Maths05

IIT-JEE2005-M-1
FIIIITJEE Solutions to IITJEE––2005 Mains Paper
Mathematics
Time: 2 hours
Note: Question number 1 to 8 carries 2 marks each, 9 to 16 carries 4 marks each and 17 to 18 carries
6 marks each.
Q1. A person goes to office either by car, scooter, bus or train probability of which being 1, 3 , 2
7 7 7
and 1
 
  =
 
 
  =
 
 
  =
 
 
  =
 
 
  ×
P L .P C 1 7
  =   = =  
  × + × + × + ×
1 − 2x +
5x
3x 2x 1
 π π −   
1 − 2x +
5x
3x 2x 1
 −   
 
+
−
+
−
 π π   π π − −  ∪      
FIITJEE Ltd. ICES House, Sarvapriya Vihar (Near Hauz Khas Bus Term.), New Delhi - 16, Ph : 2686 5182, 26965626, 2685 4102, 26515949 Fax : 26513942
7
respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is 2 , 1, 4
9 9 9
and 1
9
respectively. Given that he reached office in time, then what is the probability that he travelled
by a car.
Sol. Let C, S, B, T be the events of the person going by car, scooter, bus or train respectively.
Given that P(C) = 1
7
, P(S) = 3
7
, P(B) = 2
7
, P(T) = 1
7
Let L be the event of the person reaching the office in time.
⇒ P L 7
C 9
, P L 8
S 9
, P L 5
B 9
, P L 8
T 9
⇒
( )
( )
C C 7 9 1 P
L P L 1 7 3 8 2 5 8 1 7
7 9 7 9 7 9 9 7
.
Q2. Find the range of values of t for which 2 sin t =
2
2
− −
, t ∈ ,
2 2
.
Sol. Let y = 2 sin t
so, y =
2
2
− −
⇒ (3y − 5)x2 − 2x(y − 1) − (y + 1) = 0
since x ∈ R − 1, 1
3
, so D ≥ 0
⇒ y2 − y − 1 ≥ 0
or y ≥ 1 5
2
and y ≤ 1 5
2
or sin t ≥ 1 5
4
and sin t ≤ 1 5
4
Hence range of t is , 3 ,
2 10 10 2
.
Q3. Circles with radii 3, 4 and 5 touch each other externally if P is the point of intersection of tangents to
these circles at their points of contact. Find the distance of P from the points of contact.

IIT-JEE2005-M-2
Sol. Let A, B, C be the centre of the three circles.
Clearly the point P is the in–centre of the ΔABC, and
hence
Δ s (s − a)(s − b)(s − c) − − −
r =
= =
(s a)(s b)(s c)
s s s
Now 2s = 7 + 8 + 9 = 24 ⇒ s = 12.
Hence r = 5. 4.3 5
12
= .
A
3
4
C
B
3
•P
4
5
5
Q4. Find the equation of the plane containing the line 2x – y + z – 3 = 0, 3x + y + z = 5 and at a distance of
1
6
from the point (2, 1, – 1).
Sol. Let the equation of plane be (3λ + 2)x + (λ − 1)y + (λ + 1)z − 5λ – 3 = 0
⇒
6 λ + 4 + λ − 1 − λ − 1 − 5 λ −
3 =
1
(3 λ + 2) 2 + ( λ − 1) 2 + ( λ +
1) 2
6
⇒ 6(λ – 1)2 = 11λ2 + 12λ + 6 ⇒ λ = 0, − 24
5
.
⇒ The planes are 2x – y + z – 3 = 0 and 62x + 29y + 19z − 105 = 0.
Q5. If |f(x1) – f(x2)| < (x1 – x2)2, for all x1, x2 ∈R. Find the equation of tangent to the curve y = f(x) at the
point (1, 2).
Sol. |f (x1) – f (x2)| < (x1 – x2)2
⇒
f(x ) −
f(x )
1 2
lim < lim | x −
x |
→ x x →
x x 1 2 1 2 1 −
2
x 1 x 2
⇒ |f′ (x)| < δ ⇒ f′ (x) = 0.
Hence f (x) is a constant function and P (1, 2) lies on the curve.
⇒ f (x) = 2 is the curve.
Hence the equation of tangent is y – 2 = 0.
Q6. If total number of runs scored in n matches is n 1
 + 
  4


(2n+1 – n – 2) where n > 1, and the runs scored
in the kth match are given by k. 2n+1–k, where 1 ≤ k ≤ n. Find n.
Sol. Let Sn =
n
n 1 k
= Σ
k 1
k.2 + −
=
n
= Σ
n 1 k
2 + k.2−
k 1
2 .2 1 1 n
= n 1
 − −   n n1

2 2
+
+
(sum of the A.G.P.)
= 2[2n+1 – 2 – n]
n +
⇒ 1 2
4
= ⇒ n = 7.
Q7. The area of the triangle formed by the intersection of a line parallel to x-axis and passing through
P (h, k) with the lines y = x and x + y = 2 is 4h2. Find the locus of the point P.
Sol. Area of triangle = 1
2
. AB. AC = 4h2
and AB = 2 |k – 1| = AC
⇒ 4h2 = 1
2
. 2. (k – 1)2
⇒ k – 1 = ± 2h.
⇒ locus is y = 2x + 1, y = – 2x + 1.
y = x
A(1,1)
B(k,k) C(2−k,k) y =k
P(h,k)
X
x+y=2
O

IIT-JEE2005-M-3
π  e  1    2 sin  cos x  + 3 cos  1  cos x

     ∫ sin x dx.
Q8. Evaluate |cos x|
0
2 2
π  e  1    2 sin  cos x  + 3 cos  1  cos x

     ∫ sin x dx
Sol. I = |cos x|
0
2 2
= 6
/ 2
e sin x cos 1 cos x dx

  0, if f(2a − x) = − f(x)

 2a
  =      − =     
∫ cos x
 
 
( )   ( )
0
2
π
∵
∫ a
∫ 0
2 f x dx, if f(2a x) f(x)
0
f x dx
Let cos x = t
1
I = 6
e cos t dt
∫
t
 
 
  0
2
         +   − 
     
= 24 ecos 1 e sin 1 1
5 2 2 2
.
Q9. Incident ray is along the unit vector ˆv and the reflected ray is
along the unit vector wˆ . The normal is along unit vector aˆ
outwards. Express wˆ in terms of aˆ and vˆ .
ˆv
â
wˆ
Sol. vˆ is unit vector along the incident ray and wˆ
is the unit vector along the reflected ray.
Hence â is a unit vector along the external
bisector of vˆ and wˆ . Hence
wˆ − vˆ = λaˆ
⇒ 1 + 1 – wˆ ⋅ vˆ = λ2
or 2 – 2 cos 2θ = λ2
or λ = 2 sin θ
where 2θ is the angle between vˆ and wˆ .
â
wˆ
2θ
ˆv
(90-θ)
mirror
Hence wˆ − vˆ = 2sinθaˆ = 2cos(900 − θ)aˆ = −(2aˆ ⋅ vˆ )aˆ
⇒ wˆ = vˆ − 2(aˆ ⋅ vˆ )aˆ .
Q10. Tangents are drawn from any point on the hyperbola
x2 y2 1
9 4
− = to the circle x2 + y2 = 9. Find the
locus of mid–point of the chord of contact.
Sol. Any point on the hyperbola
x2 y2 1
9 4
− = is (3 secθ, 2 tanθ).
Chord of contact of the circle x2 + y2 = 9 with respect to the point (3 sec θ, 2tan θ) is
3 secθ.x + 2 tanθ.y = 9 ….(1)
Let (x1, y1) be the mid–point of the chord of contact.
⇒ equation of chord in mid−point form is xx1 + yy1 = x1
2 + y1
2 ….(2)
Since (1) and (2) represent the same line,
3sec 2 tan 9
x y x y
2 2
θ θ
= =
+
1 1 1 1
9x
1
⇒ secθ = 3 ( x 2 1 + y
2
)
1
9y
1
, tanθ = 2 ( x 2 1 + y
2
)
1
2 2
1 1
81x 81y
− =
Hence ( ) ( )
2 2 2 2 2 2
1 1 1 1
1
9 x + y 4 x +
y
⇒ the required locus is
x 2 y 2  2 2 x + − = y
2 
9 4  9

 
.

IIT-JEE2005-M-4
Q11. Find the equation of the common tangent in 1st quadrant to the circle x2 + y2 = 16 and the ellipse
x2 y2 1
25 4
+ = . Also find the length of the intercept of the tangent between the coordinate axes.
Sol. Let the equations of tangents to the given circle and the ellipse respectively be
y = mx + 4 1+ m2
and y = mx + 25m2 + 4
Since both of these represent the same common tangent,
4 1+ m2 = 25m2 + 4
⇒ 16(1 + m2) = 25m2 + 4
⇒ m = ± 2
3
The tangent is at a point in the first quadrant ⇒ m < 0.
⇒ m = 2
− , so that the equation of the common tangent is
3
y = 2 x 4 7
− + .
3 3
 
 
 
It meets the coordinate axes at A (2 7, 0) and B 0, 4 7
3
⇒ AB = 14
3
.
Q12. If length of tangent at any point on the curve y = f(x) intercepted between the point and the x–axis is of
length 1. Find the equation of the curve.
Sol. Length of tangent =
2 y 1 dx
 
+   dy


⇒ 1 =
   2
  +   
   
y2 1 dx
dy
⇒
2
dy = ±
y
dx 1 −
y
⇒
1 y2
∫ −
dy = ± x + c
.
y
Writing y = sin θ, dy = cos θ dθ and integrating, we get the equation of the curve as
2
2 1 − 1 −
y
1 y ln x c
− + = ± + .
y
Q13. Find the area bounded by the curves x2 = y, x2 = – y and y2 = 4x – 3.
Sol. The region bounded by the given curves
x2 = y, x2 = −y and y2 = 4x – 3 is
symmetrical about the x–axis. The parabolas x2 = y
and y2 = 4x – 3 touch at the point (1, 1).
Moreover the vertex of the curve
y2 = 4x – 3 is at 3, 0
 
  4


.
Hence the area of the region
=
 1 1

 ∫ 2
− ∫
− 
 
2 x dx 4x 3dx
0 3/ 4
x2 =y
x2 =−y
(1, 1)
(3/4, 0)
x
y2 =4x−3
y
     − −       
= (( ) ) 3 1 3 / 2 1
3 / 4
2 x 1 4x 3
3 6
0
= 2 1 1 1
 −  =  
3 6 3
. sq. units.

IIT-JEE2005-M-5
Q14. If one of the vertices of the square circumscribing the circle |z – 1| = 2 is 2 + 3 i. Find the other
vertices of square.
Sol. Since centre of circle i.e. (1, 0) is also the
mid–point of diagonals of square
⇒ 1 2
z z 3i
= ⇒ = −
0 2
z +
z
2
z 1
− ± π
and 3 i / 2
z 1
1
e
=
−
⇒ other vertices are
z3, z4 = (1− 3) + i and (1+ 3) − i .
z0(1,0)
z1(2, √3)
z4
y
z2
z3
x
O
Q15. If f (x – y) = f (x). g (y) – f (y). g (x) and g (x – y) = g (x). g (y) + f (x). f (y) for all x, y ∈ R. If right hand
derivative at x = 0 exists for f (x). Find derivative of g (x) at x = 0.
Sol. f(x – y) = f (x) g (y) – f (y) g (x) … (1)
Put x = y in (1), we get
f (0) = 0
put y = 0 in (1), we get
g (0) = 1.
Now, f′ (0+) =
lim f(0 h) f(0)
→ + h
h 0
+ −
=
lim f(0)g( h) g(0)f( h) f(0)
→ + h
h 0
− − − −
=
lim f( h)
→ + h
h 0
−
−
(∵f (0) = 0)
=
lim f(0 h) f(0)
→ + h
h 0
− −
−
= f′ (0–).
Hence f (x) is differentiable at x = 0.
Put y = x in g (x – y) = g (x). g (y) + f (x). f (y).
Also f2 (x) + g2 (x) = 1
⇒ g2 (x) = 1 – f2 (x)
⇒ 2g′ (0) g (0) = – 2f (0) f′ (0) = 0 ⇒ g′ (0) = 0.
Q16. If p(x) be a polynomial of degree 3 satisfying p(−1) = 10, p(1) = −6 and p(x) has maximum at x = −1
and p′(x) has minima at x = 1. Find the distance between the local maximum and local minimum of the
curve.
Sol. Let the polynomial be P (x) = ax3 + bx2 + cx + d
According to given conditions
P (–1) = –a + b – c + d = 10
P (1) = a + b + c + d = –6
Also P′ (–1) = 3a – 2b + c = 0
and P″ (1) = 6a + 2b = 0 ⇒ 3a + b = 0
Solving for a, b, c, d we get
P (x) = x3 – 3x2 – 9x + 5
⇒ P′ (x) = 3x2 – 6x – 9 = 3(x + 1)(x – 3)
⇒ x = –1 is the point of maximum and x = 3 is the point of minimum.
Hence distance between (–1, 10) and (3, –22) is 4 65 units.
Q17. f(x) is a differentiable function and g (x) is a double differentiable function such that |f (x)| ≤ 1 and
f′(x) = g (x). If f2 (0) + g2 (0) = 9. Prove that there exists some c ∈ (– 3, 3) such that g (c). g″(c)< 0.
Sol. Let us suppose that both g (x) and g″ (x) are positive for all x ∈ (−3, 3).
Since f2 (0) + g2 (0) = 9 and −1 ≤ f (x) ≤ 1, 2 2 ≤ g (0) ≤ 3.
From f′ (x) = g (x), we get
x
f (x) =
∫ + f (−3).
3
g(x)dx
−

IIT-JEE2005-M-6
Moreover, g″ (x) is assumed to be positive
⇒ the curve y = g (x) is open upwards.
If g (x) is decreasing, then for some value of x
x
∫ > area of the rectangle (0 – (–3))2 2
3
g(x)dx
−
⇒ f (x) > 2 2 × 3 − 1 i.e. f (x) > 1 which is a contradiction.
x
If g (x) is increasing, for some value of x
∫ > area of the rectangle (3 – 0))2 2
3
g(x)dx
−
x
If g(x) is minimum at x = 0, then
∫ > area of the rectangle (3 – 0)2 2
3
g(x)dx
−
Hence g (x) and g″ (x) cannot be both positive throughout the interval (−3, 3).
Similarly we can prove that g(x) and g″(x) cannot be both negative throughout the interval (−3, 3).
Hence there is atleast one value of c ∈ (−3, 3) where g (x) and g″ (x) are of opposite sign
⇒ g (c) . g″ (c) < 0.
Alternate:
3 3
∫ g(x)dx = ∫ f ′(x)dx = f (3) – f (0)
0 0
⇒
3
∫g(x)dx < 2 ……(1)
0
In the same way
0
∫ < ……(2)
−
3
g(x)dx 2
⇒
3 0
∫ g(x)dx + ∫ g(x)dx < 4
……(3)
−
0 3
From (f(0))2 + (g (0))2 = 9
we get
22 < g (0) < 3 ……(4)
or –3 < g (0) < –2 2 ……(5)
Case I: 2 2 < g (0) < 3
Let g (x) is concave upward ∀ x (–3, 3) then
the area
0 3
∫ g(x)dx + ∫ g(x)dx >
6 2
−
3 0
which is a contradiction from equation (3).
∴ g (x) will be concave downward for some c
∈ (–3, 3) i.e. g″ (c) < 0 ……(6)
also at that point c
(0, 2 2 )
(0, 3)
(–3, 0) (3, 0)
g (c) will be greater than 2 2
⇒ g (c) > 0 ……(7)
From equation (6) and (7)
g (c) . g″ (c) < 0 for some c ∈ (–3, 3).

IIT-JEE2005-M-7
Case II: –3 < g (0) < –2 2
Let g (x) is concave downward ∀ x (–3, 3)
then the area
0 3
∫ g(x)dx + ∫ g(x)dx >
6 2
−
3 0
which is a contradiction from equation (3).
∴ g (x) will be concave upward for some
c ∈ (–3, 3) i.e. g″ (c) > 0 ……(8)
also at that point c
g (c) will be less than –2 2
(–3, 0) (3, 0)
(0, –2 2 )
(0, –3)
⇒ g (c) < 0 ……(9)
From equation (8) and (9)
g (c) . g″ (c) < 0 for some c ∈ (–3, 3).
Q18. If
2 2
2 2
2 2
 4a 4a 1   f( − 1)   3a + 3a
      4b 4b 1    f(1)   =  3b + 3b
      4c 4c 1   f(2)     3c + 3c

, f(x) is a quadratic function and its maximum value occurs at a
point V. A is a point of intersection of y = f (x) with x-axis and point B is such that chord AB subtends a
right angle at V. Find the area enclosed by f (x) and chord AB.
Sol. Let we have
4a2 f (–1) + 4a f (1) + f (2) = 3a2 + 3a … (1)
4b2 f (–1) + 4b f (1) + f (2) = 3b2 + 3b … (2)
4c2 f (–1) + 4c f (1) + f (2) = 3c2 + 3c … (3)
Consider a quadratic equation
4x2 f (–1) + 4x f (1) + f (2) = 3x2 + 3x
or [4f (–1) – 3] x2 + [4f (1) – 3] x + f (2) = 0 … (4)
As equation (4) has three roots i.e. x = a, b, c. It is an identity.
⇒ f (–1) = 3
4
, f (1) = 3
4
and f (2) = 0
⇒ f (x) =
(4 −
x2 )
4
… (5)
Let point A be (–2, 0) and B be (2t, – t2 + 1)
Now as AB subtends a right angle at the vertex
V (0, 1)
1 t2 1
2 2t
−
× =− ⇒ t = 4
⇒ B ≡ (8, – 15)
8 2
∴ Area =
 − + 
 + 
  ∫ = 125
4 x 3x 6 dx
4 2 −
2
3
sq. units.
A(-2,0)
V(0,1)
(2,0)
X
B(8,-15)
3x+2y+6=0

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