Aieee 2003 maths solved paper by fiitjee

AIEEE−CBSE−ENG−03
1. A function f from the set of natural numbers to integers defined by
f (n) =






−
−
evenisnwhen,
2
n
oddiswhen,
2
1n
is
(A) one−one but not onto (B) onto but not one−one
(C) one−one and onto both (D) neither one−one nor onto
2. Let z1 and z2 be two roots of the equation z2
+ az + b = 0, z being complex. Further, assume
that the origin, z1 and z2 form an equilateral triangle, then
(A) a2
= b (B) a2
= 2b
(C) a2
= 3b (D) a2
= 4b
3. If z and ω are two non−zero complex numbers such that |zω| = 1, and Arg (z) − Arg (ω) =
2
π
,
then ωz is equal to
(A) 1 (B) − 1
(C) i (D) − i
4. If
x
i1
i1






−
+
= 1, then
(A) x = 4n, where n is any positive integer
(B) x = 2n, where n is any positive integer
(C) x = 4n + 1, where n is any positive integer
(D) x = 2n + 1, where n is any positive integer
5. If
32
32
32
c1cc
b1bb
a1aa
+
+
+
= 0 and vectors (1, a, a2
) (1, b, b2
) and (1, c, c2
) are non−coplanar, then the
product abc equals
(A) 2 (B) − 1
(C) 1 (D) 0
6. If the system of linear equations
x + 2ay + az = 0
x + 3by + bz = 0
x + 4cy + cz = 0
has a non−zero solution, then a, b, c
(A) are in A. P. (B) are in G.P.
(C) are in H.P. (D) satisfy a + 2b + 3c = 0
7. If the sum of the roots of the quadratic equation ax2
+ bx + c = 0 is equal to the sum of the
squares of their reciprocals, then
a
b
,
c
a
and
b
c
are in
(A) arithmetic progression (B) geometric progression
(C) harmonic progression (D) arithmetic−geometric−progression
8. The number of real solutions of the equation x2
− 3 |x| + 2 = 0 is
(A) 2 (B) 4
(C) 1 (D) 3

9. The value of ‘a’ for which one root of the quadratic equation
(a2
− 5a + 3) x2
+ (3a − 1) x + 2 = 0 is twice as large as the other, is
(A)
3
2
(B) −
3
2
(C)
3
1
(D) −
3
1
I10. If A = 





ab
ba
and A2
= 





αβ
βα
, then
(A) α = a2
+ b2
, β = ab (B) α = a2
+ b2
, β = 2ab
(C) α = a2
+ b2
, β = a2
− b2
(D) α = 2ab, β = a2
+ b2
11. A student is to answer 10 out of 13 questions in an examination such that he must choose at
least 4 from the first five questions. The number of choices available to him is
(A) 140 (B) 196
(C) 280 (D) 346
12. The number of ways in which 6 men and 5 women can dine at a round table if no two
women are to sit together is given by
(A) 6! × 5! (B) 30
(C) 5! × 4! (D) 7! × 5!
13. If 1, ω, ω2
are the cube roots of unity, then
∆ =
nn2
n2n
n2n
1
1
1
ωω
ωω
ωω
is equal to
(A) 0 (B) 1
(C) ω (D) ω2
14. If n
Cr denotes the number of combinations of n things taken r at a time, then the expression
n
Cr+1 + n
Cr−1 + 2 × n
Cr equals
(A) n+2
Cr (B) n+2
Cr+1
(C) n+1
Cr (D) n+1
Cr+1
15. The number of integral terms in the expansion of 2568
)53( + is
(A) 32 (B) 33
(C) 34 (D) 35
16. If x is positive, the first negative term in the expansion of (1 + x)27/5
is
(A) 7th
term (B) 5th
term
(C) 8th
term (D) 6th
term
17. The sum of the series
43
1
32
1
21
1
⋅
+
⋅
−
⋅
− ……… upto ∞ is equal to
(A) 2 loge 2 (B) log2 2 − 1
(C) loge 2 (D) loge 





e
4
18. Let f (x) be a polynomial function of second degree. If f (1) = f (− 1) and a, b, c are in A. P.,
then f′ (a), f′ (b) and f′ (c) are in
(A) A.P. (B) G.P.
(C) H. P. (D) arithmetic−geometric progression

19. If x1, x2, x3 and y1, y2, y3 are both in G.P. with the same common ratio, then the points (x1, y1)
(x2, y2) and (x3, y3)
(A) lie on a straight line (B) lie on an ellipse
(C) lie on a circle (D) are vertices of a triangle
20. The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of
side a, is
(A) a cot 




 π
n
(B)
2
a
cot 




 π
n2
(C) a cot 




 π
n2
(D)
4
a
cot 




 π
n2
21. If in a triangle ABC a cos2






2
C
+ c cos2






2
A
=
2
b3
, then the sides a, b and c
(A) are in A.P. (B) are in G.P.
(C) are in H.P. (D) satisfy a + b = c
22. In a triangle ABC, medians AD and BE are drawn. If AD = 4, ∠ DAB =
6
π
and ∠ ABE =
3
π
,
then the area of the ∆ ABC is
(A)
3
8
(B)
3
16
(C)
3
32
(D)
3
64
23. The trigonometric equation sin−1
x = 2 sin−1
a, has a solution for
(A)
2
1
< |a| <
2
1
(B) all real values of a
(C) |a| <
2
1
(D) |a| ≥
2
1
24. The upper
4
3
th portion of a vertical pole subtends an angle tan−1
5
3
at point in the horizontal
plane through its foot and at a distance 40 m from the foot. A possible height of the vertical
pole is
(A) 20 m (B) 40 m
(C) 60 m (D) 80 m
25. The real number x when added to its inverse gives the minimum value of the sum at x equal
to
(A) 2 (B) 1
(C) − 1 (D) − 2
26. If f : R → R satisfies f (x + y) = f (x) + f (y), for all x, y ∈ R and f (1) = 7, then ∑=
n
1r
)r(f is
(A)
2
n7
(B)
2
)1n(7 +
(C) 7n (n + 1) (D)
2
)1n(n7 +

27. If f (x) = xn
, then the value of f (1) −
!n
)1(f)1(
...
!3
)1(f
!2
)1(f
!1
)1(f nn
−
++
′′′
−
′′
+
′
is
(A) 2n
(B) 2n−1
(C) 0 (D) 1
28. Domain of definition of the function f (x) = 2
x4
3
−
+ log10 (x3
− x), is
(A) (1, 2) (B) (− 1, 0) ∪ (1, 2)
(C) (1, 2) ∪ (2, ∞) (D) (− 1, 0) ∪ (1, 2) ∪ (2, ∞)
29.
[ ]
[ ]3
2/x
x2
2
x
tan1
xsin1
2
x
tan1
lim
−π











+
−











−
π→
is
(A)
8
1
(B) 0
(C)
32
1
(D) ∞
30. If
x
)x3log()x3log(
lim
0x
−−+
→
= k, the value of k is
(A) 0 (B) −
3
1
(C)
3
2
(D) −
3
2
31. Let f (a) = g (a) = k and their nth
derivatives fn
(a), gn
(a) exist and are not equal for some n.
Further if
)x(f)x(g
)a(g)x(f)a(g)a(f)x(g)a(f
lim
ax −
+−−
→
= 4, then the value of k is
(A) 4 (B) 2
(C) 1 (D) 0
32. The function f (x) = log (x + 1x2
+ ), is
(A) an even function (B) an odd function
(C) a periodic function (D) neither an even nor an odd function
33. If f (x) =





=
≠






+−
0x,0
0x,xe
x
1
|x|
1
then f (x) is
(A) continuous as well as differentiable for all x
(B) continuous for all x but not differentiable at x = 0
(C) neither differentiable nor continuous at x = 0
(D) discontinuous everywhere
34. If the function f (x) = 2x3
− 9ax2
+ 12a2
x + 1, where a > 0, attains its maximum and minimum
at p and q respectively such that p2
= q, then a equals
(A) 3 (B) 1
(C) 2 (D)
2
1

35. If f (y) = ey
, g (y) = y; y > 0 and F (t) = ∫
t
0
f (t − y) g (y) dy, then
(A) F (t) = 1 − e−t
(1 + t) (B) F (t) = et
− (1 + t)
(C) F (t) = t et
(D) F (t) = t e−t
36. If f (a + b − x) = f (x), then ∫
b
a
x f (x) dx is equal to
(A) ∫ −
+
b
a
dx)xb(f
2
ba
(B) ∫
+
b
a
dx)x(f
2
ba
(C) ∫
−
b
a
dx)x(f
2
ab
(D) ∫ −+
+
b
a
dx)xba(f
2
ba
37. The value of
xsinx
dttsec
lim
2
x
0
2
0x
∫
→
is
(A) 3 (B) 2
(C) 1 (D) 0
38. The value of the integral I = ∫
1
0
x (1 − x)n
dx is
(A)
1n
1
+
(B)
2n
1
+
(C)
1n
1
+
−
2n
1
+
(D)
1n
1
+
+
2n
1
+
39. 5
333
n5
444
n n
n......321
lim
n
n......321
lim
++++
−
++++
∞→∞→
is
(A)
30
1
(B) zero
(C)
4
1
(D)
5
1
40. Let
dx
d
F (x) = 







x
e xsin
, x > 0. If
3
xsin
4
1
e
x
3
∫ dx = F (k) − F (1), then one of the possible values
of k, is
(A) 15 (B) 16
(C) 63 (D) 64
41. The area of the region bounded by the curves y = |x − 1| and y = 3 − |x| is
(A) 2 sq units (B) 3 sq units
(C) 4 sq units (D) 6 sq units
42. Let f (x) be a function satisfying f′ (x) = f (x) with f (0) = 1 and g (x) be a function that satisfies
f (x) + g (x) = x2
. Then the value of the integral ∫
1
0
f (x) g (x) dx, is

(A) e −
2
5
2
e2
− (B) e +
2
3
2
e2
−
(C) e −
2
3
2
e2
− (D) e +
2
5
2
e2
+
43. The degree and order of the differential equation of the family of all parabolas whose axis is
x−axis, are respectively
(A) 2, 1 (B) 1, 2
(C) 3, 2 (D) 2, 3
44. The solution of the differential equation (1 + y2
) + (x − ytan 1
e
−
)
dx
dy
= 0, is
(A) (x − 2) = k ytan 1
e
−
− (B) 2x ytan2 1
e
−
+ k
(C) x ytan 1
e
−
= tan−1
y + k (D) x ytan2 1
e
−
= ytan 1
e
−
+ k
45. If the equation of the locus of a point equidistant from the points (a1, b1) and (a2, b2) is (a1 −
a2) x + (b1 − b2) y + c = 0, then the value of ‘c’ is
(A) )baba(
2
1 2
1
2
1
2
2
2
2 −−+ (B) 2
2
2
1
2
2
2
1 bbaa −++
(C) )bbaa(
2
1 2
2
2
1
2
2
2
1 −−+ (D) 2
2
2
2
2
1
2
1 baba −−+
46. Locus of centroid of the triangle whose vertices are (a cos t, a sin t), (b sin t, − b cos t) and
(1, 0), where t is a parameter, is
(A) (3x − 1)2
+ (3y)2
= a2
− b2
(B) (3x − 1)2
+ (3y)2
= a2
+ b2
(C) (3x + 1)2
+ (3y)2
= a2
+ b2
(D) (3x + 1)2
+ (3y)2
= a2
− b2
47. If the pair of straight lines x2
− 2pxy − y2
= 0 and x2
− 2qxy − y2
= 0 be such that each pair
bisects the angle between the other pair, then
(A) p = q (B) p = − q
(C) pq = 1 (D) pq = − 1
48. a square of side a lies above the x−axis and has one vertex at the origin. The side passing
through the origin makes an angle α (0 < α <
4
π
) with the positive direction of x−axis. The
equation of its diagonal not passing through the origin is
(A) y (cos α − sin α) − x (sin α − cos α) = a
(B) y (cos α + sin α) + x (sin α − cos α) = a
(C) y (cos α + sin α) + x (sin α + cos α) = a
(D) y (cos α + sin α) + x (cos α − sin α) = a
49. If the two circles (x − 1)2
+ (y − 3)2
= r2
and x2
+ y2
− 8x + 2y + 8 = 0 intersect in two distinct
points, then
(A) 2 < r < 8 (B) r < 2
(C) r = 2 (D) r > 2
50. The lines 2x − 3y = 5 and 3x − 4y = 7 are diameters of a circle having area as 154 sq units.
Then the equation of the circle is
(A) x2
+ y2
+ 2x − 2y = 62 (B) x2
+ y2
+ 2x − 2y = 47
(C) x2
+ y2
− 2x + 2y = 47 (D) x2
+ y2
− 2x + 2y = 62
51. The normal at the point (bt1
2
, 2bt1) on a parabola meets the parabola again in the point (bt2
2
,
2bt2), then

(A) t2 = − t1 −
1t
2
(B) t2 = − t1 +
1t
2
(D) t2 = t1 −
1t
2
(D) t2 = t1 +
1t
2
52. The foci of the ellipse 2
22
b
y
16
x
+ = 1 and the hyperbola
25
1
81
y
144
x 22
=− coincide. Then the
value of b2
is
(A) 1 (B) 5
(C) 7 (D) 9
53. A tetrahedron has vertices at O (0, 0, 0), A (1, 2, 1), B (2, 1, 3) and C (− 1, 1, 2). Then the
angle between the faces OAB and ABC will be
(A) cos−1






35
19
(B) cos−1






31
17
(C) 300
(D) 900
54. The radius of the circle in which the sphere x2
+ y2
+ z2
+ 2x − 2y − 4z − 19 = 0 is cut by the
plane x + 2y + 2z + 7 = 0 is
(A) 1 (B) 2
(C) 3 (D) 4
55. The lines
k
4z
1
3y
1
2x
−
−
=
−
=
−
and
1
5z
2
4y
k
1x −
=
−
=
−
are coplanar if
(A) k = 0 or − 1 (B) k = 1 or − 1
(C) k = 0 or − 3 (D) k = 3 or − 3
56. The two lines x = ay + b, z = cy + d and x = a′y + b′, z = c′y + d′ will be perpendicular, if and
only if
(A) aa′ + bb′ + cc′ + 1 = 0 (B) aa′ + bb′ + cc′ = 0
(C) (a + a′) (b + b′) + (c + c′) = 0 (D) aa′ + cc′ + 1 = 0
57. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x2
+ y2
+ z2
+ 4x − 2y
− 6z = 155 is
(A) 26 (B) 11
13
4
(C) 13 (D) 39
58. Two systems of rectangular axes have the same origin. If a plane cuts them at distances a,
b, c and a′, b′, c′ from the origin, then
(A) 222222
c
1
b
1
a
1
c
1
b
1
a
1
′
+
′
+
′
+++ = 0 (B) 222222
c
1
b
1
a
1
c
1
b
1
a
1
′
−
′
+
′
+−+ = 0
(C) 222222
c
1
b
1
a
1
c
1
b
1
a
1
′
−
′
−
′
+−− = 0 (D) 222222
c
1
b
1
a
1
c
1
b
1
a
1
′
−
′
−
′
−++ = 0
59. c,b,a
rrr
are 3 vectors, such that 0cba
rrrr
=++ , |c|,2|b|,1|a|
rrr
== = 3, then accbba
rrrrrr
⋅+⋅+⋅ is
equal to
(A) 0 (B) − 7
(C) 7 (D) 1
60. If v,u
rr
and w
r
are three non−coplanar vectors, then ( )wv()vu()wvu
rrrrrrr
−×−⋅−+ equals
(A) 0 (B) wvu
rrr
×⋅

(C) vwu
rrr
×⋅ (D) 3 wvu
rrr
×⋅
61. Consider points A, B, C and D with position vectors kˆ10jˆ6iˆ,kˆ7jˆ4iˆ7 +−+− , kˆ4jˆ3iˆ +−−
and kˆ5jîˆ5 +− respectively. Then ABCD is a
(A) square (B) rhombus
(C) rectangle (D) parallelogram but not a rhombus
62. The vectors kˆ4iˆ3AB += , and kˆ4jˆ2iˆ5AC +−= are the sides of a triangle ABC. The length
of the median through A is
(A) 18 (B) 72
(C) 33 (D) 288
63. A particle acted on by constant forces kˆ3jîˆ4 −+ and kˆjîˆ3 −+ is displaced from the point
kˆ3jˆ2iˆ ++ to the point kˆjˆ4iˆ5 ++ . The total work done by the forces is
(A) 20 units (B) 30 units
(C) 40 units (D) 50 units
64. Let jîˆv,jîû −=+=
rr
and kˆ3jˆ2iˆw ++=
r
. If nˆ is unit vector such that nû ⋅
r
= 0 and nˆv ⋅
r
= 0,
then |nˆw| ⋅
r
is equal to
(A) 0 (B) 1
(C) 2 (D) 3
65. The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of
the set is increased by 2, then the median of the new set
(A) is increased by 2 (B) is decreased by 2
(C) is two times the original median (D) remains the same as that of the original set
66. In an experiment with 15 observations on x, then following results were available:
∑ 2
x = 2830, ∑x = 170
One observation that was 20 was found to be wrong and was replaced by the correct value
30. Then the corrected variance is
(A) 78.00 (B) 188.66
(C) 177.33 (D) 8.33
67. Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The
probability that Mr. A selected the winning horse is
(A)
5
4
(B)
5
3
(C)
5
1
(D)
5
2
68. Events A, B, C are mutually exclusive events such that P (A) =
3
1x3 +
, P (B) =
4
x1−
and
P (C) =
2
x21−
. The set of possible values of x are in the interval
(A) 





2
1
,
3
1
(B) 





3
2
,
3
1
(C) 





3
13
,
3
1
(D) [0, 1]

69. The mean and variance of a random variable having a binomial distribution are 4 and 2
respectively, then P (X = 1) is
(A)
32
1
(B)
16
1
(C)
8
1
(D)
4
1
70. The resultant of forces P
r
and Q
r
is R
r
. If Q
r
is doubled then R
r
is doubled. If the direction of
Q
r
is reversed, then R
r
is again doubled. Then P2
: Q2
: R2
is
(A) 3 : 1 : 1 (B) 2 : 3 : 2
(C) 1 : 2 : 3 (D) 2 : 3 : 1
71. Let R1 and R2 respectively be the maximum ranges up and down an inclined plane and R be
the maximum range on the horizontal plane. Then R1, R, R2 are in
(A) arithmetic−geometric progression (B) A.P.
(C) G.P. (D) H.P.
72. A couple is of moment G
r
and the force forming the couple is P
r
. If P
r
is turned through a
right angle, the moment of the couple thus formed is H
r
. If instead, the forces P
r
are turned
through an angle α, then the moment of couple becomes
(A) G
r
sin α − H
r
cos α (B) H
r
cos α + G
r
sin α
(C) G
r
cos α − H
r
sin α (D) H
r
sin α − G
r
cos α
73. Two particles start simultaneously from the same point and move along two straight lines,
one with uniform velocity u
r
and the other from rest with uniform acceleration f
r
. Let α be the
angle between their directions of motion. The relative velocity of the second particle with
respect to the first is least after a time
(A)
f
sinu α
(B)
u
cosf α
(C) u sin α (D)
f
cosu α
74. Two stones are projected from the top of a cliff h meters high, with the same speed u so as
to hit the ground at the same spot. If one of the stones is projected horizontally and the other
is projected at an angle θ to the horizontal then tan θ equals
(A)
gh
u2
(B) 2g
h
u
(C) 2h
g
u
(D) u
gh
2
75. A body travels a distances s in t seconds. It starts from rest and ends at rest. In the first part
of the journey, it moves with constant acceleration f and in the second part with constant
retardation r. The value of t is given by
(A) 2s 





+
r
1
f
1
(B)
r
1
f
1
s2
+
(C) )rf(s2 + (D) 





+
r
1
f
1
s2

Solutions
1. Clearly both one − one and onto
Because if n is odd, values are set of all non−negative integers and if n is an even, values
are set of all negative integers.
Hence, (C) is the correct answer.
2. z1
2
+ z2
2
− z1z2 = 0
(z1 + z2)2
− 3z1z2 = 0
a2
= 3b.
5.
2
2
2
2
2
2
cc1
bb1
aa1
1cc
1bb
1aa
+ = 0
(1 + abc)
1cc
1bb
1aa
2
2
2
= 0
⇒ abc = − 1.
Hence, (B) is the correct answer
4.
2
)i1(
i1
i1 2
+
=
−
+
= i
x
i1
i1






−
+
= ix
⇒ x = 4n.
Hence, (A) is the correct answer.
6. Coefficient determinant =
cc41
bb31
aa21
= 0
⇒ b =
ca
ac2
+
.
Hence, (C) is the correct answer
8. x2
− 3 |x| + 2 = 0
(|x| − 1) (|x| − 2) = 0
⇒ x = ± 1, ± 2.
7. Let α, β be the roots
α + β = 22
11
β
+
α
α + β =
)(
222
β+α
αβ−β+α
2
2
c
ac2b
a
b −
=





−
⇒ 2a2
c = b (a2
+ bc)

⇒
b
c
,
a
b
,
c
a
are in H.P.
10. A = 





ab
ba
A2
= 





ab
ba






ab
ba
= 





+
+
22
22
baab2
ab2ba
⇒ α = a2
+ b2
, β = 2ab.
Hence, (B) is the correct answer.
9. β = 2α
3α =
3a5a
1a3
2
+−
−
2α2
=
6a5a
2
2
+−
6a5a
1
)3a5a(a
)1a3(
222
2
++
=
+−
−
⇒ a =
3
2
.
Hence, (A) is the correct answer
12. Clearly 5! × 6!
(A) is the correct answer
11. Number of choices = 5
C4 × 8
C6 + 5
C5 × 8
C5
= 140 + 56.
13. ∆ =
nn2n
n2n2n
n2nn2n
11
11
1
ωω+ω+
ωω+ω+
ωωω+ω+
= 0
Since, 1 + ωn
+ ω2n
= 0, if n is not a multiple of 3
Therefore, the roots are identical.
14. n
Cr+1 + n
Cr−1 + n
Cr + n
Cr
= n+1
Cr+1 + n+1
Cr
= n+2
Cr+1.
17.
43
1
32
1
21
1
⋅
+
⋅
−
⋅
− ………
= 1 −
4
1
3
1
3
1
2
1
2
1
−++− − ………

= 1 − 2 





−+− .........
4
1
3
1
2
1
= 2 





+−+− .........
4
1
3
1
2
1
1 − 1
= 2 log 2 − log e
= log 





e
4
.
Hence, (D) is the correct answer.
15. General term = 256
Cr ( 3 )256−r
[(5)1/8
]r
From integral terms, or should be 8k
⇒ k = 0 to 32.
18. f (x) = ax2
+ bx + c
f (1) = a + b + c
f (− 1) = a − b + c
⇒ a + b + c = a − b + c also 2b = a + c
f′ (x) = 2ax + b = 2ax
f′ (a) = 2a2
f′ (b) = 2ab
f′ (c) = 2ac
⇒ AP.
19. Result (A) is correct answer.
20. (B)
21. a
2
b3
2
Acos1
c
2
Ccos1
=




 +
+




 +
⇒ a + c + b = 3b
a + c = 2b.
26. f (1) = 7
f (1 + 1) = f (1) + f (1)
f (2) = 2 × 7
only f (3) = 3 × 7
∑=
n
1r
)r(f = 7 (1 + 2 + ……… + n)
= 7
2
)1n(n +
.
25. (B)
23. −
4
π
≤
42
xsin2
π
≤
−
4
π
≤ sin−1
(a) ≤
4
π

2
1
≤ |a| ≤
2
1
.
Hence, (D) is the correct answer
27. LHS = 1 −
!3
)2n)(1n(n
!2
)1n(n
!1
n −−
−
−
+ + ………
= 1 − n
C1 + n
C2 − ………
= 0.
30.
3
2
1
x3
1
x3
1
lim
0x
=−
+
+
→
.
28. 4 − x2
≠ 0
⇒ x ≠ ± 2
x3
− x > 0
⇒ x (x + 1) (x − 1) > 0.
Hence (D) is the correct answer.
29.
2
2/x
)x2(
2
x
4
4
)xsin1(
2
x
4
tan
lim
−π





−
π
−





−
π
π→
=
32
1
.
32. f (− x) = − f (x)
1. sin (θ + α) =
40
x
sin a =
140
x
⇒ x = 40.
θ
40
x/4
3x/4
tan−1
(3/4)
34. f (x) = 0 at x = p, q
6p2
+ 18ap + 12a2
= 0
6q2
+ 18aq + 12a2
= 0
f″ (x) < 0 at x = p
and f″ (x) > 0 at x = q.
30. Applying L. Hospital’s Rule
)a(f)a(g
)a(f)a(g)a(g)a(f
lim
a2x ′−′
′−′
→
= 4

))a(f)a(g(
))a(ff)a(g(k
′−′
′−′
= 4
k = 4.
36. ∫
b
a
x f (x) dx
= ∫ −+
b
a
)xba( f (a + b − x) dx.
33. f′ (0)
f′ (0 − h) = 1
f′ (0 + h) = 0
LHD ≠ RHD.
37.
xsinx
)xtan(
lim
2
0x→
=





→
x
xsin
x
)xtan(
lim
2
2
0x
= 1.
Hence (C) is the correct answer.
38. ∫
1
0
x (1 − x)n
dx = ∫
1
0
n
x (1 − x)
= ∫
+
−
1
0
1nn
)xx( =
2n
1
1n
1
+
−
+
.
35. F (t) = ∫
t
0
f (t − y) f (y) dy
= ∫
t
0
f (y) f (t − y) dy
= ∫
t
0
y
e (t − y) dy
= xt
− (1 + t).
34. Clearly f″ (x) > 0 for x = 2a ⇒ q = 2a < 0 for x = a ⇒ p = a
or p2
= q ⇒ a = 2.
40. F′ (x) = x
xsin
3
e

= ∫
xsin
e
x
3
dx = F (k) − F (1)
= ∫
64
1
xsin
x
e
dx = F (k) − F (1)
= ∫ ′
64
1
F (x) dx = F (k) − F (1)
F (64) − F (1) = F (k) − F (1)
⇒ k = 64.
41. Clearly area = 2 2 × 2
= sq units
(2,1)
(1,0)
(−1,2)
45. Let p (x, y)
(x − a1)2
+ (y − b1)2
= (x − a2)2
+ (y − b2)2
(a1 − a2) x + (b1 − b2) y +
2
1
)aabb( 2
1
2
2
2
1
2
2 −+− = 0.
46. x =
3
1tsinbtcosa ++
, y =
3
1tcosbtsina +−
2
3
1
x 





− + y2
=
9
ba 22
+
.
43. Equation y2
= 4a 9x − h)
2yy1 = 4a ⇒ yy1 = 2a
yy2 = y1
2
= 0.
Hence (B) is the correct answer.
42. ∫
1
0
)x(f [x2
− f (x)] dx
solving this by putting f′ (x) = f (x).
50. Intersection of diameter is the point (1, − 1)
πs2
= 154
⇒ s2
= 49
(x − 1)2
+ (y + 1)2
= 49
47. (D)
49.
dy
dx
(1 + y2
) = )xe( ysin 1
−
−

dy
dx
+ α
+ y1
x
= 2
ysub
y1
e
1
+
−−
52. 2
2
2
2
5
9
y
5
12
x






−






= 1
⇒ e1 =
4
5
ae2 =
16
b
1
2
− × 4 = 3
⇒ b2
= 7.
54. (C)
5
3
4
69. np = 4
npq = 2
q =
2
1
, p =
2
1
n = 8
p (x = 1) = 8
C1
8
2
1






=
32
1
.
49. (x − 1)2
+ (y − 3)2
= r2
(x − 4)2
+ (y + 2)2
− 16 − 4 + 8 = 0
(x − 4)2
+ (y + 2)2
= 12.
67. Select 2 out of 5
=
5
2
.
65. 0 ≤
2
x21
4
x1
3
1x3 −
+
−
+
+
≤ 1
12x + 4 + 3 − 3x + 6 − 12x ≤ 1
0 ≤ 13 − 3x ≤ 12
3x ≤ 13
⇒ x ≥
3
1
x ≤
3
13
.

3. Arg
2
z π
=





ω
|zω| = 1
z ω = − i or + i.

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