2
2 –
y
b
2
2 = 1 is given by SS1
= T2
7. Equations of normals in different forms :
(a) Point form : The equation of normal to the hyperbola
x
a
2
2 –
y
b
2
2 = 1 at (x1
, y1
) is
a x
x
2
1
+
b y
y
2
1
= a2
+ b2
.
(b) Parametric form : The equation of normal at
(a sec θ , b tan θ ) to the hyperbola
x
a
2
2 –
y
b
2
2 = 1 is ax cos θ + by cot θ = a2
+ b2
PAGE # 87 PAGE # 88

E D U C A T I O N S
E D U C A T I O N S
(c) Slope form : The equation of the normal to the
hyperbola
x
a
2
2 –
y
b
2
2 = 1 in terms of the slope m of
the normal is y = mx m
m a b
a b m
( )2 2
2 2 2
+
−
(d) Condition for normality : If y = mx + c is the normal
of
x
a
2
2 –
y
b
2
2 = 1,
then c = m
m a b
a b m
( )2 2
2 2 2
+
−
or c2
=
m a b
a m b
( )
( )
2 2 2
2 2 2
+
−
, which
is condition of normality.
(e) Points of contact : Co-ordinates of points of contact
are ±
− −
F
HG
I
KJa
a b m
mb
a b m
2
2 2 2
2
2 2 2
, m .
8. The equation of director circle of hyperbola
x
a
2
2 –
y
b
2
2 = 1 is x2
+ y2
= a2
– b2
.
9. Equation of chord of contact of the tangents drawn from
the external point (x1
, y1
) to the hyperbola is given by
xx
a
1
2 –
yy
b
1
2 = 1.
10. The equation of chord of the hyperbola
x
a
2
2 –
y
b
2
2 = 1
whose mid point is (x1
, y1
) is T = S1
.
11. Equation of chord joining the points P(a sec φ 1
, b tan φ 1
)
and Q(a sec φ 2
, b tan φ 2
) is
x
a
cos
φ φ1 2
2
−F
HG I
KJ –
y
b
sin
φ φ1 2
2
+F
HG I
KJ = cos
φ φ1 2
2
+F
HG I
KJ .
12. Equation of polar of the point (x1
, y1
) w.r.t. the hyperbola
is given by T = 0.
The pole of the line l x + my + n = 0 w.r.t.
x
a
2
2 –
y
b
2
2 = 1 is −
F
HG
I
KJa
n
b m
n
2 2
l
,
13. The equation of a diameter of the hyperbola
x
a
2
2 –
y
b
2
2 = 1 is y =
b
a m
2
2 x.
14. The diameters y = m1
x and y = m2
x are conjugate if
m1
m2
=
b
a
2
2
15. Asymptotes of a hyperbola :
* The equations of asymptotes of the hyperbola
x
a
2
2 –
y
b
2
2 = 1 are y = ±
b
a
x.
Asymptote to a curve touches the curve at infinity.
* The asymptote of a hyperbola passes through the
centre of the hyperbola.
PAGE # 89 PAGE # 90

E D U C A T I O N S
E D U C A T I O N S
* The combined equation of the asymptotes of the
hyperbola
x
a
2
2 –
y
b
2
2 = 1 is
x
a
2
2 –
y
b
2
2 = 0.
* The angle between the asymptotes of
x
a
2
2 –
y
b
2
2 = 1 is 2 tan–1
y
b
2
2 or 2 sec–1
e.
* A hyperbola and its conjugate hyperbola have the
same asymptotes.
* The bisector of the angles between the asymptotes
are the coordinate axes.
* Equation of hyperbola – Equation of asymptotes =
Equation of asymptotes – Equation of conjugate
hyperbola = constant.
16. Rectangular or Equilateral Hyperbola :
* A hyperbola for which a = b is said to be rectangular
hyperbola, its equation is x2
– y2
= a2
* xy = c2
represents a rectangular hyperbola with
asymptotes x = 0, y = 0.
* Eccentricity of rectangular hyperbola is 2 and angle
between asymptotes of rectangular hyperbola is 90º.
* Parametric equation of the hyperbola xy = c2
are
x = ct, y =
c
t
, where t is a parameter.
* Equation of chord joining t1
, t2
on xy = c2
is
x + y t1
t2
= c(t1
+ t2
)
* Equation of tangent at (x1
, y1
) to xy
= c2
is
x
x1
+
y
y1
= 2.
Equation of tangent at t is x + yt2
= 2ct
* Equation of normal at (x1
, y1
) to xy = c2
is
xx1
– yy1
= x1
2
– y1
2
* Equation of normal at t on xy = c2
is
xt3
– yt – ct4
+ c = 0.
(This results shows that four normal can be drawn
from a point to the hyperbola xy = c2
)
* If a triangle is inscribed in a rectangular hyperbola
then its orthocentre lies on the hyperbola.
* Equation of chord of the hyperbola xy = c2
whose
middle point is given is T = S1
* Point of intersection of tangents at t1
& t2
to the
hyperbola xy = c2
is
2 21 2
1 2 1 2
c t t
t t
c
t t+ +
F
HG
I
KJ,
PAGE # 91 PAGE # 92

E D U C A T I O N S
E D U C A T I O N S
MEASURES OF CENTRAL TENDENCY
AND DISPERSION
1. Arithmetic mean :
(i) For ungrouped data (individual series) x =
x x x
n no of terms
n1 2+ + +......
( . )
=
Σ
i
n
ix
n
=1
(ii) For grouped data (continuous series)
(a) Direct method x =
Σ
Σ
i
n
i i
i
n
i
fx
f
=
=
1
1
, where xi
, i = 1 .... n
be n observations and fi
be their corresponding
frequencies
(b) short cut method : x = A +
Σ
Σ
fd
f
i i
i
,
where A = assumed mean, di
= xi
– A = deviation
for each term
2. Properties of A.M.
(i) In a statistical data, the sum of the deviation of items
from A.M. is always zero.
(ii) If each of the n given observation be doubled, then
their mean is doubled
(iii) If x is the mean of x1
, x2
, ...... xn
. The mean of ax1
, ax2
.....axn
is a x where a is any number different from
zero.
(iv) Arithmetic mean is independent of origin i.e. it is x
effected by any change in origin.
3. Geometric Mean :
(i) For ungrouped data
G.M. = (x1
x2
x3
.....xn
)1/n
or G.M. = antilog
1
1
n
xi
i
n
log
=
∑
F
HG
I
KJ
(ii) For grouped data
G.M. = x x xf f
n
f Nn
1 2
1
1 2
....e j , where N = fi
i
n
=
∑1
= antilog
f x
f
i
i
n
i
i
i
n
=
=
∑
∑
F
H
GGGG
I
K
JJJJ
1
1
log
4. Harmonic Mean - Harmonic Mean is reciprocal of arith-
metic mean of reciprocals.
(i) For ungrouped data H.M. =
n
xii
n
1
1=
∑
(ii) For grouped data H.M. =
f
f
x
i
i
n
i
ii
n
=
=
∑
∑
F
HG I
KJ
1
1
5. Relation between A.M., G.M and H.M.
A.M. ≥ G.M. ≥ H.M.
Equality holds only when all the observations in the series
are same.
PAGE # 93 PAGE # 94

E D U C A T I O N S
E D U C A T I O N S
6. Median :
(a) Individual series (ungrouped data) : If data is raw,
arrange in ascending or descending order and n be
the no. of observations.
If n is odd, Median = Value of
n
th
+F
HG I
KJ1
2
observation
If n is even, Median =
1
2
[Value of
n
th
2
F
HG I
KJ + value of
n
th
2
1+
F
HG I
KJ ] observation.
(b) Discrete series : First find cumulative frequencies of
the variables arranged in ascending or descending
order and
Median =
n
th
+F
HG I
KJ1
2
observation, where n is cumulative
frequency.
(c) Continuous distribution (grouped data)
(i) For series in ascending order
Median = l +
N
C
f
2
−
F
HG I
KJ
× i
Where l = Lower limit of the median class.
f = Frequency of the median class.
N = Sum of all frequencies.
i = The width of the median class
C = Cumulative frequency of the
class preceding to median class.
(ii) For series in descending order
Median = u -
N
C
f
2
−
F
HG I
KJ
× i
where u = upper limit of median class.
7. Mode :
(i) For individual series : In the case of individual series,
the value which is repeated maximum number of times
is the mode of the series.
(ii) For discrete frequency distribution series : In the case
of discrete frequency distribution, mode is the value of
the variate corresponding to the maximum frequency.
(iii) For continuous frequency distribution : first find the
model class i.e. the class which has maximum frequency.
For continuous series
Mode = l 1
+
f f
f f f
1 0
1 0 22
−
− −
L
NM O
QP × i
Where l 1
= Lower limit of the model class.
f1
= Frequency of the model class.
f0
= Frequency of the class preceding model
class.
f2
= Frequency of the class succeeding model
class.
i = Size of the model class.
8. Relation between Mean, Mode & Median :
(i) In symmetrical distribution : Mean = Mode = Median
(ii) In Moderately symmetrical distribution : Mode = 3 Me-
dian – 2 Mean
PAGE # 95 PAGE # 96

E D U C A T I O N S
E D U C A T I O N S
Measure of Dispersion :
The degree to which numerical data tend to spread about
an average value is called variation or dispersion.
Popular methods of measure of dispersion.
1. Mean deviation : The arithmetic average of deviations
from the mean, median or mode is known as mean deviation.
(a) Individual series (ungrouped data)
Mean deviation =
Σ| |x S
n
−
Where n = number of terms, S = deviation of variate
from mean mode, median.
(b) Continuous series (grouped data).
Mean deviation =
Σ
Σ
f x s
f
| |−
=
Σf x s
N
| |−
Note : Mean deviation is the least when measured from the
median.
2. Standard Deviation :
S.D. (σ) is the square root of the arithmetic mean of the
squares of the deviations of the terms from their A.M.
(a) For individual series (ungrouped data)
σ =
Σ( )x x
N
− 2
where x = Arithmetic mean of
the series
N = Total frequency
(b) For continuous series (grouped data)
(i) Direct method σ =
Σf x x
N
i i( )− 2
Where x = Arithmetic mean of series
xi
= Mid value of the class
fi
= Frequency of the corresponding xi
N = Σ f = Total frequency
(ii) Short cut method
σ =
Σ Σfd
N
fd
N
2 2
−
F
HG I
KJ or σ =
Σ Σd
N
d
N
2 2
−
F
HG I
KJ
Where d = x – A = Derivation from assumed mean A
f = Frequency of item (term)
N = Σf = Total frequency.
Variance – Square of standard direction
i.e. variance = (S.D.)2
= (σ)2
Coefficient of variance = Coefficient of S.D. × 100
=
σ
x
× 100
PAGE # 97 PAGE # 98

E D U C A T I O N S
E D U C A T I O N S
MATRICES AND DETERMINANTS
MATRICES :
1. Matrix - A system or set of elements arranged in a rectan-
gular form of array is called a matrix.
2. Order of matrix : If a matrix A has m rows & n columns then
A is of order m × n.
The number of rows is written first and then number of col-
umns. Horizontal line is row & vertical line is column
3. Types of matrices : A matrix A = (aij
)m×n
A matrix A = (aij
)mxn
over the field of complex numbers is
said to be
Name Properties
A row matrix if m = 1
A column matrix if n = 1
A rectangular matrix if m ≠ n
A square matrix if m = n
A null or zero matrix if aij
= 0 ∀ i j. It is denoted by O.
A diagonal matrix if m = n and aij
= 0 for i ≠ j.
A scalar matrix if m = n and aij
= 0 for i ≠ j
= k for i = j
i.e. a11
= a22
....... = ann
= k (cons.)
Identity or unit matrix if m = n and aij
= 0 for i ≠ j
= 1 for i = j
Upper Triangular matrix if m = n and aij
= 0 for i > j
Lower Triangular matrix if m = n and aij
= 0 for i < j
Symmetric matrix if m = n and aij
= aji
for all i, j
or AT
= A
Skew symmetric matrix if m = n and aij
= – aji
∀ i, j
or AT
= – A
4. Trace of a matrix : Sum of the elements in the principal
diagonal is called the trace of a matrix.
trace (A ± B) = trace A ± trace B
trace kA = k trace A
trace A = trace AT
trace In
= n when In
is identity matrix.
trace On
= O On
is null matrix.
trace AB ≠ trace A trace B.
5. Addition & subtraction of matrices : If A and B are two
matrices each of order same, then A + B (or A – B) is defined
and is obtained by adding (or subtracting) each element of B
from corresponding element of A
6. Multiplication of a matrix by a scalar :
KA = K (aij
)m×n
= (Ka)m×n
where K is constant.
Properties :
(i) K(A + B) = KA + KB
(ii) (K1
K2
)A = K1
(K2
A) = K2
(K1
A)
(iii) (K1
+ K2
)A = K1
A + K2
A
7. Multiplication of Matrices : Two matrices A & B can be
multiplied only if the number of columns in A is same as the
number of rows in B.
Properties :
(i) In general matrix multiplication is not commutative i.e.
AB ≠ BA.
(ii) A(BC) = (AB)C [Associative law]
PAGE # 99 PAGE # 100

E D U C A T I O N S
E D U C A T I O N S
(iii) A.(B + C) = AB + AC [Distributive law]
(iv) If AB = AC ⇒/ B = C
(v) If AB = 0, then it is not necessary A = 0 or B = 0
(vi) AI = A = IA
(vii) Matrix multiplication is commutative for +ve integral
i.e. Am+1
= Am
A = AAm
8. Transpose of a matrix :
A' or AT
is obtained by interchanging rows into columns or
columns into rows
Properties :
(i) (AT
)T
= A
(ii) (A ± B)T
= AT
± BT
(iii) (AB)T
= BT
AT
(iv) (KA)T
= KAT
(v) IT
= I
9. Some special cases of square matrices :A square matrix
is called
(i) Orthogonal matrix : if AAT
= In
= AT
A
(ii) Idempotent matrix : if A2
= A
(iii) Involutory matrix : if A2
= I or A–1
= A
(iv) Nilpotent matrix : if ∃ p ∈ N such that Ap
= 0
(v) Hermitian matrix : if Aθ
= A i.e. aij
= aji
(vi) Skew - Hermitian matrix : if A = –Aθ
DETERMINANT:
1. Minor & cofactor : If A = (aij
)3×3
, then minor of a11
is
M11
=
a a
a a
22 23
32 33
and so.
cofactor of an element aij
is denoted by Cij
or Fij
and is equal
to (–1)i+j
Mij
or Cij
= Mij
, if i = j
= –Mij
, if i ≠ j
Note : |A| = a11
F11
+ a12
F12
+ a13
F13
and a11
F21
+ a12
F22
+ a13
F23
= 0
2. Determinant : if A is a square matrix then determinant of
matrix is denoted by det A or |A|.
expansion of determinant of order 3 × 3
⇒
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= a1
b c
b c
2 2
3 3
–b1
a c
a c
2 2
3 3
+ c1
a b
a b
2 2
3 3
or = –a2
b c
b c
1 1
3 3
+ b2
a c
a c
1 1
3 3
– c2
a b
a b
1 1
3 3
Properties :
(i) |AT
| = |A|
(ii) By interchanging two rows (or columns), value of de-
terminant differ by –ve sign.
(iii) If two rows (or columns) are identical then |A| = 0
(iv) |KA| = Kn
det A, A is matrix of order n × n
PAGE # 101 PAGE # 102

E D U C A T I O N S
E D U C A T I O N S
(v) If same multiple of elements of any row (or column) of
a determinant are added to the corresponding elements
of any other row (or column), then the value of the
new determinant remain unchanged.
(vi) Determinant of :
(a) A nilpotent matrix is 0.
(b) An orthogonal matrix is 1 or – 1
(c) A unitary matrix is of modulus unity.
(d) A Hermitian matrix is purely real.
(e) An identity matrix is one i.e. |In
| = 1, where In
is a
unit matrix of order n.
(f) A zero matrix is zero i.e. |0n
| = 0, where 0n
is a zero
matrix of order n
(g) A diagonal matrix = product of its diagonal elements.
(h) Skew symmetric matrix of odd order is zero.
3. Multiplication of two determinants :
Multiplication of two second order determinants is defined as
follows.
a b
a b
1 1
2 2
×
l
l
1 1
2 2
m
m =
a b a m b m
a b a m b m
1 1 1 2 1 1 1 2
2 1 2 2 2 1 2 2
l l
l l
+ +
+ +
If order is different then for their multiplication, express them
firstly in the same order.
MATRICES AND DETERMINANTS :
1. Adjoint of a matrix :
adj A = (Cij
)T
, where Cij
is cofactor of aij
Properties :
(i) A(adj A) = (adjA) A = |A|In
(ii) |adj A| = |A|n–1
(iii) (adjAB) = (adjB) (adjA)
(iii) (adj AT
) = (adjA)T
(iv) adj(adjA) = |A|n–2
(v) (adj KA) = Kn–1
(adj A)
2. Inverse of a matrix :
(i) A–1
exists if A is non singular i.e. |A| ≠ 0
(ii) A–1
=
adjA
A| |
, |A| ≠ 0
(iii) A–1
A = In
= A A–1
(iv) (AT
)–1
= (A–1
)T
(v) (A–1
)–1
= A
(vi) |A–1
| = |A|–1
=
1
| |A
(vii) If A & B are invertible square matrices then
(AB)–1
= B–1
A–1
3. Rank of a matrix :
A non zero matrix A is said to have rank r, if
(i) Every square sub matrix of order (r + 1) or more is
singular
(ii) There exists at least one square submatrix of order r
which is non singular.
PAGE # 103 PAGE # 104

E D U C A T I O N S
E D U C A T I O N S
4. Homogeneous & non homogeneous system of linear
equations :
A system of equations Ax = B is called a homogeneous sys-
tem if B = 0. If B ≠ 0, then it is called non homogeneous
system equations.
5. (a) Solution of non homogeneous system of linear
equations :
(i) Cramer's rule : Determinant method
The non homogeneous system Ax = B, B ≠ 0 of n
equations in n variables is -
Consistent (with unique solution) if |A| ≠ 0 and
for each i = 1, 2, ........ n,
xi
=
det
det
A
A
i
, where Ai
is the matrix obtained
from A by replacing ith
column with B.
Inconsistent (with no solution) if |A| = 0 and
at least one of the det (Ai
) is non zero.
Consistent (With infinite many solution), if
|A| = 0 and all det (Ai
) are zero.
(ii) Matrix method :
The non homogeneous system Ax = B, B ≠ 0 of
n equations in n variables is -
Consistent (with unique solution) if |A| ≠ 0 i.e.
if A is non singular, x = A–1
B.
Inconsistent (with no solution), if |A| = 0 and
(adj A) B is a non null matrix.
Consistent (with infinitely many solutions), if
|A| = 0 and (adj A) B is a null matrix.
PAGE # 105 PAGE # 106
(b) Solution of homogeneous system of linear equa-
tions :
The homogeneous system Ax = B, B = 0 of n equations
in n variables is
(i) Consistent (with unique solution) if |A| ≠ 0 and
for each i = 1, 2, ......... n
xi
= 0 is called trivial solution.
(ii) Consistent (with infinitely many solution),
if |A| = 0
(a) |A| = |Ai
| = 0 (for determinant method)
(b) |A| = 0, (adj A) B = 0 (for matrix method)
NOTE : A homogeneous system of equations is never
inconsistent.

E D U C A T I O N S
E D U C A T I O N S
FUNCTION
1. Modulus function :
|x| =
x x
x x
x
,
,
,
>
− <
=
R
S|
T|
0
0
0 0
Properties :
(i) |x| ≠ ± x
(ii) |xy| = |x||y|
(iii)
x
y
=
| |
| |
x
y
(iv) |x + y| ≤ |x| + |y|
(v) |x – y| ≥ |x| – |y| or ≤ |x| + |y|
(vi) ||a| – |b|| ≤ |a – b| for equality a.b ≥ 0.
(vii) If a > 0
|x| = a ⇒ x = ± a
|x| = –a ⇒ no solution
|x| > a ⇒ x < – a or x > a
|x| ≤ a ⇒ –a ≤ x ≤ a
|x| < –a ⇒ No solution.
|x| > –a ⇒ x ∈ R
2. Logarithmic Function :
(i) logb
a to be defined a > 0, b > 0, b ≠ 1
(ii) loga
b = c ⇒ b = ac
(iii) loga
b > c
⇒ b > ac
, a > 1
or b < ac
, 0 < a < 1
(iv) loga
b > loga
c
⇒ b > c, if a > 1
or b < c, if 0 < a < 1
Properties :
(i) loga
1 = 0
(ii) loga
a = 1
(iii) a a blog = b
if k > 0, k = b b klog
(iv) loga
b1
+ loga
b2
+ ...... + loga
bn
= loga
(b1
b2
........bn
)
(v) loga
b
c
F
HG I
KJ = loga
b – loga
c
(vi) Base change formulae
loga
b =
log
log
c
c
b
a or loga
b =
1
logb a
(vii) logam bn
=
n
m
loga
b
(viii) loga
1
b
F
HG I
KJ = – loga
b = log1/a
b
(ix) log1/a
b
c
F
HG I
KJ = loga
c
b
F
HG I
KJ
(x) a b clog = c b alog
3. Greatest Integer function :
f(x) = [x], where [.]denotes greatest integer function equal
or less than x.
i.e., defined as [4.2] = 4, [–4.2] = –5
Period of [x] = 1
PAGE # 107 PAGE # 108

E D U C A T I O N S
E D U C A T I O N S
Properties :
(i) x – 1 < [x] ≤ x
(ii) [x + I] = [x] + I
[x + y] ≠ [x] + [y]
(iii) [x] + [–x] = 0, x ∈ I
= –1, x ∉ I
(iv) [x] = I, where I is an integer x ∈ [I, I + 1)
(v) [x] ≥ I, x ∈ [I, ∞ )
(vi) [x] ≤ I, x ∈ (– ∞ , I + 1]
(vii) [x] > I, [x] ≥ I + 1, x ∈ [I + 1, ∞ )
(viii) [x] < I, [x] ≤ I – 1, x ∈ (–∞ , I)
4. Fractional part function :
f(x) = {x} = difference between number & its integral part
= x – [x].
Properties :
(i) {x}, x ∈ [0, 1)
(ii) {x + I} = {x}
{x + y} ≠ {x} + {y}
(iii) {x} + {–x} = 0, x ∈ I
= 1, x ∉ I
(iv) [{x}] = 0, {{x}} = {x}, {[x]} = 0
5. Signum function :
f(x) = sgn (x) =
− ∈
=
∈
R
S
||
T
||
−
+
1
0 0
1
,
,
,
x R
x
x R
or f(x) =
| |x
x
, x ≠ 0
= 0, x = 0
6. Definition :
Let A and B be two given sets and if each element a ∈ A is
associated with a unique element b ∈ B under a rule f, then
this relation (mapping) is called a function.
Graphically - no vertical line should intersect the graph of
the function more than once.
Here set A is called domain and set of all f images of the
elements of A is called range.
i.e., Domain = All possible values of x for which f(x) exists.
Range = For all values of x, all possible values of f(x).
Table : Domain and Range of some standard functions -
Functions Domain Range
Polynomial function R R
Identity function x R R
Constant function K R (K)
Reciprocal function
1
x
R0
R0
x2
, |x| (modulus function) R R+
∪{x}
x3
, x|x| R R
Signum function
| |x
x
R {-1, 0, 1}
x +|x| R R+
∪{x}
x -|x| R R-
∪{x}
[x] (greatest integer function) R 1
x - {x} R [0, 1]
x [0, ∞) [0, ∞]
ax
(exponential function) R R+
log x (logarithmic function) R+
R
PAGE # 109 PAGE # 110

E D U C A T I O N S
E D U C A T I O N S
Trigonometric Domain Range
Functions
sin x R [-1, 1]
cos x R [-1, 1]
tan x R- ± ±
RST
UVW
π π
2
3
2
, ,... R
cot x R- {0,± π, ± 2π ,...} R
sec x R - ± ±
RST
UVW
π π
2
3
2
, ,... R - (-1,1)
cosec x R- {0, ± π , ± 2π} R - (-1,1)
Inverse Domain Range
Trigo Functions
sin -1
x (-1, 1]
−L
NM O
QPπ π
2 2
,
cos-1
x [-1,1] [0, π]
tan-1
x R
−F
HG I
KJπ π
2 2
,
cot-1
x R (0, π)
sec-1
x R -(-1,1) [0, π ]-
π
2
RST
UVW
cosec-1
x R - (-1,1) −
L
NM O
QPπ π
2 2
, - {0}
7. Kinds of functions :
(i) One-one (injection) function - f : A → B is one-one if
f(a) = f(b) ⇒ a = b
or a ≠ b
⇒ f(a) ≠ f(b), a, b ∈ A
Graphically-no horizontal line intersects with the graph
of the function more than once.
(ii) Onto function (surjection) - f : A → B is onto if
R (f) = B i.e. if to each y ∈ B ∃ x ∈ A s.t. f(x) = y
(iii) Many one function : f : A → B is a many one function
if there exist x, y ∈ A s.t. x ≠ y
but f(x) = f(y)
Graphically - atleast one horizontal line intersects with
the graph of the function more than once.
(iv) Into function : f is said to be into function if R(f) < B
(v) One-one-onto function (Bijective) - A function which
is both one-one and onto is called bijective function.
8. Inverse function : f–1
exists iff f is one-one & onto both
f–1
: B → A, f–1
(b) = a ⇒ f(a) = b
9. Transformation of curves :
(i) Replacing x by (x – a) entire graph will be shifted parallel
to x-axis with |a| units.
If a is +ve it moves towards right.
a is –ve it moves toward left.
Similarly if y is replace by (y – a), the graph will be
shifted parallel to y-axis,
upward if a is +ve
downward if a is –ve.
PAGE # 111 PAGE # 112

E D U C A T I O N S
E D U C A T I O N S
(ii) Replacing x by –x, take reflection of entire curve is y-
axis.
Similarly if y is replaced by –y then take reflection of
entire curve in x-axis.
(iii) Replacing x by |x|, remove the portion of the curve
corresponding to –ve x (on left hand side of y-axis)
and take reflection of right hand side on LHS.
(iv) Replace f(x) by |f(x)|, if on L.H.S. y is present and
mode is taken on R.H.S. then portion of the curve below
x-axis will be reflected above x-axis.
(v) Replace x by ax (a > 0), then divide all the value on x-
axis by a.
Similarly if y is replaced by ay (a > 0) then divide all the
values of y-axis by a.
10. Even and odd function : A function is said to be
(i) Even function if f(–x) = f(x) and
(ii) Odd function if f(–x) = –f(x).
11. Properties of even & odd function :
(a) The graph of an even function is always symmetric
about y-axis.
(b) The graph of an odd function is always symmetric
about origin.
(c) Product of two even or odd function is an even
function.
(d) Sum & difference of two even (odd) function is an
even (odd) function.
(e) Product of an even or odd function is an odd function.
(f) Sum of even and odd function is neither even nor
odd function.
(g) Zero function i.e. f(x) = 0 is the only function which
is even and odd both.
(h) If f(x) is odd (even) function then f'(x) is even (odd)
function provided f(x) is differentiable on R.
(i) A given function can be expressed as sum of even
& odd function.
i.e. f(x) =
1
2
[f(x) + f(–x)] +
1
2
[f(x) – f(–x)]
= even function + odd function.
12. Increasing function :A function f(x) is an increasing function
in the domain D if the value of the function does not decrease
by increasing the value of x.
13. Decreasing function :A function f(x) is a decreasing function
in the domain D if the value of function does not increase by
increasing the value of x.
14. Periodic function: Function f(x) will be periodic if a +ve real
number T exist such that
f(x + T) = f(x), ∀ x ∈ Domain.
There may be infinitely many such T which satisfy the above
equality. Such a least +ve no. T is called period of f(x).
(i) If a function f(x) has period T, then
Period of f(xn + a) = T/n and
Period of (x/n + a) = nT
(ii) If the period of f(x) is T1
& g(x) has T2
then the period
of f(x) ± g(x) will be L.C.M. of T1
& T2
provided it
satisfies definition of periodic function.
(iii) If period of f(x) & g(x) are same T, then the period of
af(x) + bg(x) will also be T.
PAGE # 113 PAGE # 114

E D U C A T I O N S
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510PAGE # 115 PAGE # 116
Function Period
sin x, cos x 2π
sec x, cosec x
tan x, cot x π
sin (x/3) 6π
tan 4x π/4
cos 2πx 1
|cos x| π
sin4
x + cos4
x π/2
2 cos
x −F
HG I
KJπ
3
6π
sin3 x + cos3
x 2π/3
sin3
x + cos4
x 2π
sin
sin
x
x5
2π
tan2
x – cot2
x π
x – [x] 1
[x] 1
NON PERIODIC FUNCTIONS :
x , x2
, x3
, 5
cos x2
x + sin x
x cos x
cos x
15. Composite function :
If f : X → Y and g : Y → Z are two function, then the
composite function of f and g, gof : X → Z will be defined as
gof(x) = g(f(x)), ∀ x ∈ X
In general gof ≠ fog
If both f and g are bijective function, then so is gof.

E D U C A T I O N S
E D U C A T I O N S
LIMIT
1. Limit of a function : lim
x a→ f(x) = l (finite quantity)
2. Existence of limit : lim
x a→ f(x) exists iff lim
x a→ − f(x) = lim
x a→ + f(x) =l
3. Indeterminate forms :
0
0
,
∞
∞
, ∞ – ∞ , ∞ × 0, ∞0
, 0∞
, 1∞
4. Theorems on limits :
(i) lim
x a→ (k f(x)) = k lim
x a→ f(x), k is a constant.
(ii) lim
x a→
(f(x) ± g(x)) = lim
x a→
f(x) ± Lim
x a→
g(x)
(iii) lim
x a→ f(x).g(x) = lim
x a→ f(x). Lim
x a→ g(x)
(iv) lim
x a→
f x
g x
( )
( )
=
lim ( )
lim ( )
x a
x a
f x
g x
→
→
, provided lim
x a→ g(x) ≠ 0
(v) lim
x a→ f(g(x)) = f lim ( )
x a
g x
→
FH IK, provided value of
g(x) function f(x) is continuous.
(vi) lim
x a→ [f(x) + k] = lim
x a→ f(x) + k
(vii) lim
x a→ log(f(x)) = log lim ( )
x a
f x
→
FH IK
(viii) lim
x a→ (f(x))g(x)
= lim ( )
lim ( )
x a
g x
f x
x a
→
L
NM O
QP
→
5. Limit of the greatest integer function :
Let c be any real number
Case I : If c is not an integer, then lim
x c→ [x] = [c]
Case II: If c is an integer, then lim
x c→ − [x] = c – 1, lim
x c→ + [x] = c
and lim
x c→
[x] = does not exist
6. Methods of evaluation of limits :
(i) Factorisation method : If lim
x a→
f x
g x
( )
( )
is of
0
0
form
then factorize num. & devo. separately and cancel the
common factor which is participating in making
0
0
form.
(ii) Rationalization method :If we have fractional powers
on the expression in num, deno or in both, we rationalize
the factor and simplify.
(iii) When x → ∞ :Divide num. & deno. by the highest power
of x present in the expression and then after removing
the indeterminate form, replace
1
x
,
1
2
x
,.. by 0.
(iv) lim
x a→
x a
x a
n n
−
−
= nan–1
(v) By using standard results (limits) :
(a) lim
x → 0
sinx
x
= 1 =
lim
x → 0
x
xsin
(b) lim
x → 0
tanx
x
= 1 = lim
x → 0
x
xtan
(c) lim
x → 0 sinx = 0
(d) lim
x → 0 cosx = lim
x → 0
1
cos x
= 1
PAGE # 117 PAGE # 118

E D U C A T I O N S
E D U C A T I O N S
(e) lim
x → 0
sinx
x
0
=
π
180
(f) lim
x → 0
sin−1
x
x
= 1 = lim
x → 0
x
xsin−1
(g) lim
x → 0
tan−1
x
x
= 1 = lim
x → 0
x
xtan−1
(h) lim
x → 0
a
x
x
− 1
= loge
a
(i) lim
x → 0
e
x
x
− 1
= 1
(j) lim
x → 0
log( )1 + x
x
= 1
(k) lim
x→0
log ( )a x
x
1 +
=
1
loga
(l) lim
x → 0
( )1 1+ −x
x
n
= n
(m) lim
x → ∞
sinx
x
= lim
x → ∞
cos x
x
= 0
(n) lim
x → ∞
sin
1
1
x
x
= 1
(o) lim
x→0
(1 + x)1/x
= e = lim
x→∞
1
1
+
F
HG I
KJx
x
(p) lim
x → 0 (1 + ax)1/x
= ea
= lim
x → ∞
1 +
F
HG I
KJa
x
x
(vi) By substitution :
(a) If x → a, then we can substitute
x = a + t ⇒ t = x – a
If x → a, t → 0.
(b) When x → – ∞ substitute x = – t ⇒ t → ∞
(c) When x → ∞ substitute t =
1
x
⇒ t → 0+
(vii) By using some expansion :
ex
= 1 + x +
x2
2!
+
x3
3!
+ .....
e–x
= 1 – x +
x2
2!
–
x3
3!
+ .....
log(1 + x) = x –
x2
2
+
x3
3
– ......
log(1 – x) = –x –
x2
2
–
x3
3
–.....
ex ln a
= ax
= 1 + xloge
a +
( log )
!
x ae
2
2
+
( log )
!
x ae
3
3
+ ......
sinx = x –
x3
3!
+
x5
5!
–.......
cosx = 1 –
x2
2!
+
x4
4!
–......
tanx = x +
x3
3
+
2
15
x5
+ .....
(1 + x)n
= 1 + nx +
n n( )
!
−1
2
x2
+ .....
7. Sandwich Theorem : In the neighbour hood of x = a
f(x) < g(x) < h(x)
lim
x a→
f(x) = lim
x a→
h(x) = l, then lim
x a→
g(x) = l.
⇒ l < lim
x a→
g(x) < l.
PAGE # 119 PAGE # 120

E D U C A T I O N S
E D U C A T I O N S
DIFFERENTIATION
1. SOME STANDARD DIFFERENTIATION :
Function Derivative Function Derivative
A cons. (k) 0 xn
nxn – 1
loga
x
1
x aelog
loge
x
1
x
ax
ax
loge
a ex
ex
sin x cos x cos x –sin x
tan x sec2
x cot x –cosec2
x
cosec x –cosec x cot x sec x sec x tan x
sin–1
x
1
1 2
− x
,–1<x<1 cos–1
x –
1
1 2
− x
,–1<x<1
sec–1
x
1
1 2
| |x x−
,|x|>1 cosec–1
x –
1
1 2
| |x x−
,–1|x|>|
tan–1
x
1
1 2
+ x
, x ∈ R cot–1
x –
1
1 2
+ x
, x ∈R
[x] 0, x ∉ I |x|
x
x| |
, x ≠ 0
NOTE :
d
dx
[x] does not exist at any integral Point.
2. FUNDAMENTAL RULES FOR DIFFERENTIATION :
(i)
d
dx
f(x) = 0 if and only if f(x) = constant
(ii)
d
dx
cf x( )c h = c
d
dx
f(x), where c is a constant.
(iii)
d
dx
f x g x( ) ( )±c h =
d
dx
f(x) ±
d
dx
g(x)
(iv)
d
dx
(uv) = u
dv
dx
+ v
du
dx
, where u & v are functions
of x. (Product rule)
or
d
dx
(uvw) = vw
du
dx
+ uw
dv
dx
+ uv
dw
dx
.
(v) If
d
dx
f(x) = φ(x), then
d
dx
f (ax + b)
= a φ(ax + b)
(vi)
d
dx
u
v
F
HG I
KJ =
v
du
dx
u
dv
dx
v
−
2
(quotient rule)
(vii) If y = f(u), u = g(x) [chain rule or differential co-
efficient of a function of a function]
then
dy
dx
=
dy
du
×
du
dx
llly If y = f(u), u = g(v), v = h(x), then
PAGE # 121 PAGE # 122

E D U C A T I O N S
E D U C A T I O N S
dy
dx
=
dy
du
×
du
dv
×
dv
dx
i.e if y = un
⇒
dy
dx
= nun–1
du
dx
OR
(viii) Differentiation of composite functions
Suppose a function is given in form of fog(x) or
f[g(x)], then differentiate applying chain rule
i.e.,
d
dx
f[g(x)] = f'g(x) . g'(x)
(ix)
d
dx
1
u
F
HG I
KJ =
−1
2
u
du
dx
, u ≠ 0
(x)
d
dx
|u| =
u
u| |
du
dx
, u ≠ 0
(xi) Logarithmic Differentiation : If a function is in the
form (f(x))g(x)
or
f x f x
g x g x
1 2
1 2
( ) ( )....
( ) ( ).... We first take log on
both sides and then differentiate.
(a) loge
(mn) = loge
m + loge
n
(b) loge
m
n
= logm – loge
n
(c) loge
(m)n
= nloge
m (d) logn
m logm
n = 1
(e) logan xm
=
m
n
loga
x (f) aloga x
= x
(g) loge
e = 1 (h) logn
m =
log
log
e
e
m
n
(xii) Differentiation of implicit function : If f (x, y) = 0,
differentiate w.r.t. x and collect the terms containing
dy
dx
at one side and find
dy
dx
.
[The relation f(x, y) = 0 in which y is not expressible
explicitly in terms of x are called implicit functions]
(xiii) Differentiation of parametric functions : If x = f(t)
and y = g(t), where t is a parameter, then
dy
dx
=
dy
dt
dx
dt
=
g t
f t
'( )
'( )
(xiv) Differentiation of a function w.r.t. another func-
tion : Let y = f(x) and z = g(x), then differentiation
of y w.r.t. z is
dy
dz
=
dy dx
dz dx
/
/
=
f x
g x
'( )
'( )
(xv) Differentiation of inverse Trigonometric functions
using Trigonometrical Transformation : To solve
the problems involving inverse trigonometric functions
first try for a suitable substitution to simplify it and
then differentiate. If no such substitution is found
then differentiate directly by using trigonometrical
formula frequently.
PAGE # 123 PAGE # 124

E D U C A T I O N S
E D U C A T I O N S
3. Important Trigonometrical Formula :
(i) sin2x = 2sinx. cosx =
2
1 2
tan
tan
x
x+
(ii) cos2x =
1
1
2
2
−
+
tan
tan
x
x
= 2 cos2
x – 1 = 1 – 2 sin2
x
(viii) tan2x =
2
1 2
tan
tan
x
x−
(iii) sin3x = 3sinx – 4sin3
x
(vi) cos3x = 4cos3
x – 3cosx
(ix) tan3x =
3
1 3
3
2
tan tan
tan
x x
x
−
−
(x) sin–1
x + cos–1
x = π /2
(xi) sec–1
x + cosec–1
x = π /2
(xii) tan–1
x + cot–1
x = π /2
(xiii) tan–1
x ± tan–1
y = tan–1
x y
xy
±F
HG I
KJ1 m
(xiv) sin–1
x ± sin–1
y = sin–1
x y y x1 12 2
− ± −F
H
I
K
(xv) cos–1
x ± cos–1
y = cos–1
xy x ym 1 12 2
− −FH IK
(xvi) sin–1
sin (x) = x, for –
π
2
≤ x ≤
π
2
cos–1
(cos x) = x, for 0 ≤ x ≤ π
tan–1
(tan x) = x, for –
π
2
< x <
π
2
(xvii) sin–1
(–x) = –sin–1
x, tan–1
(–x) = – tan–1
x,
cos–1
(–x) = π – cos–1
x
(xviii) sin–1
1
x
F
HG I
KJ = cosec–1
x, cos–1
1
x
F
HG I
KJ = sec–1
x,
tan–1
1
x
F
HG I
KJ = cot–1
x, cot–1
1
x
F
HG I
KJ = tan–1
x,
sec–1
1
x
F
HG I
KJ = cos–1
x, cosec–1
1
x
F
HG I
KJ = sin–1
x
(xix) sin–1
(cos θ ) = sin–1 sin
π
θ
2
−
F
HG I
KJF
HG
I
KJ =
π
2
– θ
cos–1
(sin θ ) = cos–1 cos
π
θ
2
−
F
HG I
KJF
HG I
KJ =
π
2
– θ
tan–1
(cot θ ) = tan–1 tan
π
θ
2
−
F
HG I
KJF
HG
I
KJ =
π
2
– θ
PAGE # 125 PAGE # 126

E D U C A T I O N S
E D U C A T I O N S
4. Some Useful Substitutions :
Part A
Expression Substitution Formula Result
3x – 4x3
x = sinθ 3sinθ – 4sin3
θ sin3θ
4x3
– 3x x = cosθ 4cos3
θ – 3cosθ cos3θ
3
1 3
3
2
x x
x
−
−
x = tanθ
3
1 3
3
2
tan tan
tan
θ θ
θ
−
−
tan3θ
2
1 2
x
x+
x = tanθ
2
1 2
tan
tan
θ
θ+
sin2θ
2
1 2
x
x−
x = tanθ
2
1 2
tan
tan
θ
θ−
tan2θ
1 – 2x2
x = sinθ 1 – 2sin2
θ cos2θ
2x2
– 1 x = cosθ 2cos2
θ – 1 cos2θ
1 – x2
x = sinθ 1 – sin2
θ cos2
θ
x = cosθ 1 – cos2
θ sin2
θ
x2
– 1 x = secθ sec2
θ – 1 tan2
θ
x = cosecθ cosec2
θ – 1 cot2
θ
1 + x2
x = tanθ 1 + tan2
θ sec2
θ
x = cotθ 1 + cot2
θ cosec2
θ
Part B
Expression Substitution
a2
+ x2
x = a tan θ or x = a cot θ
a x
a x
+
−
or
a x
a x
−
+
x = a tanθ
a2
– x2
x = a sinθ or x = a cos θ
a x
a x
+
−
or
a x
a x
−
+
x = a cosθ
x2
– a2
x = a sec θ or x=acosec θ
a x
a x
2 2
2 2
+
−
or
a x
a x
2 2
2 2
−
+
x2
= a2
cosθ
5. Successive differentiations or higher order derivatives :
(a) If y = f(x) then
dy
dx
= f'(x) is called the first deriva-
tive of y w.r.t. x
⇒
d y
dx
2
2 =
d
dx
dy
dx
F
HG I
KJ =
d
dx
f x'( )c h
is called the second derivative of y w.r.t. x
PAGE # 127 PAGE # 128

E D U C A T I O N S
E D U C A T I O N S
(f) If y = sin (ax + b), then yn
= an
sin ax b
n
+ +
F
HG I
KJπ
2
If y = cos (ax + b), then yn
= an
cos ax b
n
+ +
F
HG I
KJπ
2
6. nth
Derivatives of Some Functions :
(i)
d
dx
x
n
n
n
e j = n!
(ii)
d
dx
x
n
n
sinc h = sin x
n
+
F
HG I
KJπ
2
(iii)
d
dx
n
n (cos x) = cos x
n
+
F
HG I
KJπ
2
(iv)
d
dx
n
n (emx
) = mn
emx
(v)
d
dx
n
n (log x) = (– 1)n–1
(n–1)! x–n
NOTE : If u = g(x) is such that g'(x) = K (constant)
then
d
dx
n
n f g x( )c h = Kn
d
du
f u
n
n
u g x
( )
( )
L
N
MM
O
Q
PP =
PAGE # 129 PAGE # 130
llly
d y
dx
3
3 =
d
dx
2
2 f x'( )c h etc......
Thus, This process can be continued and we can
obtain derivatives of higher order
Note : To obtain higher order derivative of parametric
functions we use chain rule
i.e. if x = 2t, y = t2
⇒
dy
dx
= t
⇒
d y
dx
2
2 =
d
dx
dy
dx
F
HG I
KJ =
d
dx
(t) = 1.
dt
dx
=
1
t
(b) If y = (ax + b)m
m ∉ I, then
yn
= m(m–1) (m–2) ..... (m–n+1) (ax + b)m–n
.an
(c) If m ∈I, then
ym
= m! am
and ym+1
= 0
(d) If y =
1
ax b+
, then yn
=
( ) !
( )
−
+ +
1
1
n
n
n
ax b
an
(e) If y = log (ax + b), then yn
=
( ) ( )!
( )
− −
+
−
1 11n
n
n
ax b
an

E D U C A T I O N S
E D U C A T I O N S
7. Differentiation of Infinite Series : method is illustrated
with the help of example
if y = xxx− −∞
then function becomes y = xy
now taking log
on both sides
i.e logy = y log x, differentiating both sides w.r.t. x
we get
1
y
dy
dx
= y
1
x
+ logx
dy
dx
⇒
dy
dx
=
y
x
y
x
1
−
F
HG I
KJlog
=
y
x y x
2
1( log )−
8. L-hospital rule :
if as x → a f(x) & g(x) either both → 0 or both → ∞, then
lim
x a→
f x
g x
( )
( )
= lim
x a→
f x
g x
'( )
'( )
(a) it can be applied only on 0/0 or ∞/∞ form
(b) Numerator & denominator are differentiated separately
not
u
v
formulae.
(c) If R.H.S. exist or d'not exist because value → ∞, then
L.H rule can be applied.
But if value fluctuate on R.H.S. then L.H. rule can't
be applied.
If it is applied continuously then at each step 0/0 or
∞/∞ should be checked.
9. Differentiation of Determinant :
∆ =
R
R
R
1
2
3
= |C1
C2
C3
|
∆' =
R
R
R
'1
2
3
+
R
R
R
1
2
3
' +
R
R
R
1
2
3'
= |C'1
C2
C3
| + |C1
C'2
C3
| + |C1
C2
C'3
|

E D U C A T I O N S
E D U C A T I O N S
APPLICATION OF DERIVATIVES
TANGENT AND NORMAL :
1. Geometrically f'(a) represents the slope of the tangent to
the curve y = f(x) at the point (a, f(a))
2. If the tangent makes an angle ψ (say) with +ve x direction
then
f'(x) =
dy
dx x y
F
HG I
KJ( , )1 1
= tan ψ = slope of the tangent.
3. If the tangent is parallel to x-axis, ψ = 0
⇒
dy
dx x y
F
HG I
KJ( , )1 1
= 0.
4. If the tangent is perpendicular to x-axis, ψ =
π
2
⇒
dy
dx x y
F
HG I
KJ( , )1 1
→ ∞
5. If the tangent line makes equal angle with the axes, then
dy
dx x y
F
HG I
KJ( , )1 1
= ± 1.
6. Equation of the tangent to the curve y = f(x) at a point
(x1
, y1
) is
y – y1
=
dy
dx x y
F
HG I
KJ( , )1 1
(x – x1
)
7. Length of intercepts made on axes by the tangent :
x – intercept = x1
–
y
dy
dx x y
1
1 1
F
HG I
KJ
R
S
||
T
||
U
V
||
W
||( , )
y – intercept = y1
– x1
dy
dx x y
F
HG I
KJ( , )
1 1
8. Length of perpendicular from origin to the tangent :
=
y x
dy
dx
dy
dx
x y
x y
1 1
2
1 1
1 1
1
−
F
HG I
KJ
+
F
HG I
KJ
( , )
( , )
9. Slope of the normal = –
1
Slope of the genttan
= –
dx
dy x y
F
HG I
KJ( , )1 1
10. If normal makes an angle of φ with +ve direction of x-axis,
then
dy
dx
= – cot φ .
PAGE # 133 PAGE # 134

E D U C A T I O N S
E D U C A T I O N S
11. If the normal is parallel to x-axis ⇒
dy
dx x y
F
HG I
KJ( , )1 1
= 0.
12. If the normal is perpendicular to x-axis ⇒
dy
dx x y
F
HG I
KJ( , )1 1
= 0.
13. If normal is equally inclined from both the axes or cuts
equal intercept then
dy
dx
F
HG I
KJ = ± 1.
14. The equation of the normal to the curve y = f(x) at a point
(x1
, y1
) is
y – y1
= –
1
1 1
dy
dx x y
F
HG I
KJ( , )
(x – x1
)
15. Length of intercept made on axes by the normal :
x – intercept = x1
+ y1
dy
dx x y
F
HG I
KJ( , )1 1
y – intercept = y1
+ x1
dx
dy x y
F
HG I
KJ( , )1 1
16. Length of perpendicular from origin to normal :
=
x y
dy
dx
dy
dx
x y
1 1
2
1 1
1
+
F
HG I
KJ
+
F
HG I
KJ
( , )
17. Angle of intersection of the two curves :
tanθ = ±
dy
dx
dy
dx
dy
dx
dy
dx
F
HG I
KJ −
F
HG I
KJ
−
F
HG I
KJ F
HG I
KJ
1 2
1 2
1
where
dy
dx
F
HG I
KJ1
is the slope of first
curve &
dy
dx
F
HG I
KJ2
of second. If both curves intersect orthogo-
nally then
dy
dx
F
HG I
KJ1
dy
dx
F
HG I
KJ2
= –1
18. Length of tangent, normal, subtangent & subnormal :
Length of tangent =
y
dy
dx
dy
dx
1
2
+
F
HG I
KJ
Length of normal = y 1
2
+
F
HG I
KJdy
dx
Length of sub-tangent =
y
dy dx/
Length of sub-normal = y
dy
dx
PAGE # 135 PAGE # 136

E D U C A T I O N S
E D U C A T I O N S
MONOTONICITY, MAXIMA & MINIMA :
1. A function is said to be monotonic function in a domain if it is
either monotonic increasing or monotonic decreasing in that
domain
2. At a point function f(x) is monotonic increasing if f'(a) > 0
At a point function f(x) is monotonic decreasing if f'(a) < 0
3. In an interval [a, b], a function f(x) is
Monotonic increasing if f'(x) ≥ 0
Monotonic decreasing if f'(x) ≤ 0
constant if f'(x) = 0 ∀ x ∈ (a, b)
Strictly increasing if f'(x) > 0
Strictly decreasing if f'(x) < 0
4. Maximum & Minimum Points :
Maxima : A function f(x) is said to be maximum at x =
a, if there exists a very small +ve number h, such that
f(x) < f(a), ∀ x ∈ (a – h, a + h), x ≠ a.
Minima : A function f(x) is said to be minimum at x = b,
if there exists a very small +ve number h, such that
f(x) > f(b), ∀ x ∈ (b – h, b + h), x ≠ b.
Remark :
(a) The maximum & minimum points are also known as
extreme points.
(b) A function may have more than one maximum &
minimum points.
5. Conditions for Maxima & Minima of a function :
(i) Necessary condition : A point x = a is an extreme
point of a function f(x) if f'(a) = 0, provided f'(a)
exists.
(ii) Sufficient condition :
(a) The value of the function f(x) at x = a is
maximum if f'(a) = 0 and f"(a) < 0.
(b) The value of the function f(x) at x = a is
minimum if f'(a) = 0 and f"(a) > 0.
6. Working rule for finding local maxima & Local Minima :
(i) Find the differential coefficient of f(x) w.r.to x, i.e.
f'(x) and equate it to zero.
(ii) Solve the equation f'(x) = 0 and let its real roots
(critical points) be a, b, c ......
(iii) Now differentiate f'(x) w.r.to x and substitute the
critical points in it and get the sign of f"(x) for each
critical point.
(iv) If f"(a) < 0, then the value of f(x) is maximum at
x = 0 and if f"(a) > 0, then the value of f(x) is
minimum at x = a. Similarly by getting the sign of f"(x)
for other critical points (b, c, ......) we can find the
points of maxima and minima.
7. Absolute (Greatest and Least) values of a function in
a given interval :
(i) A minimum value of a function f(x) in an interval [a,
b] is not necessarily its greatest value in that interval.
Similarly a minimum value may not be the least value
of the function.
(ii) If a function f(x) is defined in an interval [a, b], then
greatest or least values of this function occurs either
at x = a or x = b or at those values of x for which
f'(x) = 0.
Thus greatest value of f(x) in interval [a, b]
= max [f(a), f(b), f(c), f(d)]
Least value of f(x) in interval [a, b]
= min. [f(a), f(b), f(c), f(d)]
Where x = c, x = d are those points for which f'(x) = 0.
PAGE # 137 PAGE # 138

E D U C A T I O N S
E D U C A T I O N S
8. Some Geometrical Results :
In Usual Notations Results
Area of equilateral
3
4
(side)2
.
and its perimeter 3 (side)
Area of square (side)2
Perimeter 4(side)
Area of rectangle l × b
Perimeter 2(l × b)
Area of trapezium
1
2
(sum of parallel sides)
× (distance between them)
Area of circle πr2
Perimeter 2πr
Volume of sphere
4
3
πr3
Surface area of sphere 4πr2
Volume of cone
1
3
πr2
h
Surface area of cone πrl
Volume of cylinder πr2
h
Curved surface area 2πrh
Total surface area 2πr(h + r)
Volume of cuboid l × b × h
Surface area of cuboid 2(lb + bh + hl)
Area of four walls 2(l × b) h
Volume of cube l3
Surface area of cube 6l2
Area of four walls of cube 4l2
ROLLE'S THEOREM & LAGRANGES THEOREM:
1. Rolle's Theorem : If f(x) is such that
(a) It is continuous on [a, b]
(b) It is differentiable on (a, b) and
(c) f(a) = f(b), then there exists at least one point
c ∈ (a, b) such that f'(c) = 0.
2. Mean value theorem [Lagrange's theorem] :
(i) If f(x) is such that
(a) It is continuous on [a, b]
(b) It is differentiable on (a, b), then
there exists at least one c ∈ (a, b) such that
f b f a
b a
( ) ( )−
−
= f'(c)
(ii) If for c in lagrange's theorem (a < c < b) we can say
that c = a + θ h where 0 < θ < 1 and h = b – a
the theorem can be written as
f(a + h) = f(a) + h f'(a + θh), 0 < θ < 1, h = b – a
PAGE # 139 PAGE # 140

E D U C A T I O N S
E D U C A T I O N S
INDEFINITE INTEGRATION
1. (i) If
d
dx
F(x) = f(x), then f x dxc hz = F(x) + c
Here m rdxz is the notation of integration, f(x) is the
integrand, c is any real no. (integrating constant)
(ii)
d
dx
f x dxc hz = f(x)
(iii) f x dx'c hz = f(x) + c, c ∈ R
(iv) k f x dxc hz = k zf(x) dx
(v) ( )f x g x dxc h c h±z = f x dx g x dxc h c h± zz
2. FUNDAMENTAL FORMULAE :
Function Integration
x dxn
z x
n
n+
+
1
1
+ c, n ≠ –1
ax b
n
+zc h dx
1
a
.
ax b
n
n
+
+
+
c h 1
1
+ c, n ≠ –1
1
x
z dx log|x| + c
1
ax b+z dx
1
a
(log|ax + b|) + c
ex
z dx ex
+ c
ax
z dx
a
a
x
elog
+ c
sinz x dx –cos x + c
cos xz dx sin x + c
sec2
xz dx tan x + c
cosec x2
z dx – cot x + c
sec tanx x dxz sec x + c
cos cotecx xz dx –cosec x + c
tanxz dx –log|cos x| + c = log|sec x| + c
cot xz dx log|sin x| + c = –log|cosec x| + c
sec xz dx log|sec x + tan x|+c = log tan
π
4 2
+
F
HG I
KJx
+c
cosec xz dx log|cosec x – cot x|+c = log tan
x
2
+c
dx
x1 2
−
z sin–1
x + c = –cos–1
x + c
dx
a x2 2
−
z sin–1
x
a
+ c = –cos–1
x
a
+ c
dx
x1 2
+
z tan–1
x + c = –cot–1
x + c
dx
a x2 2
+
z 1
a
tan–1
x
a
+ c =
−1
a
cot–1
x
a + c
dx
x x| | 2
1−
z sec–1
x + c = –cosec–1
x + c
dx
x x a| | 2 2
−
z 1
a
sec–1
x
a
+ c =
−1
a
cosec–1
x
a + c
PAGE # 141 PAGE # 142

E D U C A T I O N S
E D U C A T I O N S
3. INTEGRATION BY SUBSTITUTION :
By suitable substitution, the variable x in f x dxc hz is changed
into another variable t so that the integrand f(x) is changed
into F(t) which is some standard integral. Some following
suggestions will prove useful.
Function Substitution Integration
f ax b dx+z c h ax + b = t
1
a
F(ax + b) + c
f x f x dxc h c h'z f(x) = t
f x
c
c hd i2
2
+
f x x dxφ φc hd i c hz φ(x) = t f t dtc hz
f x
f x
dx
'c h
c hz f(x) = t log|f(x)| + c
f x f x dx
n
c hd i c h'z f(x) = t
f x
n
n
( )c h +
+
1
1
+ c, n ≠ – 1
f x
f x
'c h
c hz dx f(x) = t 2[f(x)]1/2
+ c
SOME RECOMMENDED SUBSTITUTION :
Function Substitution
a x2 2
− ,
1
2 2
a x−
, a2
– x2
x = a sin θ or a cos θ
x a2 2
+ ,
1
2 2
x a+
, x2
+ a2
x = a tanθ or x = a sinhθ
x a2 2
− ,
1
2 2
x a−
, x2
– a2
x = a sec θ
or x = a cosh θ
x
a x+
,
a x
x
+
,
x a x+c h.
1
x a x+c h x = a tan2
θ
x
a x−
,
a x
x
−
,
x a x−c h,
1
x a x−c h x = a sin2
θ
x
x a−
,
x a
x
−
,
x x a−c h,
1
x a x−c h
x = a sec2
θ
a x
a x
−
+
,
a x
a x
+
−
x = a cos 2θ
x
x
−
−
α
β
, x x− −α βc h c h,(β > α) x = α cos2
θ + β sin2
θ
PAGE # 143 PAGE # 144

E D U C A T I O N S
E D U C A T I O N S
IMPORTANT RESULTS USING STANDARD SUBSTITUTIONS :
1
2 2
x a−
z 1
2a
log
x a
x a
−
+
+ c
=
−1
a
coth–1
x
a + c when x > a
1
2 2
a x−
z dx
1
2a
log
a x
a x
+
−
+ c
=
1
a
tanh–1
x
a + c, when x < a
dx
x a2 2
−
z log{|x + x a2 2
− |} + c
= cosh–1
x
a
F
HG I
KJ + c
dx
x a2 2
+
z log{|x + x a2 2
+ |} + c
= sinh–1
x
a
F
HG I
KJ + c
a x2 2
−z dx
1
2
x a x2 2
− +
1
2
a2
sin–1
x
a
F
HG I
KJ + c
x a2 2
−z dx
1
2
x x a2 2
− –
1
2
a2
log {|x + x a2 2
− |} + c
x a2 2
+z dx
1
2
x x a2 2
+ +
1
2
a2
log {|x + x a2 2
+ |} + c
INTEGRATION OF FUNCTIONS USING ABOVE STANDARD
RESULTS :
Function Method
1
2
ax bx c
dx
+ +z or Express : ax2
+ bx + c =
1
2
ax bx c
dx
+ +
z or a x
b
a
ac b
a
+
F
HG I
KJ +
−L
N
MM
O
Q
PP2
4
4
2 2
2
( )ax bx c2
+ +z dx then use appropriate formula
px q
ax bx c
dx
+
+ +z 2 or Express : px + q
px q
ax bx c
dx
+
+ +
z 2 or = λ
d
dx
(ax2
+ bx + c) + µ
( ) ( )px q ax bx c+ + +z 2
dx evaluate λ & µ by equat
ing coefficient of x and
constant, the integral reduces to
known form
P x
ax bx c
dx
( )
2
+ +
z , Apply division rule and express it
where P(x) is a in form Q(x) +
R x
ax bx c
c h
2
+ +
polynomial of degree The integral reduces to known
2 or more form
PAGE # 145 PAGE # 146

E D U C A T I O N S
E D U C A T I O N S
1
2 2
a x b x c
dx
sin cos+ +z Divide numerator & denominator
by cos2
x,
or
1
2
a x b x
dx
sin cos+
zc h then put tanx = t & solve.
dx
a x b x c
dx
sin cos+ +z Replace sin x =
2 2
1 22
tan /
tan /
x
x+
,
cos x =
1 2
1 2
2
2
−
+
tan /
tan /
x
x
then put tan x/2 = t and
replace 1 + tan2
x/2 = sec2
x/2
a x b x
c x d x
dx
sin cos
sin cos
+
+z Express : num. = λ(deno.) +
µ
d
dx
(deno.) Evaluate λ & µ. Thus
integral reduces to known form.
a x b x c
p x q x r
dx
sin cos
sin cos
+ +
+ +z Express : Num. = λ(deno.) +
µ
d
dx
(deno.) + ν Evaluate λ, µ, ν.
Thus integral reduces to known
form.
x a
x kx a
dx
2 2
4 2 4
±
+ +
z Divide numerator & denominator
by x2
and put x
a
x
±
F
HG
I
KJ
2
= t, the
integral becomes one of standard
forms.
x
x kx a
dx
2
4 2 4
+ +
z Divide numerator & denominator
by 2 and then add & sub. a2
.
Thus the form reduces as above.
dx
x kx a4 2 2
+ +
z Divide num & deno. by 2a2
and
then add & sub x2
. Thus the form
reduces to the known form.
4. INTEGRATION BY PARTS :
when integrand involves more than one type of functions
the formula of integration by parts is used to integrate the
product of the functions i.e.
(i) u dx.υz = u. υ dxz –
du
dx
dxυzz FH IK
L
NM O
QPdx
or 1 2st fun nd fun dx. . .c h c hz
= (1st fun) 2nd fun dx.z –
d
dx
st fun nd fun dx dx1 2. .
F
HG I
KJL
NM O
QPzz e j
(ii) Rule to choose the first function : first fun. should
be choosen in the following order of preference (ILATE).
[The fun. on the left is normally chosen as first
function]
I – Inverse trigonometric function
L – Logarithmic function
A – Algebraic function
T – Trigonometric function
E – Exponential function
PAGE # 147 PAGE # 148

E D U C A T I O N S
E D U C A T I O N S
(iii) (a) e f x f x dxx
c h c h+z ' = ex
f(x) + c
(b) e mf x f x dxmx
c h c h+z ' = emx
f(x) + c
(c) e f x
f x
m
dxmx
c h c h+
L
N
MM
O
Q
PPz '
=
e f x
m
mx
c h + c.
(iv) xf x f x dx'c h c h+z = x f(x) + c.
NOTE : Breaking (iii) & (iv) integral into two integrals.
Integrate one integral by parts and keeping other integral
as it is by doing so we get the result (integral).
(v) e bx dxax
sinz and e bx c dxax
sin +z c h
=
e
a b
ax
2 2
+
(a sin bx – b cos bx) + k and
e
a b
ax
2 2
+
[a sin (bx + c) – b cos(bx + c)] + k1
(vi) e bx dxax
cosz and e bx c dxax
cos +z c h
=
e
a b
a bx b bx
ax
2 2
+
+cos sinc h + k
and
e
a b
a bx c b bx c
ax
2 2
+
+ + +cos sinc h c h + k1
.
5. INTEGRATION OF RATIONAL ALGEBRAIC FUNCTIONS
USING PARTIAL FRACTION :
Every Rational fun. may be represented in the form
P x
Q x
c h
c h,
where P(x), Q(x) are polynomials.
If degree of numerator is less than that of denominator,
the rational fun. is said to be proper other wise it is
improper. If deg (num.) ≥ deg(deno.) apply division rule
i.e.
f x
g x
c h
c h = q(x) +
r x
g x
c h
c h , for integrating
r x
g x
c h
c h , resolve the
fraction into partial factors. The following table illustrate
the method.
Types of proper Types of partial
rational functions fractions
px q
x a x b
+
− −c h c h, a ≠ b
A
x a−
+
B
x b−
px qx r
x a x b x c
2
+ +
− − −c hc hc h,
A
x a−
+
B
x b−
+
C
x c−
a, b, c are distinct
px qx r
x a x b
2
2
+ +
− −c h c h, a ≠ b
A
x a−
+
B
x a−c h2
+
C
x b−
px qx r
x a x bx c
2
2
+ +
− + +c h e j, where
A
x a−
+
Bx C
x bx c
+
+ +2
x2
+ bx + c can
not be factorised
px qx rx s
x ax b x cx d
3 2
2 2
+ + +
+ + + +e j e j,
Ax B
x ax b
+
+ +2 +
Cx D
x cx d
+
+ +2
where x2
+ ax + b,
x2
+ cx + d can not
be factorised
PAGE # 149 PAGE # 150

E D U C A T I O N S
E D U C A T I O N S
6. INTEGRATION OF IRRATIONAL ALGEBRAIC FUNCTIONS :
(i) If integrand is a function of x & (ax + b)1/n
then put
(ax + b) = tn
(ii) If integrand is a function of x, (ax + b)1/n
and
(ax + b)1/m
then put (ax + b) = tp
where p = (L.C.M. of m & n).
(iii) To evaluate
dx
linear linear
z put linear = t
(iv) To evaluate
dx
quad linear.
z put linear = t2
(v) To evaluate
dx
linear quadratic.
z put linear = 1/t
or
dx
linear quadraticc h2
.
z
or
x dx
linear quadraticc h2
.
z
(vi) To evaluate
dx
pure quad pure quad.
z put
pure quad = t
(vii) To evaluate
dx
pure quad pure quad.
z put x =
1
t
and
then is the resulting integral, put pure quad = u
(viii) To evaluate
dx
quad quad.
z or
linear
quad quad.
z dx
and if the quadratic not under the square root can
be resolved into real linear factors, then resolve
1
quadratic
or
linear
quadratic
F
HG I
KJ into partial fractions and
split the integral into two, each of which is of the
form :
dx
linear quad.
z
7. INTEGRATION USING TRIGONOMETRICAL IDENTITIES :
(A) To evaluate trigonometric functions transform the
function into standard integrals using trigonometric
identities as
(i) sin2
mx =
1 2
2
− cos mx
(ii) cos2
mx =
1 2
2
+ cos mx
(iii) sin mx = 2sin
mx
2
cos
mx
2
(iv) sin3
mx =
3 3
4
sin sinmx mx−
(v) cos3
mx =
3 3
4
cos cosmx mx+
(vi) tan2
mx = sec2
mx – 1
(vii) cot2
mx = cosec2
mx – 1
(viii) 2 cos A cos B = cos (A + B) + cos (A – B)
(ix) 2 sin A cos B = sin (A + B) + sin (A – B)
(x) 2 sin A sin B = cos (A – B) – cos (A + B)
PAGE # 151 PAGE # 152

E D U C A T I O N S
E D U C A T I O N S
(B) sin cosm m
x xdxz .
(i) if m is odd put cos x = t
(ii) if m is even put sin x = t
(iii) if m & n both odd put sin x or cos x as t
(iv) if m & n both even use the formula of sin2
x & cos2
x
(v) if m & n rational no. &
m n+ − 2
2
is –ve integer
put tan x = t
8. INTEGRATION BY SUCCESSIVE REDUCTION (REDUCTION
FORMULA) :
x e dxn ax
z , n ∈ N
1
a
xn
eax
–
n
a
In–1
where In–1
= x e dxn ax−
z 1
x x dxn
sinz –xn
cos x + nxn–1
sin x – n(n – 1) In–2
sinn
x dxz –
sin cosn
x
n
−1
+
n
n
− 1
In–2
cosn
x dxz cos sinn
x x
n
−1
+
n
n
− 1
In–2
tann
x dxz tanx
n
n
c h −
−
1
1
– In–2
cotn
x dxz –
cot x
n
n
c h −
−
1
1
– In–2
secn
x dxz sec tann
x x
n
−
−
2
1
+
n
n
−
−
2
1
In–2
cosec x dxn
z –
cos cotec x x
n
n−
−
2
1
+
n
n
−
−
2
1
In–2
sin cosm n
x x dxz cos sin ,
n m
m nx x n I
m n
− +
−+ −
+
1 1
21c h
c h
–sinm–1
x cosn+1
x + (m – 1) Im–2,n
NOTE : These formulae are specifically useful when m & n
are both even nos.
PAGE # 153 PAGE # 154

E D U C A T I O N S
E D U C A T I O N S
DEFINITE INTEGRATION
1. Definite Integration :
If f x dxc hz = F(x) + c, then
f x dx
a
b
c hz = F x c
a
b
c h+ = F(b) – F(a) is called definite integral
of f(x) w.r.t. x from x = a to x = b Here a is called lower
limit and b is called upper limit.
Remarks :
* To evaluate definite integral of f(x). First obtain the
indefinite integral of f(x) and then apply the upper
and lower limit.
* For integration by parts in definite integral we use
following rule.
uv dx
a
b
z = u v dx
a
b
.z{ } –
du
dx
v dx dx
a
b
. .zzF
HG I
KJ
* When we use method of substitution. We note that
while changing the independent variable in a definite
integral, the limits of integration must also we changed
accordingly.
PROPERTIES OF DEFINITE INTEGRAL :
I. f x dx
a
b
c hz = f t dt
a
b
c hz
PAGE # 155 PAGE # 156
II. f x dx
a
b
c hz = – f x dx
b
a
c hz
III. f x dx
a
b
c hz = f x dx
a
c
c hz + f x dx
c
b
c hz where a < c < b
This property is mainly used for modulus function,
greatest integer function & breakable function
IV. f x dx
b
a
c hz = f a b x dx
b
a
+ −z c h or f x dx
a
c h
0
z = f a x dx
a
−z c h
0
V. f x dx
a
a
c h
−
z = f x f x
a
c h c h+ −z0
dx
=
2
0
0
f x dx if f x is an even function
if f x is an odd function
a
c h c h
c h
zR
S|
T|
,
,
VI. f x dx
a
c h
0
2
z =
2 2
0 2
0
f x dx if f a x f x
if f a x f x
a
c h c h c h
c h c h
z − =
− = −
R
S
||
T
||
,
,
VII. If f(x) is a periodic function with period T, Then
f x dx
nT
c h
0
z = n f x dx
T
c h
0
z
and further if a ∈ R+
, then

E D U C A T I O N S
E D U C A T I O N S
f x dx
nT
a nT
c h
+
z = f x dx
a
c h
0
z , f x dx
mT
nT
c hz = (n – m) f x dx
T
c h
0
z ,
f x dx
a nT
b nT
c h
+
+
z = f x dx
a
b
c hz
VIII. If m and M are the smallest and greatest values of
a function f(x) on an interval [a, b], then
m(b – a) < f x dx
a
b
c hz < M(b – a)
IX. f x dx
a
b
c hz < | |f x dx
a
b
c hz
X. If f(x) < g(x) on [a, b], then f x dx g x dx
a
b
a
b
c h c hz z≤
2. Differentiation Under Integral Sign :
Leibnitz's Rule :
(i) If f(x) is continuous and u(x), v(x) are differentiable
functions in the interval [a, b], then,
d
dx
f t dt
u x
v x
( )
( )
( )
z =
f{v(x)}
d
dx
{v(x)} – f{u(x)}
d
dx
{u(x)}.
(ii) If the function φ(x) and ψ(x) are defined on [a, b]
and differentiable at a point x ∈ (a, b), and f(x, t)
is continuous, then,
d
dx
f x t dt
x
x
( , )
( )
( )
φ
ψ
zL
N
MM
O
Q
PP = f x t dt
x
x
( , )
( )
( )
φ
ψ
z +
d x
dx
ψ ( )RST
UVWf(x,
ψ(x)) –
d x
dx
φ( )RST
UVWf(x, φ(x)).
3. Reduction Formulae :
(i) cos
/2
n
a
x dx
π
z = sin
/2
n
x dx
0
π
z
=
n
n
n
n
if n is odd
n
n
n
n
if n is even
− −
−
− −
−
R
S|
T||
1 3
2
2
3
1
1 3
2
1
2 2
. ..... . ,
. ...... . ,
π
(ii) For integration sin cos
/2
m n
x x dx
0
π
z follow the following
steps
(a) If m is odd put cos x = t
(b) If n is odd put sin x = t
(c) If m and n are even use sin2
x = 1– cos2
x
or cos2
x = 1 – sin2
x and then use
sin
/2
n
x dx
0
π
z or cos
/2
n
x dx
0
π
z
PAGE # 157 PAGE # 158

E D U C A T I O N S
E D U C A T I O N S
(iii) e bxax−
∞
z cos
0
dx =
a
a b2 2
+
(iv) e bx dxax−
∞
z sin
0
=
b
a b2 2
+
(v) e x dxax n−
∞
z0
=
n
an
!
+ 1
(vi) sin cos
/2
n m
x x dx
0
π
z
=
m
m n
m
m n n n
if m is odd and n may be even or odd
m
m n
m
m n n
n
n
n
n
if m is enen and n is odd
m
m n
m
m n n
n
n
n
n
if m is even and n is even
−
+
−
+ − + +
−
+
−
+ − +
− −
−
−
+
−
+ − +
− −
−
L
N
MMMMMMM
O
Q
PPPPPPP
1 3
2
2
3
1
1
1 3
2
1
2
1 3
2
2
3
1 3
2
1
2
1 3
2
1
2 2
. .... . ;
. .... . . .... ;
. .... . . .... . ;
π
These formulae can be expressed as a single formula :
sin cos
/2
m n
x x dx
0
π
z
=
[( ) ( )....] [( ) ( ) .....]
( ) ( ) ....
m m n n
m n m n
− − − −
− + −
1 3 1 3
2
to be multiplied by
π
2
when m and n are both even
integers.
4. Summation of series by Definite integral or limit as a
sum :
(i) f x dx
a
b
c hz = lim
h→0
h[f(a) + f(a + h) + f(a + 2h) +.....
+f(a + (n – 1)h]
where nh = b – a.
(ii) lim
n→∞
1
1
n
f
r
nr
n
F
HG I
KJ
=
∑ = f x dxc h
0
1
z
[i.e. exp. the given series in the form
1
n
f
r
n
F
HG I
KJ∑
replace
r
n
by x and
1
n
by dx and the limit of the
sum is f x dxc h
0
1
z ]
5. Key Results :
* logsin
/2
x dx
0
π
z = logcos
/2
x dx
0
π
z =
−π
2
2log
*
f x
f x f x
sin
sin cos
/2
c h
c h c h+z0
π
dx =
f x
f x f x
cos
sin cos
/2
c h
c h c h+z0
π
dx
PAGE # 159 PAGE # 160

E D U C A T I O N S
E D U C A T I O N S
=
f x
f x f x
dx
tan
tan cot
/2
c h
c h c h+z0
π
=
f x
f x f ecx
dx
sec
sec cos
/2
c h
c h c h+z0
π
=
f ec x
f ec x f x
cos
cos sec
c h
c h c h+z dx=
f x
f x f x
dx
cot
tan cot
/2
c h
c h c h+z0
π
= π/4.
* sin sin
/2
mx nx dx
0
π
z = cos . cos
/2
mx nx dx
0
π
z
=
0
2
if m n are different ve egers
if m n
, int+
=
R
S|
T|
π
* a x dx
a
2 2
0
−z =
π
4
2
a
*
1
2 2
0 a x
dx
a
−
z =
π
2
*
x
a x
dx
a
2 2
0 −
z = a
*
x
a x
dx
a 2
2 2
0 −
z =
πa a2 2
6
3
8
+
*
x dx
a x
dx
a
2 2
3
0 +
ze j
/2
=
1
2 2
a
.
* x a x dx
a
2 2 2
0
−z =
πa4
16
* x
a x
a x
dx
a
2
2 2
0
−
+z = a3
π
4
2
3
−
F
HG I
KJ if a > 0
* 2 2
0
2
ax x dx
a
−z =
πa2
2
* If n ∈ N, then a x dx
n
a
2 2
0
−ze j =
2 4 6 2
3 5 7 2 1
2 1
. . ......
. . .....
n
n
a nc h
c h+
+
* If a < b then
(i)
dx
x a b xa
b
− −
=z π
(ii)
x a
a x
dx
a
−
+z0
=
π b a−c h
2
(iii) x a b x
a
− −z c hc h
0
dx =
π
2
2
b a−c h
PAGE # 161 PAGE # 162

E D U C A T I O N S
E D U C A T I O N S
(iv)
dx
x x a b xa
b
− −
z c hc h =
π
ab
2
ab > 0
* If a > 0 then
(i)
a x
a x
dx
a
+
−z0
=
a
2
2π +c h
(ii)
a x
a x
a
−
+z0
dx =
a
2
2π −c h
(iii)
a x
a x
a
+
−
z0
dx =
10
3
a a
(iv)
a x
a x
a
+
−z0
dx =
π
2
1+
F
HG I
KJa
* If a > 0, n ∈ N, then
(i) x e ax−
∞
z0
dx =
π
2a a
a > 0
(ii) e dxr x−
∞
z 2 2
0
=
π
2r
(r > 0)
(iii)
e e
x
dx
ax bx− −∞
−
z0
= loge
(b/a) (a, b > 0)
* If f(x) is continuous on [a, b] then there exists a
point c ∈ (a, b) s.t f x dx
a
b
c hz = f(c) [b – a]. The no.
f(c) =
1
b a
f x dx
a
b
− z c h is called the mean value of the
fun. f(x) on the interval [a, b]. The above result is
called the first mean value theorem for integrals.
* x x dx
k
−zd i
0
2
= k, where k ∈ I,
Q x – [x] is a periodic function with period 1.
* If f(x) is a periodic fun. with period T, then
f x
a
a T
c h
+
z dx is independent of a.
* log tan
/
1
0
4
+z x dxc h
π
=
π
8
2log

E D U C A T I O N S
E D U C A T I O N S
DIFFERENTIAL EQUATIONS
1. Order of a differential equation : The order of a differential
equation is the order of the highest derivative occurring in
it.
2. Degree of a differential equation : The degree of a
differential equation is the degree of the highest order
derivative occurring in it when the derivatives are made
free from the radical sign.
Eg. (i)
d y
dx
2
2 +
dy
dx
+ 5y = 0
(ii) y = x
dy
dx
+ 1
2
+
F
HG I
KJdy
dx
(iii)
d y
dx
3
3
2
F
HG
I
KJ + 1
2
+
F
HG I
KJdy
dx
+ 5y = 0
order of (i) 2 (ii) 1 & (iii) 3,
degree of (i) 1 (ii) 2 & (iii) 2
3. SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST
ORDER AND FIRST DEGREE :
(A) Differential equation of the form
dy
dx
= f(x) or
dy
dx
= f(y)
Integrate both sides i.e. dyz = f x( )z dx
or
dy
f y( )z = dxz to get its solution.
(B) Variable Separable Form : Differential equation of
the form
dy
dx
= f(x) g(y)
This can be integrated as
dy
g y( )z = z f(x) dx + c
(C) Homogeneous Equations : It is a differential equation
of the form
dy
dx
=
f x y
g x y
( , )
( , )
, where f(x, y) and g(x,
y) are homogeneous functions of x and y of the same
degree. A function f(x, y) is said to be homogeneous
of degree n if it can be written as xn
f
y
x
F
HG I
KJ or yn
f
x
y
F
HG I
KJ.
Such an equation can be solved by putting y = vx or
x = vy. After substituting y = vx or x = vy. The given
equation will have variables separable in v and x.
(D) Equations Reducible to Homogeneous form and
variable separable form
* Form
dy
dx
=
ax by c
Ax By C
+ +
+ +
........... (1)
where
a
A
≠
b
B
This is non Homogeneous
Put x = X + h and y = Y + k in (1)
∴
dy
dx
=
dY
dX
Put ah + bk + c = 0, Ah + Bk + C = 0,
find h, k
Then
dY
dX
=
aX bY
AX BY
+
+
. This is homogeneous.
Solve it and then put X = x – h, Y = y – k we shall
get the solution.
PAGE # 165 PAGE # 166

E D U C A T I O N S
E D U C A T I O N S
* Form
dy
dx
=
ax by c
Ax By C
+ +
+ +
..... (1),
where
a
A
=
b
B
= k say
∴
dy
dx
=
k By c
Ax By C
(Ax )+ +
+ +
Put Ax + By = z ⇒ A + B
dy
dx
=
dz
dx
⇒
dz
dx
= A + B
kz c
z c
+
+
This is variable separable form and can be solved.
* Form
dy
dx
= f(ax + by + c)
Put ax + by = z ⇒ a + b
dy
dx
=
dz
dx
∴
dz
dx
= a + b f(z)
This is variable separable form and can be solved.
(E) Linear equation :
* In y :
dy
dx
+ Py = Q, where P, Q are function of x
alone or constant.
its solution ye
P dxz = Q e
P dxzz dx + c
where
e
P dxz is called the integrating factor (I.F.) of
the equation.
* In x :
dx
dy
+ Rx = S, where R, S are functions of y
alone or constant.
its solution xe
R dyz = S e
R dy
.
.zz dy + c
where
e
R dy.z is called the integrating factor (I.F.) of
the equation.
(F) Equation reducible to linear form :
* Differential equation of the form
dy
dx
+ Py = Qyn
where P and Q are functions of x or constant is called
Bernoulli's equation. On dividing through out by yn
, we
get
y–n
dy
dx
+ py–n + 1
= Q
Put y–n + 1
= z
⇒ The given equation will be linear in z and can be
solved in the usual manner.
Note : In general solution of differential equation we can
take integrating constant c as tan–1
c, ec
, log c etc.
according to our convenience.
PAGE # 167 PAGE # 168

E D U C A T I O N S
E D U C A T I O N S
VECTORS
1. Types of vectors :
(a) Zero or null vector : A vector whose magnitude is
zero is called zero or null vector.
(b) Unit vector : $a =
r
a
a| |
=
Vector a
Magnitude of a
(c) Equal vector : Two vectors a and b are said to be
equal if |a| = |b| and they have the same direction.
2. Triangle law of addition : AB + BC = AC
c = a + b
c =
a
+
b
–
–
–
a
b
–
–
A
C
B
3. Parallelogram law of addition : OA + OB = OC
a + b = c
a
–
b
–
C
B
AD
where OC is a diagonal of the parallelogram OABC
4. Vectors in terms of position vectors of end points -
AB = OB – OA = Position vector of B – position vector of A
i.e. any vector = p.v. of terminal pt – p.v. of initial pt.
5. Multiplication of a vector by a scalar :
If
r
a is a vector and m is a scalar, then m
r
a is a vector and
magnitude of m
r
a = m|a|
and if
r
a = a1
$i + a2
$j + a3
$k
then m
r
a = (ma1
) $i + (ma2
)
$j + (ma3
) $k
6. Distance between two points :
Distance between points A(x1
, y1
, z1
) and B(x2
, y2
, z2
)
= Magnitude of AB
→
= ( ) ( ) ( )x x y y z z2 1
2
2 1
2
2 1
2
− + − + −
7. Position vector of a dividing point :
(i) If A( a) & B(
r
b ) be two distinct pts, the p.v. c of the
point C dividing [AB] in ratio m1
: m2
is given by
r
c =
m b m a
m m
1 2
1 2
r r
+
+
(ii) p.v. of the mid point of [AB] is
1
2
[p.v. of A + p.v. of B]
(iii) If point C divides AB in the ratio m1
: m2
externally,
then p.v. of C is c =
m b m a
m m
1 2
1 2
−
−
PAGE # 169 PAGE # 170

E D U C A T I O N S
E D U C A T I O N S
10. Coplanar and non coplanar vector :
(i) If a, b , c be three non coplanar non zero vector
then x a + y b + z c = 0
⇒ x = 0, y = 0, z = 0
(ii) If a, b , c be three coplanar vectors, then a vector
c can be expressed uniquely as linear combination of
remaining two vectors i.e. c = λ a + µ b
(iii) Any vector r can be expressed uniquely as inner com-
bination of three non coplanar & non zero vectors a ,
b and c i.e. r = x a + y b + z c
11. Products of vectors :
(I) Scalar or dot product of two vectors :
(i) a . b = |a| |b| cosθ
(ii) Projection of a in the direction of b =
a b
b
.
| |
& Projection of b in the direction of a =
a b
a
.
| |
(iii) Component of r on a =
r a
a
.
| |2
F
HG I
KJ a
Component of r ⊥ to a = r –
r a
a
.
| |2
F
HG I
KJ a
(iv) $i . $i =
$j .
$j = $k . $k = 1
(v) $i .
$j =
$j . $k = $k . $i = 0
(iv) p.v. of centriod of triangle formed by the points A( a ),
B(
r
b ) and C (
r
c ) is
a b c+ +
3
(v) p.v. of the incentre of the triangle formed by the points
A(
r
α ), B(
r
β ) and C(
r
γ ) is
a b c
a b c
α β γ+ +
+ +
where a = |BC|, b = |CA|, c = |AB|
8. Some results :
(i) If D, E, F are the mid points of sides BC, CA & AB
respectively, then AD + BE + CF = 0
(ii) If G is the centriod of ∆ABC, then GA + GB + GC = 0
(iii) If O is the circumcentre of a ∆ABC, then
OA + OB + OC = 3 OG = OHwhere G is centriod and
H is orthocentre of ∆ABC.
(iv) If H is orthocentre of ∆ABC, then
HA + HB + HC = 3HG = OH
9. Collinearity of three points :
(i) Three points A, B and C are collinear if AB = λ AC for
some non zero scalar λ.
(ii) The necessary and sufficient condition for three points
with p.v. a , b , c to be collinear is that there exist
three scalars l, m, n all non zero such that
l a + m b + n c = 0, l + m + n = 0
PAGE # 171 PAGE # 172

E D U C A T I O N S
E D U C A T I O N S
(vi) If a and b are like vectors, then a. b = | a|| b | and
If a and b are unlike vectors, then a. b = –| a|| b |
(vii) a , b are ⊥ ⇔ a. b = 0
(viii) ( a . b ). b is not defined
(ix) ( a ± b )2
= a2
± 2 a . b + b2
(x) | a + b | = | a| + | b | ⇒ a || b
(xi) | a + b |2
= |a|2
+ |b|2
⇒ a ⊥ b
(xii) | a + b | = | a – b | ⇒ a ⊥ b
(xiii) work done by the force :
work done = F . d , where F is force vector and d
is displacement vector.
(II) Vector or cross product of two vectors :
(i) a × b = |a| |b| sinθ $n
(ii) if a, b are parallel ⇔ a × b = 0
(iii) a × b = –( b × a)
(iv) $n = ±
a b
a b
×
×| |
(v) let a = a1
$i + a2
$j + a3
$k & b = b1
$i + b2
$j + b3
$k , then
a × b =
$ $ $i j k
a a a
b b b
1 2 3
1 2 3
(vi) a × a = 0
(vii) a × ( b × c ) = ( a × b ) × c
(viii) a × ( b + c ) = ( a × b ) + ( a × c )
(ix) $i × $i =
$j ×
$j = $k × $k = 0, $i ×
$j = $k ,
$j × $k = $i , $k × $i =
$j
(x) Area of triangle :
(a)
1
2
AB AC×
(b) If a , b , c are p.v. of vertices of ∆ABC,
then =
1
2
|( a × b ) + ( b × c ) + ( c × a)|
(xi) Area of parallelogram :
(a) If a & b are two adjacent sides of a parallelo-
gram, then area = | a × b |
(b) If a and b are two diagonals of a parallelogram,
then area =
1
2
| a × b |
(xii) Moment of Force :
Moment of the force F acting at a point A about O is
Moment of force = OA ×F = r × F
(xiii) Lagrange's identity : | a × b |2
=
a a a b
a b b b
. .
. .
(III) Scalar triple product :
(i) If
r
a = a1
$i + a2
$j + a3
$k ,
r
b = b1
$i + b2
$j + b3
$k and
r
c = c1
$i + c2
$j + c3
$k then
PAGE # 173 PAGE # 174

E D U C A T I O N S
E D U C A T I O N S
(
r
a ×
r
b ).
r
c = [
r
a
r
b
r
c ] =
a a a
b b b
c c c
1 2 3
1 2 3
1 2 3
and [
r
a
r
b
r
c ] = volume of the parallelopiped whose
coterminus edges are formed by
r
a ,
r
b ,
r
c
(ii) [
r
a
r
b
r
c ] = [
r
b
r
c
r
a ] = [
r
c
r
a
r
b ],
but [
r
a
r
b
r
c ] = – [
r
b
r
a
r
c ] = – [
r
a
r
c
r
b ] etc.
(iii) [
r
a
r
b
r
c ] = 0 if any two of the three vectors
r
a ,
r
b ,
r
c are collinear or equal.
(iv) (
r
a ×
r
b ).
r
c =
r
a .(
r
b ×
r
c ) etc.
(v) [ $i $j $k ] = 1
(vi) If λ is a scalar, then [λ
r
a
r
b
r
c ] = λ[
r
a
r
b
r
c ]
(vii) [
r
a +
r
d
r
b
r
c ] = [
r
a
r
b
r
c ] + [
r
d
r
b
r
c ]
(viii)
r
a ,
r
b ,
r
c are coplanar ⇔ [
r
a
r
b
r
c ] = 0
(ix) Volume of tetrahedron ABCD is
1
6
| AB
→
× AC
→
. AD
→
|
(x) Four points with p.v.
r
a ,
r
b ,
r
c ,
r
d will be coplanar if
[
r
d
r
b
r
c ] + [
r
d
r
c
r
a ] + [
r
d
r
a
r
b ] = [
r
a
r
b
r
c ]
(xi) Four points A, B, C, D are coplanar if
[ AB
→
AC
→
AD
→
] = 0
(xii) (a) [
r
a +
r
b
r
b +
r
c
r
c +
r
a ] = 2[
r
a
r
b
r
c ]
(b) [
r
a –
r
b
r
b –
r
c
r
c –
r
a ] = 0
(c) [
r
a ×
r
b
r
b ×
r
c
r
c ×
r
a ] = [
r
a
r
b
r
c ]2
(d) If
r
a ,
r
b ,
r
c are coplanar, then so are
r
a ×
r
b ,
r
b ×
r
c ,
r
c ×
r
a and
r
a +
r
b ,
r
b +
r
c ,
r
c +
r
a and
r
a –
r
b ,
r
b –
r
c ,
r
c –
r
a are also coplanar.
(IV) Vector triple Product :
If
r
a ,
r
b ,
r
c be any three vectors, then (
r
a ×
r
b ) ×
r
c
and
r
a × (
r
b ×
r
c ) are known as vector triple product
and is defined as
(
r
a ×
r
b ) ×
r
c = (
r
a .
r
c )
r
b – (
r
b .
r
c )
r
a
and
r
a × (
r
b ×
r
c ) = (
r
a .
r
c )
r
b – (
r
a .
r
b )
r
c
Clearly in general
r
a × (
r
b ×
r
c ) ≠ (
r
a ×
r
b ) ×
r
c but
(
r
a ×
r
b ) ×
r
c =
r
a × (
r
b ×
r
c ) if and only if
r
a ,
r
b
&
r
c are collinear
12. Application of Vector in Geometry :
(i) Direction cosines of
r
r ai bj ck= + +$ $ $ are
a
r
b
r
c
r| |
,
| |
,
| |
r r r .
(ii) Incentre formula : The position vector of the incentre
of ∆ ABC is
aa bb cc
a b c
r r r
+ +
+ +
.
(iii) Orthocentre formula : The position vector of the
orthocentre of ∆ ABC is
r r r
a A b B c C
A B C
tan tan tan
tan tan tan
+ +
+ +
(iv) Vector equation of a straight line passing through a
fixed point with position vector
r
a and parallel to a
given vector
r
b is
r r r
r a b= + λ .
PAGE # 175 PAGE # 176

E D U C A T I O N S
E D U C A T I O N S
(v) The vector equation of a line passing through two
points with position vectors
r
a and
r
b is
r r r r
r a b a= + −λe j.
(vi) Shortest distance between two parallel lines : Let l1
and l2
be two lines whose equations are l1
:
r r r
r a b= +1 1λ and l2
:
r r r
r a b= +2 2µ respectively.
Then, shortest distance
PQ =
b b a a
b b
b b a a
b b
1 2 2 1
1 2
1 2 2 1
1 2
× −
×
=
−
×
c h c h c h.
| | | |
shortest distance between two parallel lines : The
shortest distance between the parallel lines
r r r
r a b= +1 λ
and
r r r
r a b= +2 µ is given by d =
| |
| |
r r r
r
a a b
b
2 1− ×c h .
If the lines
r r r
r a b= +1 1λ and
r r r
r a b= +2 2µ intersect,
then the shortest distance between them is zero.
Therefore, [b b a a1 2 2 1−c h] = 0
⇒ [
r r r r
a a b b2 1 1 2−c h ] = 0 ⇒
r r r r
a a b b2 1 1 2− ×c h e j. = 0.
(vii) Vector equation of a plane normal to unit vector
r
n and
at a distance d from the origin is
r
r n. $ = d.
If
r
n is not a unit vector, then to reduce the equation
r r
r n. = d to normal form we divide both sides by |
r
n |
to obtain
r
r
rr
n
n
.
| |
=
d
n| |
r or
r
rr n
d
n
. $
| |
= .
(viii) The equation of the plane passing through a point
having position vector
r
a and parallel to
r
b and
r
c is
r r r r
r a b c= + +λ µ or [
r r r
r bc ] = [
r r r
abc ], where λ and µ are
scalars.
(ix) Vector equation of a plane passing through a point
r r r
abc is
r r r r
r s t a sbt c= − − + +1c h
or
r r r r r r r
r b c c a a b. × + × + ×e j = [
r r r
abc ].
(x) The equation of any plane through the intersection
of planes
r r
r n. 1 = d1
and
r r
r n. 2 = d2
is
r r
r n n. 1 2+ λc h = d1
+ λd2
, where λ is an arbitrary
constant.
(xi) The perpendicular distance of a point having position
vector
r
a from the plane
r r
r n. = d is given by
p =
| . |
| |
r r
r
a n d
n
−
.
(xii) An angle θ between the planes
r r
r n d1 1 1. = and
r r
r n d2 2 2. = is given by cos θ = ±
n n
n n
1 2
1 2
.
| || | .
(xiii) The equation of the planes bisecting the angles
between the planes
r r
r n1 1. = d1
and
r r
r n2 2. = d2
are
| . |
| |
| . |
| |
r r
r
r r
r
r n d
n
r n d
n
1 1
1
2 2
2
−
=
−
(xiv) The plane
r r
r n. = d touches the sphere |
r r
r a− | = R,
if
| . |
| |
r r
r
a n d
n
−
= R.
(xv) If the position vectors of the extremities of a diam-
eter of a sphere are
r
a and
r
b , then its equation is
(
r r
r a− ).( r r
r b− ) = 0 or |
r
r |2
–
r r r r r
r a b a b. .− +e j = 0.
PAGE # 177 PAGE # 178

E D U C A T I O N S
E D U C A T I O N S
THREE DIMENSIONAL GEOMETRY
1. Points in Space :
(i) Origin is (0, 0, 0)
(ii) Equation of x-axis is y = 0, z = 0
(iii) Equation of y-axis is z = 0, x = 0
(iv) Equation of z-axis is x = 0, y = 0
(v) Equation of YOZ plane is x = 0
(vi) Equation of ZOX plane is y = 0
(vii) Equation of XOY plane is z = 0
2. Distance formula :
(i) Distance between two points A(x1
, y1
, z1
) and B(x2
,
y2
, z2
) is given by
AB = ( ) ( ) ( )x x y y z z2 1
2
2 1
2
2 1
2
− + − + −
(ii) Distance between origin (0, 0, 0) & point (x, y, z)
= x y z1
2
1
2
1
2
+ +
(iii) Distance of a point p(x, y, z) from coordinate axes
OX, OY, OZ is given by
y z2 2
+ , z x2 2
+ and x y2 2
+
3. Section formula :
The coordinates of a point which divides the join of (x1
,
y1
, z1
) and (x2
, y2
, z2
) in the ratio m : n
* Internally are
mx nx
m n
my ny
m n
mz nz
m n
2 1 2 1 2 1+
+
+
+
+
+
F
HG I
KJ, ,
* Externally are
mx nx
m n
my ny
m n
mz nz
m n
2 1 2 1 2 1−
−
−
−
−
−
F
HG I
KJ, ,
* Coordinates of the centroid of a triangle are
x x x y y y z z z1 2 3 1 2 3 1 2 3
3 3 3
+ + + + + +F
HG I
KJ, ,
* Coordinates of centroid of a tetrahedron
x x x x y y y y z z z z1 2 3 4 1 2 3 4 1 2 3 4
4 4 4
+ + + + + + + + +F
HG I
KJ, ,
Note :
* Area of triangle is given by ∆ = ∆ ∆ ∆x y z
2 2 2
+ +
Where ∆x
=
1
2
y z
y z
y z
1 1
2 2
3 3
1
1
1
and so.
* Condition of collinearity
x x
x x
1 2
2 3
−
− =
y y
y y
1 2
2 3
−
− =
z z
z z
1 2
2 3
−
−
* Volume of tetrahedron =
1
6
x y z
x y z
x y z
x y z
1 1 1
2 2 2
3 3 3
4 4 4
1
1
1
1
4. Direction cosines and direction ratios of a line :
* If α , β , γ are the angles which a directed line
segment makes with the +ve direction of the coordinate
axes, then l = cos α , m = cos β , n = cos γ are
called direction cosines of the line and cos2
α + cos2
β + cos2 γ = 1 i.e. l 2
+ m2
+ n2
= 1, where 0 ≤
α , β , γ ≤ π
PAGE # 179 PAGE # 180

E D U C A T I O N S
E D U C A T I O N S
* If l , m, n are direction cosines of a line and a, b,
c are proportional to l , m, n respectively, then a, b,
c are called direction ratios of the line and
l
a
=
m
b
=
n
c
= ±
l2 2 2
2 2 2
+ +
+ +
m n
a b c
= ±
1
2 2 2
a b c+ +
.
* Direction cosines of x-axis are 1, 0, 0, similarly direction
cosines of y-axis and z-axis are respectively 0, 1, 0
and 0, 0, 1.
* If l , m, n are d.c.s of a line OP and (x, y, z) are
coordinates of P then x = l r, y = mr and z = nr where
r = OP.
* Direction cosines of PQ = r, where P is (x1
, y1
, z1
) and
Q(x2
, y2
, z2
) are
x x
r
2 1−
,
y y
r
2 1−
,
z z
r
2 1−
* If a, b, c are direction no. of a line, then
a2
+ b2
+ c2
need not to be equal to 1.
Note : Direction cosines of a line are unique but the
direction ratios of line are not unique.
If P(x1
, y1
, z1
) & Q(x2
, y2
, z2
) be two points and L be a
line with d.c.'s l , m, n, then projection of [PQ] on
L = l (x2
– x1
) + m(y2
– y1
) + n(z2
– z1
)
5. Straight line in space :
* Equation of a straight line passing through a fixed
point and having d.r.'s a, b, c is
x x
a
− 1
=
y y
b
− 1
=
z z
c
− 1
(is the symmetrical
form)
* Equation of a line passing through two points is
x x
x x
−
−
1
2 1
=
y y
y y
−
−
1
2 1
=
z z
z z
−
−
1
2 1
* The angle θ between the lines whose d.c.'s are l 1
,
m1
, n1
and l 2
, m2
, n2
is given by
cos θ = l 1 l 2
+ m1
m2
+ n1
n2
.
The lines are || if
l
l
1
2
=
m
m
1
2
=
n
n
1
2
and
The lines are ⊥ if l 1 l 2
+ m1
m2
+ n1
n2
= 0
* The angle θ between the lines whose d.r.s are a1
, b1
,
c1
and a2
, b2
, c2
is given by
cos θ = ±
a a b b c c
a b c a b c
1 2 1 2 1 2
1
2
1
2
1
2
2
2
2
2
2
2
+ +
+ + + +
The lines are || if
a
a
1
2
=
b
b
1
2
=
c
c
1
2
and
The lines are ⊥ if a1
a2
+ b1
b2
+ c1
c2
= 0
* Length of the projection of PQ upon AB with d.c.,
l , m, n
= (x2
– x1
) l + (y2
– y1
)m + (z2
– z1
)n, where
p(x1
, y1
, z1
) and Q(x2
, y2
, z2
).
* Two straight lines in space (not in same plane) which
are neither parallel nor intersecting are called skew
lines.
* Shortest distance between two skew lines,
x x− 1
1l =
y y
m
− 1
1
=
z z
n
− 1
1
and
x x− 2
2l =
y y
m
− 2
2
=
z z
n
− 2
2
is given
PAGE # 181 PAGE # 182

E D U C A T I O N S
E D U C A T I O N S
s.d. = ±
x x y y z z
m n
m n
m n m n n n m m
2 1 2 1 2 1
1 1 1
2 2 2
1 2 2 1
2
1 2 1 2
2
1 2 1 2
2
− − −
− + − + −
l
l
l l l l( ) ( ) ( )
* Two straight lines are coplanar if they are intersecting
or parallel
condition
x x y y z z
m n
m n
2 1 2 1 2 1
1 1 1
2 2 2
− − −
l
l
= 0
6. Plane : A plane is a surface such that if two points are
taken in it, straight line joining them lies wholly in the
surface.
* Ax + By + Cz + D = 0 represents a plane whose
normal has d.c.s proportional to A, B, C.
* Equation of plane through origin is given by Ax + By
+ Cz = 0.
* Equation of plane passing through a point (x1
, y1
, z1
)
is A(x – x1
) + B(y – y1
) + C(z – z1
) = 0, where A,
B, C are d.r.'s of a normal to the plane.
* Equation of plane through the intersection of two
planes
P ≡ a1
x + b1
y + c1
z + d1
= 0 and
Q ≡ a2
x + b2
y + c2
z + d2
= 0 is P + λ Q = 0.
* Equation of plane which cuts off intercepts a, b, c
respectively on the axes x, y and z is
x
a
+
y
b
+
z
c
= 1.
* Normal form of the equation of plane is l x + my +
nz = p, where l , m, n are the d.c.'s of the normal
to the plane and p is the length of perpendicular from
the origin.
* ax + by + cz + k = 0 represents a plane || to the
plane ax + by + cz + d = 0 and ⊥ to the line
x
a
=
y
b
=
z
c
.
* Equation of plane through three non collinear points
is
x y z
x y z
x y z
x y z
1
1
1
1
1 1 1
2 2 2
3 3 3
= 0
or
x x y y z z
x x y y z z
x x y y z z
− − −
− − −
− − −
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
= 0
* The angle between the two planes is given by
cos θ = ±
a a b b c c
a b c a b c
1 2 1 2 1 2
1
2
1
2
1
2
2
2
2
2
2
2
+ +
+ + + +
where θ is the angle between the normals.
plane are ⊥ if a1
a2
+ b1
b2
+ c1
c2
= 0
plane are || if
a
a
1
2
=
b
b
1
2
=
c
c
1
2
= 0.
PAGE # 183 PAGE # 184

E D U C A T I O N S
E D U C A T I O N S
7. Line and Plane :
If ax + by + cz + d = 0 represents a plane and
x x− 1
l
=
y y
m
− 1
=
z z
n
− 1
represents a straight line, then
* The line is ⊥ to the plane if
a
l
=
b
m
=
c
n
* The line is || to the plane if a l + bm + cn = 0.
* The line lies in the plane if
a l + bm + cn = 0 and ax1
+ by1
+ cz1
+ d = 0
* The angle θ between the line and the plane is given
by
sin θ =
a bm cn
a b c m n
l
l
+ +
+ + + +2 2 2 2 2 2
* General equation of the plane containing the line
x x− 1
l
=
y y
m
− 1
=
z z
n
− 1
is
A(x – x1
) + B(y – y1
) + C(z – z1
) = 0. where
A l + Bm + Cn = 0.
* Length of the perpendicular from a point (x1
, y1
, z1
)
to the line
x − α
l
=
y
m
− β
=
z
n
− γ
is given by
p2
= (x1
– α )2
+ (y1
– β )2
+ (z1
– γ )2
– [ l (x1
– α )
+ m(y1
– β ) + n(z1
– γ )]2
PAGE # 185 PAGE # 186
* If AP be the ⊥ from A to the given plane, then it
is || to the normal, so that its equation is
x
a
− α
=
y
b
− β
=
z
c
− γ
= r (say)
Any point P on it is (ar + α , br + β , cr + γ )
* Length of the ⊥ from P(x1
, y1
, z1
) to a plane
ax + by + cz + d = 0 is given by
p =
ax by cz d
a b c
1 1 1
2 2 2
+ + +
+ +
* Distance between two parallel planes
(ax + by + cz + d1
= 0, ax + by + cz + d2
= 0) is
given by
d d
a b c
2 1
2 2 2
−
+ +
* Two points A(x1
, y1
, z1
) and B(x2
, y2
, z2
) lie on the
same or different sides of the plane
ax + by + cz + d = 0, according as the expression
ax1
+ by1
+ cz1
+ d and ax2
+ by2
+ cz2
+ d are of
same or different sign.
* Bisector of the angles between the planes
a1
x + b1
y + c1
z + d1
= 0
and a2
x + b2
y + c2
z + d2
= 0 are
a x b y c z d
a b c
1 1 1 1
1
2
1
2
1
2
+ + +
+ +
= ±
a x b y c z d
a b c
2 2 2 2
2
2
2
2
2
2
+ + +
+ +
if a1
a2
+ b1
b2
+ c1
c2
is –ve then origin lies in the acute
angle between the planes provided d1
and d2
are of
same sign.

Maths formulae booklet (iit jee)

More Related Content

What's hot (16)

Similar to Maths formulae booklet (iit jee) (10)

Recently uploaded (20)

Maths formulae booklet (iit jee)