53664886 AIEEE Practice

(1)

(2)

(3)

(4)

(5)

(6)

f (n) =











−

even

is

n

when

,

2 n

odd

is

when

,

2

1 n

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) a}2_{= 2b}

(C) a2_{= 3b} _{(D) a}2_{= 4b}

3. If z and ω are two non−zero complex numbers such that |zω| = 1, and Arg (z) − Arg (ω) = 2 π_{, then}_z_ω_{is equal to} (A) 1 (B) − 1 (C) i (D) − i 4. If x i 1 i 1       − + _{= 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 3 2 3 2 3 2 c 1 c c b 1 b b a 1 a a + + +

= 0 and vectors (1, a, a2_{) (1, b, b}2_{) and (1, c, c}2_{) 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

(7)

(C) 1 (D) 3

9. The value of ‘a’ for which one root of the quadratic equation

(a2₋_{5a + 3) x}2_{+ (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 = _      a b b a and A2₌       α β β α , then (A) α = a2_{+ b}2_,_β_{= ab} _(B)_α_{= a}2_{+ b}2_,_β_{= 2ab} (C) α = a2_{+ b}2_,_β_{= a}2₋_b2 _(D)_α_{= 2ab,}_β_{= a}2_{+ b}2

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}

∆ = n n 2 n 2 n n 2 n 1 1 1 ω ω ω ω ω ω is equal to (A) 0 (B) 1 (C) ω (D) ω2 14. If n_C

r denotes the number of combinations of n things taken r at a time, then the

expression n_C r+1 + nCr−1 + 2 ×nCr equals (A) n+2_C r (B) n+2Cr+1 (C) n+1_C r (D) n+1Cr+1

15. The number of integral terms in the expansion of ₍ ₃₊8₅₎256_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) 5}th_term

(C) 8th_term _{(D) 6}th_term

17. The sum of the series

4 3 1 3 2 1 2 1 1 ⋅ + ⋅ − ⋅ − ……… upto ∞ is equal to

(8)

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 n 2 (C) a cot       π n 2 (D) ₄ a _cot _      π n 2

21. If in a triangle ABC a cos2 _

     2 C + c cos2 _      2 A = 2 b

3 _{, 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

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∑

=1 r ) r ( f is (A) 2 n 7 _(B) 2 ) 1 n ( 7 + (C) 7n (n + 1) (D) 2 ) 1 n ( n 7 +

27. If f (x) = xn_{, then the value of f (1)}₋

! n ) 1 ( f ) 1 ( ... ! 3 ) 1 ( f ! 2 ) 1 ( f ! 1 ) 1 ( f ₋ n n + + ′ ′ ′ − ′ ′ + ′ is (A) 2n _{(B) 2}n−1 (C) 0 (D) 1

28. Domain of definition of the function f (x) = ₂

x 4 3 − + log10 (x 3₋_{x), is} (A) (1, 2) (B) (− 1, 0) ∪ (1, 2) (C) (1, 2) ∪ (2, ∞) (D) (− 1, 0) ∪ (1, 2) ∪ (2, ∞) 29.

[

]

[

]

3 2 / x x 2 2 x tan 1 x sin 1 2 x tan 1 lim − π             + −             − π → is (A) 8 1 _{(B) 0} (C) 32 1 (D) ∞ 30. If x ) x 3 log( ) x 3 log( lim 0 x − − + → = 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 f}n_{(a), g}n_{(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 a x − + − −

→ = 4, then the value

of k is

(A) 4 (B) 2

(C) 1 (D) 0

32. The function f (x) = log (x + x2₊1_{), is}

(A) an even function (B) an odd function

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33. If f (x) =







=

≠





0 x

,

0

0 x

,

xe

|x|

x

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_{+ 12a}2_{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

→ 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) 1 n 1 + (B) n 2 1 + (C) 1 n 1 + − n 2 1 + (D) n 1 1 + + n 2 1 + 39. 3 3₅ 3 n 5 4 4 4 n n n ... 3 2 1 lim n n ... 3 2 1 lim + + + + − + + + + ∞ → ∞ → is (A) 30 1 _{(B) zero}

(11)

40. Let dx d F (x) =     x esinx , x > 0. If sinx3 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}₋ tan1y

e − ) dx

dy_{= 0, is}

(A) (x − 2) = k

e

−

tan

−1

y

(B) 2x 2tan1y

e − + k (C) x tan1y

e − = tan−1_{y + k} _{(D) x} ₂_tan1_y

e − = tan1y

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) (a b a b ) 2 1 2 1 2 1 2 2 2 2 + − − (B) a12+a22+b12−b22 (C) (a a b b ) 2 1 2 2 2 1 2 2 2 1 + − − (D) a12+b12−a22−b22

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_{= a}2₋_b2 _{(B) (3x}₋₁₎2_{+ (3y)}2_{= a}2_{+ b}2

(C) (3x + 1)2_{+ (3y)}2_{= a}2_{+ b}2 _{(D) (3x + 1)}2_{+ (3y)}2_{= a}2₋_b2

47. If the pair of straight lines x2 ₋_2pxy₋_y2_{= 0 and x}2 ₋_2qxy₋_y2_{= 0 be such that}

each pair bisects the angle between the other pair, then

(A) p = q (B) p = − q

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side passing through the origin makes an angle α (0 < α <

4) 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_{= r}2_{and x}2_{+ y}2₋_{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_{+ y}2_{+ 2x}₋_{2y = 62} _{(B) x}2_{+ y}2_{+ 2x}₋_{2y = 47}

(C) x2_{+ y}2₋_{2x + 2y = 47} _{(D) x}2_{+ y}2₋_{2x + 2y = 62}

51. The normal at the point (bt12, 2bt1) on a parabola meets the parabola again in the

point (bt22, 2bt2), then (A) t2 = − t1− 1 t 2 (B) t2 = − t1 + 1 t 2 (D) t2 = t1− 1 t 2 (D) t2 = t1 + 1 t 2

52. The foci of the ellipse 2 ₂2

b y 16

x

+ = 1 and the hyperbola

25 1 81 y 144 x2 2 = − 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) 90}0

54. The radius of the circle in which the sphere x2_{+ y}2_{+ z}2_{+ 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 4 z 1 3 y 1 2 x − − = − = − _and 1 5 z 2 4 y k 1 x− ₌ − ₌ − _{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

(13)

57. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x2_{+ y}2 + z2_{+ 4x}₋_2y₋_{6z = 155 is} (A) 26 (B) 11 13 4 (C) 13 (D) 39

−++

= 0

59. a,b, c are 3 vectors, such that a+b+c=0, |a|=1, |b|=2,|c| = 3, then

a c c b b a⋅+⋅+⋅ is equal to (A) 0 (B) − 7 (C) 7 (D) 1

60. If u, v and w are three non−coplanar vectors, then (

) w v ( ) v u ( ) w v u+− ⋅ − × − equals (A) 0 (B) u⋅v×w (C) u⋅w×v (D) 3u⋅v×w

61. Consider points A, B, C and D with position vectors 7iˆ−4ˆj+7kˆ, iˆ−6ˆj+10kˆ

, −iˆ−3jˆ+4kˆ and 5iˆ−jˆ+5kˆ respectively. Then ABCD is a

(A) square (B) rhombus

(C) rectangle (D) parallelogram but not a rhombus

62. The vectors AB =3iˆ+4kˆ, and AC =5iˆ−2jˆ+4kˆ are the sides of a triangle ABC. The length of the median through A is

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63. A particle acted on by constant forces 4iˆ+jˆ−3kˆ and 3iˆ+jˆ−kˆ is displaced from the point iˆ+2jˆ+3kˆ to the point 5iˆ+4jˆ+kˆ. The total work done by the forces is

(A) 20 units (B) 30 units

(C) 40 units (D) 50 units

64. Let u=iˆ+jˆ,v=iˆ−jˆ and w =iˆ+2jˆ+3kˆ. If nˆ is unit vector such that u⋅nˆ = 0 and v⋅nˆ = 0, then |w⋅nˆ| 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:

∑

_x2_{= 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 1 x 3 + _{, P (B) =} 4 x 1− _and _{P (C) =} 2 x 2

1− _{. 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

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the direction of Q is reversed, then R is again doubled. Then P : Q : R 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 and the force forming the couple is P. If P is turned

through a right angle, the moment of the couple thus formed is _H . If instead, the forces _P are turned through an angle α, then the moment of couple becomes

(A) G sin α− H cos α (B) _H cos α + G sin α

(C) G cos α− H sin α (D) _H sin α − G cos α

73. Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity u and the other from rest with uniform acceleration _f . 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 sin u α _(B) u cos f α (C) u sin α (D) f cos u 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 u 2 (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 s 2 + (C) 2s(f+r) (D)       + r 1 f 1 s 2

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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. z12 + z22− z1z2 = 0

(z1 + z2)2− 3z1z2 = 0

a2_{= 3b.}

Hence, (C) is the correct answer.

5. 2 2 2 2 2 2 c c 1 b b 1 a a 1 1 c c 1 b b 1 a a + = 0 (1 + abc) 1 c c 1 b b 1 a a 2 2 2 = 0 ⇒ abc = − 1.

Hence, (B) is the correct answer

4. 2 ) i 1 ( i 1 i 1 ₊ 2 = − + _{= i} x i 1 i 1       − + _{= i}x ⇒ x = 4n.

Hence, (A) is the correct answer.

6. Coefficient determinant = c c 4 1 b b 3 1 a a 2 1 = 0 ⇒ b = c a ac 2 + .

Hence, (C) is the correct answer 8. x2₋_{3 |x| + 2 = 0}

(|x| − 1) (|x| − 2) = 0 ⇒ x = ± 1, ± 2.

Hence, (B) is the correct answer 7. Let α, β be the roots

α + β = 2 2 1 1 β + α α + β = ) ( 2 2 2 β + α αβ − β + α 2 2 c ac 2 b a b_₌ −     − ⇒ 2a2_{c = b (a}2_{+ bc)} ⇒ b c , a b , c a are in H.P.

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10. A = _      a b b a A2₌       a b b a       a b b a = _      + + 2 2 2 2 b a ab 2 ab 2 b a ⇒ α = a2_{+ b}2_,_β_{= 2ab.}

Hence, (B) is the correct answer. 9. β = 2α 3α = 3 a 5 a 1 a 3 2₋ ₊ − 2α2₌ 6 a 5 a 2 2₋ ₊ 6 a 5 a 1 ) 3 a 5 a ( a ) 1 a 3 ( 2 2 2 2 + + = + − − ⇒ a = 3 2 .

Hence, (A) is the correct answer 12. Clearly 5! × 6!

(A) is the correct answer 11. Number of choices = 5_C

4×8C6 + 5C5× 8C5

= 140 + 56.

13. ∆ = n n 2 n n 2 n 2 n n 2 n n 2 n 1 1 1 1 1 ω ω + ω + ω ω + ω + ω ω ω + ω + = 0

Since, 1 + ωn₊_ω2n_{= 0, if n is not a multiple of 3}

Therefore, the roots are identical. Hence, (A) is the correct answer 14. n_C r+1 + nCr−1 + nCr + nCr = n+1_C r+1 + n+1Cr = n+2_C r+1.

17. 4 3 1 3 2 1 2 1 1 ⋅ + ⋅ − ⋅ − ……… = 1 − 4 1 3 1 3 1 2 1 2 1 − + + − − ……… = 1 − 2 _ − + −..._ 4 1 3 1 2 1

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= 2 log 2 − log e = log       e 4 .

Hence, (D) is the correct answer. 15. General term = 256_C

r ( 3)256−r [(5)1/8]r

From integral terms, or should be 8k ⇒ k = 0 to 32.

Hence, (B) is the correct answer. 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.

Hence, (A) is the correct answer. 19. Result (A) is correct answer. 20. (B) 21. a 2 b 3 2 A cos 1 c 2 C cos 1 ₌       + +       + ⇒ a + c + b = 3b a + c = 2b.

Hence, (A) is the correct answer 26. f (1) = 7 f (1 + 1) = f (1) + f (1) f (2) = 2 × 7 only f (3) = 3 × 7

∑

= n 1 r ) r ( f = 7 (1 + 2 + ……… + n) = 7 2 ) 1 n ( n + _. 25. (B) 23. − 4 π _≤ 4 2 x sin2 _≤ π − 4 π _≤_sin−1_(a)_≤ 4 π 2 1 ≤ |a| ≤ 2 1 .

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= 1 −n_C

1 + nC2− ………

= 0.

Hence, (C) is the correct answer

30.

3

2

1

3 x

1 x

3

1 lim

0 x

=

−

+

→ .

Hence, (C) is the correct answer. 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 ) x 2 ( 2 x 4 4 ) x sin 1 ( 2 x 4 tan lim − π       −π −       −π π → = 32 1 _.

Hence, (C) is the correct answer. 32. f (− x) = − f (x)

Hence, (B) is the correct answer.

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 + 12a}2_{= 0} 6q2_{+ 18aq + 12a}2_{= 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 a 2 x ′ − ′ ′ − ′ → = 4 )) a ( f ) a ( g ( )) a ( f f ) a ( g ( k ′ − ′ ′ − ′ = 4 k = 4.

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36.

∫

b a x f (x) dx =

∫

+ − b a ) x b a ( f (a + b − x) dx. Hence, (B) is the correct answer. 33. f′ (0)

f′ (0 − h) = 1 f′ (0 + h) = 0 LHD ≠ RHD.

37. x sin x ) x tan( lim 2 0 x→ =       → x x sin x ) x tan( lim 2 2 0 x = 1.

Hence (C) is the correct answer.

38.

∫

1 0 x (1 − x)n_{dx =}

∫

1 0 n x (1 − x) =

∫

− + 1 0 1 n n _x ₎ x ( = 2 n 1 1 n 1 + − + .

Hence, (A) is the correct answer.

46. x = 3 1 t sin b t cos a + + _{, y =} 3 1 t cos b t sin a − + 2 3 1 x       − + y2₌ 9 b a2+ 2 _.

Hence, (B) is the correct answer. 43. Equation y2_{= 4a 9x}₋_h)

2yy1 = 4a ⇒ yy1 = 2a

yy2 = y12 = 0.

Hence (B) is the correct answer.

42.

∫

1 0 ) x ( f [x2 ₋_{f (x)] dx}

solving this by putting f′ (x) = f (x). Hence, (B) is the correct answer.

50. Intersection of diameter is the point (1, − 1) πs2_{= 154}

⇒ s2_{= 49}

(x − 1)2_{+ (y + 1)}2_{= 49}

Hence, (C) is the correct answer. 47. (D) 49. _dydx (1 + y2_{) =}₍_esin1y _x₎ − − dy dx + ₁₊_yα x = sub ₂y y 1 e 1 + − −

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5 9 5 12             ⇒ e1 = 4 5 ae2 = 16 b 1− 2 × 4 = 3 ⇒ b2_{= 7.}

Hence, (C) is the correct answer. 54. (C) 5 3 4 69. np = 4 npq = 2 q = 2 1 , p = 2 1 n = 8 p (x = 1) = 8_C 1 8 2 1_      = 32 1 .

Hence, (A) is the correct answer. 49. (x − 1)2_{+ (y}₋₃₎2_{= r}2 (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 .

Hence, (D) is the correct answer.

65. 0 ≤ 2 x 2 1 4 x 1 3 1 x 3 + ₊ − ₊ − _≤₁ 12x + 4 + 3 − 3x + 6 − 12x ≤ 1 0 ≤ 13 − 3x ≤ 12 3x ≤ 13 ⇒ x ≥ 3 1 x ≤ 3 13 .

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|zω| = 1

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Important Instructions:

i) The test is of 11

2hours duration.

ii) The test consists of 75 questions.

iii) The maximum marks are 225.

iv) For each correct answer you will get 3 marks and for a wrong answer you will get -1 mark.

1. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

(1) a function (2) reflexive

(3) not symmetric (4) transitive

2. The range of the function 7 x x 3 f(x) - P -= is (1) {1, 2, 3} (2) {1, 2, 3, 4, 5} (3) {1, 2, 3, 4} (4) {1, 2, 3, 4, 5, 6}

3. _{Let z, w be complex numbers such that z iw}_{+ = 0 and arg zw =}π. Then arg z equals (1) 4 p (2) 5 4 p (3)3 4 p (4) 2 p 4. If z = x – i y and _z13 _{= + , then}_{p iq}

(

2 2

)

y x p q p q æ ₊ ö ç ÷ è ø + is equal to (1) 1 (2) -2 (3) 2 (4) -1 5. If_z2_-₁₌_z2_{+ , then z lies on}₁

(1) the real axis (2) an ellipse

(3) a circle (4) the imaginary axis.

6. Let 0 0 1 A 0 1 0 . 1 0 0 -æ ö ç ÷ =_ç - _÷ ç_- ÷ è ø

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(1) A is a zero matrix (2) _A2 ₌_I

(3) _A-1_{does not exist} ₍₄₎_A _{= -}

( )

_{1 I}_{, where I is a unit matrix}

7. Let 1 1 1 A 2 1 3 1 1 1 -æ ö ç ÷ =_ç - _÷ ç ÷ è ø

( )

4 2 2 10 B 5 0 1 2 3 æ ö ç ÷ = -_ç a_÷ ç _- ÷ è ø

. If B is the inverse of matrix A, then α is

(1) -2 (2) 5

(3) 2 (4) -1

8. If a , a , a , ....,a , .... are in G.P., then the value of the determinant 1 2 3 n

n n 1 n 2

n 3 n 4 n 5

n 6 n 7 n 8

loga loga loga loga loga loga loga loga loga

+ + + + + + + + , is (1) 0 (2) -2 (3) 2 (4) 1

9. Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

(1) _x2₊_{18x 16 0}₊ ₌ ₍₂₎_x2_- _{18x 16 0}_- ₌

(3) _x2₊_{18x 16 0}_- ₌ ₍₄₎_x2_- _{18x 16 0}₊ ₌

10. If (1 – p) is a root of quadratic equation x2+px+ -

(

1 p

)

= , then its roots are0

(1) 0, 1 (2) -1, 2

(3) 0, -1 (4) -1, 1

11. Let _{S(K) 1 3 5 ...}_{= + + + +}

(

_{2K 1}_-

)

_{= +}_{3 K}2_{. Then which of the following is true?}

(1) S(1) is correct

(2) Principle of mathematical induction can be used to prove the formula (3) S(K)Þ S(K 1)+

(4) S(K)Þ S(K 1)+

12. How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?

(1) 120 (2) 480

(3) 360 (4) 240

13. The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

(1) 5 (2) 8

3

C

(3)₃8 _{(4) 21}

14. If one root of the equation _x2₊_{px 12 0}₊ _{= is 4, while the equation}_x2₊_{px q 0}_{+ =}

has equal roots, then the value of ‘q’ is (1)49

4 (2) 4

(3) 3 (4) 12

FIITJEE Ltd. ICES House, Sarvapriya Vihar (Near Hauz Khas Bus Term.), New Delhi - 16, Ph : 2686 5182, 26965626, 2685 4102, 26515949

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15. The coefficient of the middle term in the binomial expansion in powers of x of

(

)

4

1+ax and of

(

1- ax

)

6 is the same if α equals (1) 5 3 - (2) 3 5 (3) 3 10 -(4) 10 3 16. The coefficient of _{x in expansion of}n

(

_{1 x 1 x}₊

) (

_-

)

n_is

(1) (n – 1) (2)

( ) (

-1 1 nn -

)

(3)

( ) (

-1n 1- n 1-

)

2 (4)

( )

-1n 1- n 17. If n n n r 0 r 1 S C = =

å

and n n n r 0 r r t C = =

å

, then n n t S is equal to (1)1n 2 (2) 1_{n 1} 2 -(3) n – 1 (4) 2n 1 2

-18. Let T be the rth term of an A.P. whose first term is a and common difference is d. r

If for some positive integers m, n, m ≠ n, m n

1 1 T and T n m = = , then a – d equals (1) 0 (2) 1 (3) 1 mn (4) 1 1 m n+

19. The sum of the first n terms of the series ₁2_{+ × +}_{2 2}2 ₃2_{+ × +}_{2 4}2 ₅2_{+ × + is}_{2 6}2 _...

(

)

2

n n 1 2

+ _{when n is even. When n is odd the sum is}

(1)3n n 1

(

)

2 + (2) n n 12

(

)

2 + (3)

(

)

2 n n 1 4 + ₍₄₎ n n 1

(

)

2 2 é + ù ê ú ê ú ë û

20. The sum of series 1 1 1 ... 2! 4! 6!+ + + is (1)

(

)

2 e 1 2 -(2)

(

)

2 e 1 2e -(3)

(

)

2 e 1 2e -(4)

(

)

2 e 2 e

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-21. Let α, β be such that π < α - β < 3π. If sinα + sinβ = 21 65

- and cosα + cosβ = 27 65 - , then the value of cos

2 a - b is (1) 3 130 - ₍₂₎ 3 130 (3) 6 65 (4) 6 65

-22. If_u₌ _{a cos}2 2_q+_{b sin}2 2_{q +} _{a sin}2 2_q+_{b cos}2 2_{q , then the difference between the}

maximum and minimum values of _{u is given by}2

(1)2 a

(

2+b2

)

(2) _{2 a}2₊_b2

(3)

(

a b+

)

2 (4)

(

a b-

)

2

23. The sides of a triangle are sinα, cosα and 1 sin cos+ a a for some 0 < α < 2 p

. Then the greatest angle of the triangle is

(1) 60o _{(2) 90}o

(3)120o _{(4) 150}o

24. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60o_{and when he retires 40}

meter away from the tree the angle of elevation becomes 30o_{. The breadth of the}

river is

(1) 20 m (2) 30 m

(3) 40 m (4) 60 m

25. If f :R®S, defined by f(x) sinx= - 3 cos x 1+ , is onto, then the interval of S is

(1) [0, 3] (2) [-1, 1]

(3) [0, 1] (4) [-1, 3]

26. The graph of the function y = f(x) is symmetrical about the line x = 2, then (1) f(x + 2)= f(x – 2) (2) f(2 + x) = f(2 – x)

(3) f(x) = f(-x) (4) f(x) = - f(-x)

27. The domain of the function

(

)

1 2 sin x 3 f(x) 9 x - -= - is (1) [2, 3] (2) [2, 3) (3) [1, 2] (4) [1, 2) 28. If 2x 2 2 x a b lim 1 e x x ®¥ æ_{+ +} ö ₌ ç ÷

è ø , then the values of a and b, are

(1) a R, b RÎ Î (2) a = 1, b RÎ

(3) a R, b 2Î = (4) a = 1 and b = 2

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29. Letf(x) 1 tanx, x , x 0, 4x 4 2 - p é pù = ¹ _{Î ê} _ú - p ë û . If f(x) is continuous in 0, 2 , then f 4 p p é ù æ ö ç ÷ ê ú ë û è ø is (1) 1 (2) 1 2 (3) 1 2 - (4) -1 30. If _{y e}y ...to x e₌ + + ¥_{, x > 0, then}dy dx is (1) x 1 x+ (2) 1 x (3)1 x x -(4) 1 x x +

31. A point on the parabola _y2 ₌_18x_{at which the ordinate increases at twice the rate}

of the abscissa is (1) (2, 4) (2) (2, -4) (3) 9 9, 8 2 -æ ö ç ÷ è ø (4) 9 9 , 8 2 æ ö ç ÷ è ø

32. A function y = f(x) has a second order derivative f″(x) = 6(x – 1). If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x – 5, then the function is

2

33. The normal to the curve x = a(1 + cosθ), y = asinθ at ‘θ’ always passes through the fixed point

(1) (a, 0) (2) (0, a)

(3) (0, 0) (4) (a, a)

34. If 2a + 3b + 6c =0, then at least one root of the equation _ax2₊_{bx c 0}_{+ = lies in}

the interval (1) (0, 1) (2) (1, 2) (3) (2, 3) (4) (1, 3) 35. r n n n r 1 1 lim e n ®¥ =

å

is (1) e (2) e – 1 (3) 1 – e (4) e + 1 36. If sinx dx Ax Blogsin(x ) C sin(x- a) = + - a +

ò

, then value of (A, B) is

(1) (sinα, cosα) (2) (cosα, sinα) (3) (- sinα, cosα) (4) (- cosα, sinα)

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then the value of 2

1 I I is (1) 2 (2) –3 (3) –1 (4) 1

42. The area of the region bounded by the curves y = |x – 2|, x = 1, x = 3 and the x-axis is

(1) 1 (2) 2

(3) 3 (4) 4

43. The differential equation for the family of curves_x2₊_y2_- _{2ay 0}_{= , where a is an}

arbitrary constant is

(1)_2(x2_- _{y )y}2 _¢₌_xy ₍₂₎_2(x2₊_{y )y}2 _¢₌_xy

(3)_(x2_- _{y )y}2 _¢₌_2xy ₍₄₎_(x2₊_{y )y}2 _¢₌_2xy

44. The solution of the differential equation y dx + (x + x2_{y) dy = 0 is}

(1) 1 C xy

- = (2) 1 logy C

xy

- + =

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(3)_xy1 +logy C= (4) log y = Cx

45. Let A (2, –3) and B(–2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line

(1) 2x + 3y = 9 (2) 2x – 3y = 7

(3) 3x + 2y = 5 (4) 3x – 2y = 3

46. The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is –1 is

(1) x y 1 2 3+ =- and y x ₁ 2 1+ =-- (2) y x ₁ 2 3- =- and y x ₁ 2 1+ = -(3) x y 1 2 3+ = and y x ₁ 2 1+ = (4) y x ₁ 2 3- = and y x ₁ 2 1+ =

-47. If the sum of the slopes of the lines given by _x2_- _{2cxy 7y}_- 2 _{= is four times their}₀

product, then c has the value

(1) 1 (2) –1

(3) 2 (4) –2

48. If one of the lines given by _6x2_- _{xy 4cy}₊ 2 _{= is 3x + 4y = 0, then c equals}₀

(1) 1 (2) –1

(3) 3 (4) –3

49. If a circle passes through the point (a, b) and cuts the circle _x2₊_y2 ₌₄

orthogonally, then the locus of its centre is

(1)_{2ax 2by (a}₊ ₊ 2₊_b2_{+ =}_{4) 0} ₍₂₎_{2ax 2by (a}₊ _- 2₊_b2_{+ =}_{4) 0}

(3)_{2ax 2by (a}_- ₊ 2₊_b2_{+ =}_{4) 0} ₍₄₎_{2ax 2by (a}_- _- 2₊_b2_{+ =}_{4) 0}

50. A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is

(1)_{(x p)}_- 2 ₌_4qy ₍₂₎_{(x q)}_- 2 ₌_4py

(3)_{(y p)}_- 2 ₌_4qx ₍₄₎_{(y q)}_- 2 ₌_4px

51. If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of circumference 10π, then the equation of the circle is

(1)_x2₊_y2_- _{2x 2y 23 0}₊ _- ₌ ₍₂₎_x2₊_y2_- _{2x 2y 23 0}_- _- ₌

(3)_x2₊_y2₊_{2x 2y 23 0}₊ _- ₌ ₍₄₎_x2₊_y2₊_{2x 2y 23 0}_- _- ₌

52. The intercept on the line y = x by the circle _x2₊_y2_- _{2x 0}_{= is AB. Equation of the}

circle on AB as a diameter is

(1)_x2₊_y2_- _{x y 0}_- ₌ ₍₂₎_x2₊_y2_- _{x y 0}_{+ =}

(3)_x2₊_y2_{+ + =}_{x y 0} ₍₄₎_x2₊_y2_{+ -}_{x y 0}₌

53. If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas _y2 ₌_4ax_and_x2 ₌_4ay_{, then}

(31)

(1)_d2₊_{(2b 3c)}₊ 2 ₌₀ ₍₂₎_d2₊_{(3b 2c)}₊ 2₌₀

(3)_d2₊_{(2b 3c)}_- 2 ₌₀ ₍₄₎_d2₊_{(3b 2c)}_- 2 ₌₀

54. The eccentricity of an ellipse, with its centre at the origin, is1

2. If one of the directrices is x = 4, then the equation of the ellipse is

(1)_3x2₊_4y2 ₌₁ ₍₂₎_3x2₊_4y2 ₌₁₂

(3)_4x2₊_3y2 ₌₁₂ ₍₄₎_4x2₊_3y2 ₌₁

55. A line makes the same angle θ, with each of the x and z axis. If the angle β, which it makes with y-axis, is such that_sin2_{b =}_3sin2_{q, then}_{cos q equals}2

(1)2 3 (2) 1 5 (3)3 5 (4) 2 5

56. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is (1)3 2 (2) 5 2 (3)7 2 (4) 9 2

57. A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y

+ a = z and

x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by

(1) (3a, 3a, 3a), (a, a, a) (2) (3a, 2a, 3a), (a, a, a) (3) (3a, 2a, 3a), (a, a, 2a) (4) (2a, 3a, 3a), (2a, a, a)

58. If the straight lines x = 1 + s, y = –3 – λs, z = 1 + λs and x = t

2, y = 1 + t, z = 2 – t with parameters s and t respectively, are co-planar then λ equals

(1) –2 (2) –1

(3) –1

2 (4) 0

59. The intersection of the spheres _x2₊_y2₊_z2₊_{7x 2y z 13}_- _- ₌ _and

2 2 2

x +y +z - 3x 3y 4z 8+ + = is the same as the intersection of one of the sphere and the plane

(1) x – y – z = 1 (2) x – 2y – z = 1

(3) x – y – 2z = 1 (4) 2x – y – z = 1

60. Let a, br r and cr be three non-zero vectors such that no two of these are collinear. If the vector a 2br+ r is collinear with cr and b 3cr+ r is collinear with ar (λ being some non-zero scalar) then a 2b 6cr+ r+ r equals

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(1) alr (2) bl r

(3) clr (4) 0

61. A particle is acted upon by constant forces ˆ ˆ4i j 3k+ - ˆ and ˆ ˆ ˆ3i j k+ - which displace it from a point ˆi 2j 3k+ +ˆ ˆ to the point ˆ5i 4j k+ + . The work done in standard units ˆ ˆ by the forces is given by

(1) 40 (2) 30

(3) 25 (4) 15

62. If a, b, c are non-coplanar vectors and λ is a real number, then the vectors a 2b 3c, b 4c+ + l + and (2l -1)c are non-coplanar for

(1) all values of λ (2) all except one value of λ (3) all except two values of λ (4) no value of λ

63. Let u, v, w be such that u 1, v= =2, w = . If the projection v along u is equal 3 to that of w along u and v, w are perpendicular to each other then u v w- + equals

(1) 2 (2) 7

(3) 14 (4) 14

64. Let a, b and c be non-zero vectors such that(a b) c 1b c a 3

´ ´ = . If θ is the acute angle between the vectors b and c , then sin θ equals

(1)1

3 (2) 23

(3)2

3 (4) 2 23

65. Consider the following statements:

(a) Mode can be computed from histogram (b) Median is not independent of change of scale

(c) Variance is independent of change of origin and scale. Which of these is/are correct?

(1) only (a) (2) only (b)

(3) only (a) and (b) (4) (a), (b) and (c)

66. In a series of 2n observations, half of them equal a and remaining half equal –a. If the standard deviation of the observations is 2, then |a| equals

(1)1

n (2) 2

(3) 2 (4) 2

n

67. The probability that A speaks truth is4

5, while this probability for B is 3

4. The probability that they contradict each other when asked to speak on a fact is

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(1) 3 20 (2) 1 5 (3) 7 20 (4) 4 5

68. A random variable X has the probability distribution:

X: 1 2 3 4 5 6 7 8

p(X) :

0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

For the events E = {X is a prime number} and F = {X < 4}, the probability P (E ∪ F) is

(1) 0.87 (2) 0.77

(3) 0.35 (4) 0.50

69. The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is

(1) 37 256 (2) 219 256 (3)128 256 (4) 28 256

70. With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then the forces are (1) (2+ 2)N and (2- 2)N (2) (2+ 3)N and (2- 3)N (3) 2 1 2 N and 2 1 2 N 2 2 æ₊ ö æ_- ö ç ÷ ç ÷ è ø è ø (4) 1 1 2 3 N and 2 3 N 2 2 æ₊ ö æ_- ö ç ÷ ç ÷ è ø è ø

71. In a right angle ∆ABC, ∠A = 90° and sides a, b, c are respectively, 5 cm, 4 cm and 3 cm. If a force Fr has moments 0, 9 and 16 in N cm. units respectively about vertices A, B and C, then magnitude of Fr is

(1) 3 (2) 4

(3) 5 (4) 9

72. Three forces P, Q and Rr r r acting along IA, IB and IC, where I is the incentre of a ∆ ABC, are in equilibrium. Then P : Q :Rr r r is

(1)cosA : cos : cosB C

2 2 2 (2)

A B C

sin : sin : sin

2 2 2

(3)secA : sec : secB C

2 2 2 (4)

A B C

cosec : cosec : cosec

2 2 2

73. A particle moves towards east from a point A to a point B at the rate of 4 km/h and then towards north from B to C at the rate of 5 km/h. If AB = 12 km and BC = 5 km, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively

(1) 17 4 km/h and 13 4 km/h (2) 13 4 km/h and 17 4 km/h

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(3) 17 9 km/h and 13 9 km/h (4) 13 9 km/h and 17 9 km/h 74. A velocity 1

4 m/s is resolved into two components along OA and OB making angles 30° and 45° respectively with the given velocity. Then the component along OB is (1) 1 8m/s (2) 1 ( 3 1) 4 - m/s (3) 1 4m/s (4) 1_{( 6} ₂₎ 8 - m/s

75. If t1 and t2 are the times of flight of two particles having the same initial velocity

u and range R on the horizontal, then 2 2

1 2 t + is equal tot (1) 2 u g (2) 2 2 4u g (3) 2 u 2g (4) 1

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FIITJEE

AIEEE

−

2004 (MATHEMATICS)

ANSWERS

1. 3 16. 2 31. 4 46. 4 61. 1 2. 1 17. 1 32. 2 47. 3 62. 3 3. 3 18. 1 33. 1 48. 4 63. 3 4. 2 19. 2 34. 1 49. 2 64. 4 5. 4 20. 2 35. 2 50. 1 65. 3 6. 2 21. 1 36. 2 51. 1 66. 3 7. 2 22. 4 37. 4 52. 1 67. 3 8. 1 23. 3 38. 1 53. 1 68. 2 9. 4 24. 1 39. 3 54. 2 69. 4 10. 3 25. 4 40. 2 55. 3 70. 3 11. 4 26. 2 41. 1 56. 3 71. 3 12. 3 27. 2 42. 1 57. 2 72. 1 13. 4 28. 2 43. 3 58. 1 73. 1 14. 1 29. 3 44. 2 59. 4 74. 4 15. 3 30. 3 45. 1 60. 4 75. 2

(36)

FIITJEE

AIEEE

−

2004 (MATHEMATICS)

) (

2_-₁

)

₌_z4₊_{2 z}2₊₁ 2 2 z z 2zz 0 z z 0 Þ + + = Þ + =

⇒ R (z) = 0 ⇒ z lies on the imaginary axis.

6. A.A = 1 0 0 0 1 0 I 0 0 1 é ù ê _ú= ê ú ê ú ë û . 7. AB = I Þ A(10 B) = 10 I 1 1 1 4 2 2 10 0 5 1 0 0 2 1 3 5 0 0 10 5 10 0 1 0 1 1 1 1 2 3 0 0 5 0 0 1 - - a é ù é ù é ù é ù ê ú ê ú ê ú ê ú Þ _ê - _{ú ê}- a =_{ú ê} a - _ú= _ê _ú ê ú ê - ú ê +aú ê ú ë û ë û ë û ë û ifa = .5 8. n n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8

loga loga loga loga loga loga loga loga loga

+ + + + + + + + C3→ C3 – C2, C2 → C3 – C1 = n n 3 n 6

loga logr logr loga logr logr loga logr logr

+ +

= 0 (where r is a common ratio).

9. Let numbers be a, b Þ a b 18,+ = ab 4= Þ ab 16= , a and b are roots of the equation

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2

x 18x 16 0

) (

)

0 Þ - - + + =

(

Hence number of ways = 7_C

2 = 21.

14. Since 4 is one of the root of x2_{+ px + 12 = 0}_⇒_{16 + 4p + 12 = 0}_⇒_{p = –7}

and equation x2_{+ px + q = 0 has equal roots}

⇒ D = 49 – 4q = 0 ⇒ q =49 4 .

15. Coefficient of Middle term in

(

)

4 4 2

3 2

1+ax = =t C ×a Coefficient of Middle term in

(

)

6 6

( )

3

4 3 1- ax = =t C - a 4 2 6 3 2 3 C a =- C .a 6 20 3 10 -Þ - = a Þ a = 16. Coefficient of xn_{in (1 + x)(1 – x)}n_{= (1 + x)(}n_C 0 – nC1x + …….. + (–1)n –1nCn – 1 xn – 1 + (–1)n n_C n xn) = (–1)n n_C n + (–1)n –1nCn – 1

( ) (

)

n 1 1 n = - - _. 17.

(

1 T a m 1 d n = = + - ...(1)

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and T_n 1 a

(

n 1 d

)

m

= = + - ...(2) from (1) and (2) we get a 1 , d 1

mn mn

= =

Hence a – d = 0

19. If n is odd then (n – 1) is even ⇒ sum of odd terms

(

)

(

)

2 2 2 n 1 n n n 1 n 2 2 - + = + = . 20. e e 1 2 4 6 2 2! 4! 6! a ₊ - a _a _a _a = + + + +…….. 2 4 6 e e ₁ _... 2 2! 4! 6! a ₊ - a _a _a _a - = + + + put α = 1, we get

(

)

2 e 1 1 1 1 2e 2! 4! 6! -= + + +……….. 21. sin α + sin β = 21 65

- and cos α + cos β = 27 65 - . Squaring and adding, we get

2 + 2 cos (α – β) = 2 1170 (65) ⇒ _cos2 9 2 130 a - b æ _ö= ç ÷ è ø ⇒ 3 cos 2 130 a - b -æ _ö= ç ÷ è ø 3 2 2 2 p a - b p æ _< _< ö ç ÷ èQ ø.

22. _u₌ _{a cos}2 2_q+_{b sin}2 2_{q +} _{a sin}2 2_q+_{b cos}2 2_q

= a2 b2 a2 b2cos2 a2 b2 b2 a2cos2 2 2 2 2 + ₊ - _{q +} + ₊ - _q ⇒ 2 2 2 2 2 2 2 2 2 a b a b 2 u a b 2 cos 2 2 2 æ + ö æ - ö = + + _ç _{÷ ç}- _÷ q è ø è ø

min value of _u2 ₌_a2₊_b2₊_2ab

max value of u2 =2 a

(

2+b2

)

⇒ ₂ ₂

₍

₎

2

max min

u - u = -a b .

23. Greatest side is 1 sin cos+ a a , by applying cos rule we get greatest angle = 120ο_.

24. tan30° = h 40 b+ ⇒ 3h 40 b= + … ..(1) tan60° = h/b ⇒ h = 3 b ….(2) ⇒ b = 20 m h b 40 30° 60°

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25. - £2 sinx- 3 cos x 2£ ⇒ 1 sinx- £ - 3 cos x 1 3+ £ ⇒ range of f(x) is [–1, 3].

Hence S is [–1, 3].

26. If y = f (x) is symmetric about the line x = 2 then f(2 + x) = f(2 – x). 27. _{9 x}_- 2 _{> and 1 x 3 1}₀ _{- £ -} _£_⇒_{x [2, 3)}_Î 28. 2 2 1 a b 2x 2x a b _{x x} _2a x x 2 2 x x a b a b lim 1 lim 1 e a 1, b R x x x x æ ö ç ÷ _æ _ö ç ÷´ ´_ç + _÷ ç ₊ ÷ è ø ç ÷ è ø ®¥ ®¥ æ_{+ +} ö ₌ æ_{+ +} ö ₌ _Þ ₌ _Î ç ÷ ç ÷ è ø è ø 29. x 4 1 tanx 1 tanx 1 f(x) lim 4x p 4x 2 ® - -= Þ =-- p - p 30. _{y e}y ... y e x e₌ + + + ¥ ⇒ x = _ey x+ ⇒ lnx – x = y ⇒dy 1 1 1 x dx x x -= - -= . 31. Any point be 9t , 9t2 2 æ ö ç ÷ è ø; differentiating y 2_{= 18x} ⇒dy 9 1 2 (given) t 1 dx= = =y t Þ = .2 ⇒ Point is 9 9, 8 2 æ ö ç ÷ è ø 32. f″ (x) = 6(x – 1) ⇒ f′ (x) = 3(x – 1)2_{+ c} and f′ (2) = 3 ⇒ c = 0 ⇒ f (x) = (x – 1)3_{+ k and f (2) = 1}_⇒_{k = 0} ⇒ f (x) = (x – 1)3_. 33. Eliminating θ, we get (x – a)2_{+ y}2_{= a}2_.

Hence normal always pass through (a, 0).

34. Let f′(x) = _ax2₊_{bx c}₊ _⇒_{f(x) =}ax3 bx2 _{cx d}

3 + 2 + +

⇒_f(x) 1

(

_2ax3 _3bx2 _{6cx 6d}

)

6

= + + + , Now f(1) = f(0) = d, then according to Rolle’s theorem

⇒ f′(x) = _ax2₊_{bx c 0}_{+ = has at least one root in (0, 1)}

35. r n n n r 1 1 lim e n ®¥ =

å

= 1 x 0 e dx (e 1)=

-ò

36. Put x – α = t

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⇒ sin( t)dt sin cottdt cos dt sint

a +

= a + a

ò

= cosa

(

x- a +

)

sin ln sint ca + A = cos , B sina = a 37. dx cos x sinx

-ò

1 1 _dx 2 cos x 4 = p æ₊ ö ç ÷ è ø

ò

1 sec x dx 4 2 p æ ö = _ç + _÷ è ø

ò

= 1 log tan x 3 C 2 8 2 p æ ₊ ö₊ ç ÷ è ø 38.

(

)

(

)

(

)

1 1 3 2 2 2 2 1 1 x 1 dx 1 x dx x 1 dx -- -- + - +

-ò

-ò

= I2 ⇒ I2 / I1 = 2. 42. Area = 2 3 1 2 (2 x)dx- + (x 2)dx

-ò

ò

= 1. y=2 – x y = x – 2 1 2 3 43. 2x + 2yy′ - 2ay′ = 0 a = x yy+_y_{¢ (eliminating a)}¢ ⇒ (x2_{– y}2_)y_′_{= 2xy.} 45. y dx + x dy + x2_{y dy = 0.}

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2 2 d(xy) 1 dy 0 y x y + = ⇒ 1 logy C xy - + _{= .}

45. If C be (h, k) then centroid is (h/3, (k – 2)/3) it lies on 2x + 3y = 1. ⇒ locus is 2x + 3y = 9. 46. x y 1 a b+ = where a + b = -1 and 4 3 1 a b+ = ⇒ a = 2, b = -3 or a = -2, b = 1. Hencex y 1and x y 1 2 3- = -2 1+ = . 47. m1 + m2 = 2c 7 - and m1 m2 = 1 7 -m1 + m2 = 4m1m2 (given) ⇒ c = 2. 48. m1 + m2 = 1 4c, m1m2 = 6 4c and m1 = 3 4 - . Hence c = -3.

49. Let the circle be x2_{+ y}2_{+ 2gx + 2fy + c = 0}_⇒_{c = 4 and it passes through (a, b)}

⇒ a2_{+ b}2_{+ 2ga + 2fb + 4 = 0.}

Hence locus of the centre is 2ax + 2by – (a2_{+ b}2_{+ 4) = 0.}

50. Let the other end of diameter is (h, k) then equation of circle is (x – h)(x – p) + (y – k)(y – q) = 0

Put y = 0, since x-axis touches the circle

⇒ x2_{– (h + p)x + (hp + kq) = 0}_⇒_{(h + p)}2_{= 4(hp + kq)} _{(D = 0)}

⇒ (x – p)2_{= 4qy.}

51. Intersection of given lines is the centre of the circle i.e. (1, − 1) Circumference = 10π⇒ radius r = 5

⇒ equation of circle is x2_{+ y}2 ₋_{2x + 2y}₋_{23 = 0.}

52. Points of intersection of line y = x with x2_{+ y}2₋_{2x = 0 are (0, 0) and (1, 1)}

hence equation of circle having end points of diameter (0, 0) and (1, 1) is x2_{+ y}2₋_x₋_{y = 0.}

53. Points of intersection of given parabolas are (0, 0) and (4a, 4a) ⇒ equation of line passing through these points is y = x

On comparing this line with the given line 2bx + 3cy + 4d = 0, we get d = 0 and 2b + 3c = 0 ⇒ (2b + 3c)2_{+ d}2_{= 0.}

54. Equation of directrix is x = a/e = 4 ⇒ a = 2 b2_{= a}2₍₁₋_e2₎_⇒_b2_{= 3}

Hence equation of ellipse is 3x2_{+ 4y}2_{= 12.}

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55. l = cos θ, m = cos θ, n = cos β

cos2_θ_{+ cos}2_θ_{+ cos}2_β_{= 1}_⇒_{2 cos}2 _θ_{= sin}2 _β_{= 3 sin}2_θ _(given)

cos2_θ_{= 3/5.}

56. Given planes are

2x + y + 2z − 8 = 0, 4x + 2y + 4z + 5 = 0 ⇒ 2x + y + 2z + 5/2 = 0 Distance between planes = | d₂1 ₂d |2 ₂

a b c -+ + = 2 2 2 | 8 5/ 2| 7 2 2 1 2 - - ₌ + + .

57. Any point on the line x y a z t₁ (say)

1 1 1

+

= = = is (t1, t1 – a, t1) and any point on

the line 2

(

)

y x a z _t _say 2 1 1 + _{= = =} is (2t2 – a, t2, t2).

Now direction cosine of the lines intersecting the above lines is proportional to (2t2 – a – t1, t2 – t1 + a, t2 – t1).

Hence 2t2 – a – t1 = 2k , t2 – t1 + a = k and t2 – t1 = 2k

On solving these, we get t1 = 3a , t2 = a.

Hence points are (3a, 2a, 3a) and (a, a, a).

58. Given lines x 1 y 3 z 1 s and x y 1 z 2 t

1 1/ 2 1 1

+

-- -

-= = = = = =

- l l - are coplanar then plan

passing through these lines has normal perpendicular to these lines ⇒ a - bλ + cλ = 0 and a b c 0

2+ - = (where a, b, c are direction ratios of the normal to the plan)

=0 r r ⇒ 1+ 2t2 = 0 ⇒ t2 = -1/2 & t1 = -6.

Since a and cr r are non-collinear.

Putting the value of t1 and t2 in (1) and (2), we get a 2b 6c 0+ + =

r r

r r _.

61. Work done by the forcesF and F is (Fr₁ r₂ r₁+F ) dr₂ ×r, where dr is displacement According to question Fr₁+Fr₂= ˆ ˆ(4i j 3k) (3i j k) 7i 2j 4k+ - ˆ + ˆ ˆ ˆ+ - = + -ˆ ˆ ˆ

andrd (5i 4j k) (i 2j 3k) 4i 2j 2k= ˆ+ + -ˆ ˆ ˆ+ +ˆ ˆ = + -ˆ ˆ ˆ. Hence (Fr₁+F ) dr₂ ×r is 40.

63. Condition for given three vectors to be coplanar is

1 2 3 0 4 0 0 2 1 l l -= 0 ⇒ λ = 0, 1/2.

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Hence given vectors will be non coplanar for all real values of λ except 0, 1/2.

0 3 + × = ⇒ b c 1 cos 0 3 æ ₊ _{q =}ö ç ÷ è ø ⇒ cosθ = –1/3 ⇒ sinθ = 2 2 3 .

65. Mode can be computed from histogram and median is dependent on the scale. Hence statement (a) and (b) are correct.

66. xi =a for i 1, 2, .... ,n= and xi =-a for i n, ...., 2n=

S.D. =

(

)

æ _°+ ö æ _°+ ö æ _°+ ö

ç ÷ ç ÷ ç ÷

è ø è ø è ø

⇒cosA : cos : cosB C

2 2 2. A B C 90+A/2 90+B/2 90+C/2 73. Time T1 from A to B = 12 4 = 3 hrs. T2 from B to C = 5 5 = 1 hrs. Total time = 4 hrs. Average speed = 17 4 km/ hr. Resultant average velocity = 13

4 km/hr. B A C 12 5 13 74. Component along OB =

₍

₎

1_sin30 1 4 ₆ ₂ sin(45 30 ) 8 ° = -°+ ° m/s. 75. t1 = 2usin g a , t2 = 2usin g b where α + β = 900 ∴ 2 2 2 1 2 2 4u t t g + = .

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FIITJEE

SOLUTION TO AIEEE-2005

MATHEMATICS

1. If A2_{– A + I = 0, then the inverse of A is}

(1) A + I (2) A

(3) A – I (4) I – A

1. (4)

Given A2_{– A + I = 0}

A–1A2 – A–1A + A–1 – I = A–1⋅0 (Multiplying A–1_{on both sides)}

⇒ A - I + A-1_{= 0 or A}–1_{= I – A.}

2. If the cube roots of unity are 1, ω, ω2_{then the roots of the equation}

(x – 1)3_{+ 8 = 0, are} (1) -1 , - 1 + 2ω, - 1 - 2ω2 _{(2) -1 , -1, - 1} (3) -1 , 1 - 2ω, 1 - 2ω2 _{(4) -1 , 1 + 2ω, 1 + 2ω}2 2. (3) (x – 1)3_{+ 8 = 0 ⇒ (x – 1) = (-2) (1)}1/3 ⇒ x – 1 = -2 or -2ω or -2ω2 or n = -1 or 1 – 2ω or 1 – 2ω2_. 3. Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12} be a relation on the set A = {3, 6, 9, 12}. The relation is (1) reflexive and transitive only (2) reflexive only

(3) an equivalence relation (4) reflexive and symmetric only 3. (1)

Reflexive and transitive only.

e.g. (3, 3), (6, 6), (9, 9), (12, 12) [Reflexive] (3, 6), (6, 12), (3, 12) [Transitive].

4. Area of the greatest rectangle that can be inscribed in the ellipse

2 2 2 2 x y 1 a +b = is (1) 2ab (2) ab (3) ab (4) a b 4. (1)

Area of rectangle ABCD = (2acosθ) (2bsinθ) = 2absin2θ

⇒ Area of greatest rectangle is equal to 2ab

when sin2θ = 1.

(-acosθ, bsinθ) B

(-acosθ, -bsinθ)C D(acosθ, -bsinθ) A(acosθ, bsinθ)

X Y

5. The differential equation representing the family of curves y2₌_{2c x}

(

₊ _c

)

_{, where c}

> 0, is a parameter, is of order and degree as follows:

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(3) order 1, degree 3 (4) order 2, degree 2 5. (3)

y2_{= 2c(x + √c)} _…(i)

2yy′ = 2c⋅1 or yy′ = c …(ii)

⇒ y2_{= 2yy′ (x + yy′ )} _{[on putting value of c from (ii) in (i)]}

On simplifying, we get

(y – 2xy′)2_{= 4yy′}3 _…(iii)

Hence equation (iii) is of order 1 and degree 3.

6. 2 2 2

2 2 2 2 2

n

1 1 2 4 1

lim sec sec .... sec 1

n n n n n →∞  ₊ ₊ ₊     equals (1) 1sec1 2 (2) 1 cos ec1 2 (3) tan1 (4) 1tan1 2 6. (4) 2 2 2 2 2 2 2 2 2 2 n 1 1 2 4 3 9 1

lim sec sec sec .... sec 1

n n n n n n n →∞  ₊ ₊ ₊ ₊     is equal to 2 2 2 2 2 2 2 n n r r 1 r r

lim sec lim sec

n n

n n n

→∞ = →∞ ⋅

(1) sinA : sin B : sinC (2) sinA : sinB: sinC

2 2 2

(3)cosA : cosB : cosC

2 2 2 (4) cosA : cosB : cosC

7. (3)

Using Lami’s Theorem

∴P : Q : R cosA: cos : cosB C

2 2 2 = . A B C I P Q R

8. If in a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

(1) 22.0 (2) 20.5

(3) 25.5 (4) 24.0

8. (4)

Mode + 2Mean = 3 Median