**JEE MAIN (AIEEE) Past Exam Paper-2003**

**Mathematics**

1. A function f from the set of natural numbers to integers defined by is

(1) one-one but not onto

(2) onto but not one-one

(3) one-one and onto both

(4) neither one-one nor onto

2. Let z_{1} and z_{2} be two roots of the equation z_{2} + az + b = 0, z being complex. Further, assume that the origin, z_{1} and z_{2} form an equilateral triangle. Then

(1) a^{2} = b

(2) a^{2} = 2b

(3) a^{2} = 3b

(4) a^{2} = 4b

3. If z and ω are two non-zero complex numbers such that |zω| = 1, and arg(z) – arg(ω) = π/2, then is equal to

(1) 1

(2) −1

(3) i

(4) −i

4.

(1) x = 4 n, where n is any positive integer

(2) x = 2n, where n is any positive integer

(3) x = 4n + 1, where n is any positive integer

(4) x = 2n + 1, where n is any positive integer

5. If and vectors (1, a, a^{2}), (1, b, b^{2}) and (1, c, c^{2}) are non-coplanar, then the product abc equals

(1) 2

(2) −1

(3) 1

(4) 0

6. If the system of linear equations

x + 2 ay + az = 0,

x + 3 by + bz = 0

and x + 4 cy + cz = 0

has a non-zero solution, then a, b, c

(1) are in AP

(2) are in GP

(3) are in HP

(4) satisfy a + 2b + 3c = 0

7. If the sum of the roots of the quadratic equation ax^{2} + bx + c = 0 is equal to the sum of the squares of their reciprocals, then a/c, b/a and c/b are in

(1) AP

(2) GP

(3) HP

(4) arithmetic-geometric progression

8. The number of the real solutions of the equation x^{2} – 3|x| + 2 = 0 is

(1) 2

(2) 4

(3) 1

(4) 3

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

(a^{2} – 5a + 3)x^{2} + (3a – 1)x + 2 = 0 is twice as large as the other, is

(1) 2/3

(2) −2/3

(3) 1/3

(4) −1/3

10. If then

(1) α^{2} = a^{2} + b^{2}, β = ab

(2) α = a^{2} + b^{2}, β = 2ab

(3) α = a^{2} + b^{2}, β = a^{2} – b^{2}

(4) 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

(1) 140

(2) 196

(3) 280

(4) 346

12. The number of ways in which 6 men and 5 women can dine at a round table, if not tow women are to sit together, is given by

(1) 6! × 5!

(2) 30

(3) 5! × 4!

(4) 7! × 5!

13. If 1, ω, ω^{2} are the cube roots of unity, then is equal to

(1) 0

(2) 1

(3) ω

(4) ω^{2}

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}+ ^{n}C_{r−1}+ 2 × ^{n}C_{r} equals

(1) ^{n+2}C_{r}

(2) ^{n+2}C_{r+1}

(3) ^{n+1}C_{r}

(4) ^{n+1}C_{r+1}

15. The number of integral terms in the expansion of is

(1) 32

(2) 33

(3) 34

(4) 35

16. If x is positive, the first negative term in the expansion of (1 + x)^{27/5} is

(1) 7th term

(2) 5th term

(3) 8th term

(4) 6th term

17. The sum of the series upto ∞ is equal to

(1) 2 log_{e} 2

(2) log_{e} 2 – 1

(3) log_{e} 2

(4) log_{e} (4/e)

18. Let f(x) be a polynomial function of second degree. If f(1) = f(−1) and a, b, c are in AP, then f'(a), f'(b) and f'(c) are in

(1) AP

(2) GP

(3) HP

(4) arithmetic-geometric progression

19. If x_{1}, x_{2}, x_{3} and y_{1}, y_{2}, y_{3} are both in GP with the same common ratio, then the points (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3})

(1) lie on a straight line

(2) lie on an ellipse

(3) lie on a circle

(4) 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

(1) a cot (π/n)

(2) (a/2) cot (π/2n)

(3) a cot (π/2n)

(4) (a/4) cot (π/2n)

21. If in a ∆ABC then the sides a, b and c

(1) are in AP

(2) are in GP

(3) are in HP

(4) satisfy a + b = c

22. In a ∆ABC, medians AD and BE are drawn. If AD = 4, ∠DAB = π/6 and ∠ABE = π/3, then the area of the ∆ ABC is

(1) 8/3

(2) 16/3

(3) 32/3√3

(4) 64/3

23. The trigonometric equation sin^{−1}x = 2sin^{−1} a, has a solution for

(1)

(2) all real values of a

(3)

(4)

24. The upper 3/4th portion of a vertical pole subtends an angle tan^{−1} (3/5) at a 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

(1) 20 m

(2) 40 m

(3) 60 m

(4) 80 m

25. The real number x when added to its inverse gives the minimum sum at x equals

(1) 2

(2) 1

(3) −1

(4) −2

26. If f:R→R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then is

(1) 7n/2

(2)

(3) 7n(n+1)

(4)

27. If f(x) = x^{n}, then the value of is

(1) 2^{n}

(2) 2^{n – 1 }

(3) 0

(4) 1

28. Domain of definition of the function is

(1) (1, 2)

(2) (−1, 0) ∪ (1, 2)

(3) (1, 2) ∪ (2, ∞)

(4) (−1, 0) ∪ (1, 2) ∪ (2, ∞)

29.

(1) 1/8

(2) 0

(3) 1/32

(4) ∞

30. If then value of k is

(1) 0

(2) −1/3

(3) 2/3

(4) −2/3

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

then the value of k is equal to

(1) 4

(2) 2

(3) 1

(4) 0

32. The function is

(1) an even function

(2) an odd function

(3) a periodic function

(4) neither an even nor an odd function

33.

(1) continuous as well as differentiable for all x

(2) continuous for all x but not differentiable at x = 0

(3) neither differentiable nor continuous at x = 0

(4) discontinuous everywhere

34. If the function f(x) = 2x^{3} – 9ax^{2} + 12a^{2}x + 1, where a > 0, attains its maximum and minimum at p and q respectively such that p^{2} = q, then a equals

(1) 3

(2) 1

(3) 2

(4) 1/2

35. If f(y) = e^{y}, g(y) = y; y > 0 and then

(1) F(t) = 1 – e^{−t}(1 + t)

(2) F(t) = e^{t} – (1 + t)

(3) F(t) = te^{t}

(4) F(t) = te^{−t}

36. If f(a + b – x) = f(x), then is equal to

(1)

(2)

(3)

(4)

37. The value of is

(1) 3

(2) 2

(3) 1

(4) −1

38. The value of the integral is

(1)

(2)

(3)

(4)

39. is

(1) 1/30

(2) 0

(3) 1/4

(4) 1/5

40. Let

If then one of the possible values of k, is

(1) 15

(2) 16

(3) 63

(4) 64

41. The area of the region bounded by the curves y = |x – 1| and y = 3 – |x| is

(1) 2 sq unit

(2) 3 sq unit

(3) 4 sq unit

(4) 6 sq unit

42. Let f(x) be function satisfying f'(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x^{2}. Then, the value of the integral is

(1)

(2)

(3)

(4)

43. The degree and order of the differential equation of the family of all parabolas whose axis is x-axis, are respectively

(1) 2, 1

(2) 1, 2

(3) 3, 2

(4) 2, 3

44. The solution of the differential equation is

(1)

(2)

(3)

(4)

45. If the equation of the locus of a point equidistant from the points (a_{1}, b_{1}) and (a_{2}, b_{2}) is (a_{1} – a_{2}) x + (b_{1} – b_{2})y + c = 0, then the value of ‘c’ is

(1)

(2)

(3)

(4)

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 parameter, is

(1) (3x – 1)^{2} + (3y)^{2} = a^{2} – b^{2}

(2) (3x – 1)^{2} + (3y)^{2} = a^{2} + b^{2}

(3) (3x + 1)^{2} + (3y)^{2} = a^{2} + b^{2}

(4) (3x + 1)^{2} + (3y)^{2} = a^{2} – b^{2}

47. If the pair straight lines x^{2} – 2pxy – y^{2} = 0 and x^{2} – 2qxy – y^{2} = 0 be such that each pair bisects the angle between the other pair, then

(1) p = q

(2) p = −q

(3) pq = 1

(4) 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 < π/4) with the positive direction of x-axis. The equation of its diagonal not passing through the origin is

(1) y(cos α – sin α) – x(sin α – cos α) = a

(2) y(cos α + sin α) + x( sin α – cos α) = a

(3) y(cos α + sin α) + x(sin α + cos α) = a

(4) 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} + y2 – 8x + 2y + 8 = 0 intersect in two distinct points, then

(1) 2 < r < 8

(2) r < 2

(3) r = 2

(4) r > 2

50. The lines 2x – 3y = 5 and 3x – 4y = 7 are diameters of a circles having area as 154 sq unit. Then, the equation of the circle is

(1) x^{2} + y^{2}+ 2x – 2y = 62

(2) x^{2} + y^{2} + 2x – 2y = 47

(3) x^{2} + y^{2} − 2x + 2y = 47

(4) x^{2} + y^{2} − 2x + 2y = 62

51. The normal at the point on a parabola meets the parabola again in the point then

(1)

(2)

(3)

(4)

52. The foci of the ellipse and the hyperbola Then, the value of b^{2} is

(1) 1

(2) 5

(3) 7

(4) 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

(1) cos^{−1}(19/35)

(2) cos^{−1}(17/31)

(3) 30°

(4) 90°

54. The radius of the circle in which the sphere x^{2} + y^{2} + z^{2} + 2x – 2y – 4z – 19 = 0 is cut be the plane x + 2y + 2z + 7 = 0, is

(1) 1

(2) 2

(3) 3

(4) 4

55. The lines are coplanar, if

(1) k = 0 or −1

(2) k = 1 or −1

(3) k = 0 or −3

(4) 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

(1) aa’ + bb’ + cc’ + 1 = 0

(2) aa’ + bb’ + cc’ = 0

(3) (a + a’)(b + b’) + (c + c’) = 0

(4) aa’ + cc’ + 1 = 0

57. The shortest distance from the plane 12x + 4y + 3z = 327 to the sphere x^{2} + y^{2} + z^{2} + 4x – 2y – 6z = 155 is

(1) 26

(2) 11 (4/13)

(3) 13

(4) 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

(1)

(2)

(3)

(4)

59. are three vectors, such that then is equal to

(1) 0

(2) −7

(3) 7

(4) 1

60. If are three non-coplanar vectors, then

equals

(1) 0

(2)

(3)

(4)

61. Consider points A, B, C and D with position vectors and Then, ABCD is a

(1) square

(2) rhombus

(3) rectangle

(4) √17

62. The vectors are the sides of a ∆ABC. The length of the median through A is

(1) √18

(2) √72

(3) √33

(4) √288

63. A particle acted on by constant forces is displaced from the point to the point The total work done by the forces is

(1) 20 unit

(2) 30 unit

(3) 40 unit

(4) 50 unit

64. Let If is a unit vector such that then is equal to

(1) 0

(2) 1

(3) 2

(4) 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

(1) is increased by 2

(2) is decreased by 2

(3) is two times the original median

(4) remains the same as that of the original set

66. In an experiment with 15 observations on x, the following results were available

∑ x^{2} = 2830, ∑ x = 170.

One observations that was 20, was found to be wrong and was replaced by the correct value 30. Then, the corrected variance is

(1) 78.00

(2) 188.66

(3) 177.33

(4) 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

(1) 4/5

(2) 3/5

(3) 1/5

(4) 2/5

68. Events, A, B, C are mutually exclusive events such that The set of possible values of x are in the interval

(1) [1/3, 1/2]

(2) [1/3, 2/3]

(3) [1/3, 13/3]

(4) [0, 1]

69. The mean and variance of a random variable X having a binomial distribution are 4 and 2 respectively, then P(X = 1) is

(1) 1/32

(2) 1/16

(3) 1/8

(4) 1/4

70. The resultant of forces is doubled, then is doubled. If the directions of is again doubled, then P^{2} : Q^{2} : R^{2} is

(1) 3 : 1 : 1

(2) 2 : 3 : 2

(3) 1 : 2 : 3

(4) 2 : 3 : 1

71. Let R_{1} and R_{2} respectively be the maximum ranges up and down an inclined plane and R be the maximum range on the horizontal plane. Then, R_{1}, R, R_{2} are in

(1) AGP

(2) AP

(3) GP

(4) HP

72. A couple is of moment and the force forming the couple is . If is turned through a right angle, the moment of the couple thus formed is If instead, the forces is turned through an angle α, then the moment of couple becomes

(1)

(2)

(3)

(4)

73. Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity and the other from rest with uniform acceleration Let α be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time

(1)

(2)

(3) u sin α

(4)

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

(1)

(2)

(3)

(4)

75. A body travels a distance 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

(1)

(2)

(3)

(4)

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