**JEE MAIN (AIEEE) Past Exam Paper-2006**

**Mathematics**

1. If the roots of the quadratic equation x^{2} + px + q = 0 are tan 30° and tan 15° respectively, then the value of 2 + q – p is

(1) 3

(2) 0

(3) 1

(4) 2

2. The value of the integral is

(1) 3/2

(2) 2

(3) 1

(4) 1/2

3. Let W denote the words in the English dictionary. Define the relation R by R = {(x, y) ∈ W × W| the words x and y have at least one letter in common.}. Then, R is

(1) reflexive, symmetric and not transitive

(2) reflexive, symmetric and transitive

(3) reflexive, not symmetric and transitive

(4) not reflexive, symmetric and transitive

4. The number of values of x in the interval [0, 3π] satisfying the equation 2sin^{2} x + 5 sin x – 3 = 0 is

(1) 6

(2) 1

(3) 2

(4) 4

5. If A and B are square matrices of size n × n such that A^{2} – B^{2} = (A – B)(A + B), then which of the following will be always true

(1) AB = BA

(2) either of A or B is a zero matrix

(3) either of A or B is an identity matrix

(4) A = B

6. The value of is

(1) 1

(2) −1

(3) −i

(4) i

7. If where are any three vectors such that are

(1) inclined at an angle of π/6 between them

(2) perpendicular

(3) parallel

(4) inclined at angle of π/3 between them

8. All the values of m for which both roots of the equation x^{2} – 2mx + m^{2} – 1 = 0 are greater than −2 but less than 4 lie in the interval

(1) m < 3

(2) −1 < m < 3

(3) 1 < m < 4

(4) −2 < m < 0

9. ABC is triangle, right angled at A. the resultant of the forces acting along with magnitudes 1/AB and 1/AC respectively is the force along ,where D is the foot of the perpendicular from A onto BC. The magnitude of the resultant is

(1)

(2)

(3)

(4)

10. Suppose, a population A has 100 observations 101, 102, ….200 and another population B has 100 observations 151, 152, …..250. If V_{A} and V_{B} represent the variances of the two population respectively, then V_{A}/V_{B} is

(1) 9/4

(2) 4/9

(3) 2/3

(4) 1

11. is equal to

(1)

(2)

(3)

(4)

12. In an ellipse, the distances between its foci is 6 and minor axis is 8. Then, its eccentricity is

(1) 1/2

(2) 4/5

(3) 1/√5

(4) 3/5

13. The locus of the vertices of the family of parabolas is

(1) xy = ¾

(2) xy = 35/16

(3) xy = 64/105

(4) xy = 105/64

14. A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is

(1) 3x – 4y + 7 = 0

(2) 4x + 3y = 24

(3) 3x + 4y = 25

(4) x + y = 7

15. The value of a, for which the points A, B, C with position vectors respectively are the vertices of a right angled triangle with C = π/2 are

(1) −2 and −1

(2) −2 and 1

(3) 2 and −1

(4) 2 and 1

16.

(1)

(2)

(3)

(4)

17. The two lines x = ay + b, z = cy + d and x = a’y + b’, z = c’y + d’ are perpendicular to each other, if

(1) aa’ + cc’ = 1

(2)

(3)

(4) aa’ + cc’ = −1

18. At an election, a voter may vote for any number of candidates not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote, is

(1) 6210

(2) 385

(3) 1110

(4) 5040

19. If the expansion in powers of x of the function is a_{0} + a_{1}x + a_{2}x_{2} + a_{3}x_{3} + ……, then a_{n} is

(1)

(2)

(3)

(4)

20. For natural numbers m, n, if (1 – y)m (1 + y)n = 1 + a_{1}y + a_{2}y_{2} + …. and a_{1} = a_{2}= 10, then (m, n) is

(1) (35, 20)

(2) (45, 35)

(3) (35, 45)

(4) (20, 45)

21. A particle has two velocities of equal magnitude inclined to each other at an angle θ. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then θ is

(1) 120°

(2) 45°

(3) 60°

(4) 90°

22. At a telephone enquiry system the number of phone calls regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 min time intervals. The probability that there is at the most one phone call during a 10 min time period, is

(1) 5/6

(2) 6/55

(3) 6/e^{5}

(4) 6/5e

23. A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4 s prior to passing through P. If g = 10 m/s^{2}, then the height above the point P from where the body began to fall is

(1) 900 m

(2) 320 m

(3) 680 m

(4) 720 m

24. The set of points, where is differentiable, is

(1) (−∞,−1) ∪ (−1, ∞)

(2) (−∞, ∞)

(3) (0, ∞)

(4) (−∞, 0) ∪ (0, ∞)

25. Let a, b, ∈ Then,

(1) there exist more than one but finite number of B’s such that AB = BA

(2) there exists exactly one B such that AB = BA

(3) there exists infinitely many B’s such that AB = BA

(4) there cannot exist any B such that AB = BA

26. Let a_{1}, a_{2}, a_{3}, ….. cannot be terms of an AP. If equals

(1) 7/2

(2) 2/7

(3) 11/41

(4) 41/11

27. The function has a local minimum at

(1) x = −2

(2) x = 0

(3) x = 1

(4) x = 2

28. Angle between the tangents to the curve y = x2 – 5x + 6 at the points (2, 0) and (3, 0) is

(1) π/2

(2) π/6

(3) π/4

(4) π/3

29. If x is real, the maximum value of is

(1) 41

(2) 1

(3) 17/7

(4) 1/4

30. A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area enclosed by the park is

(1)

(2)

(3) πx^{2}

(4)

31. If (a, a^{2}) falls inside the angle made by the lines y = x/2, x > 0 and y = 3x, x > 0, then a belongs to

(1) (3, ∞)

(2) (1/2, 3)

(3) (−3, −1/2)

(4) (0, 1/2)

32. If x^{m} y^{n} = (x + y)^{m+n}, then dy/dx is

(1)

(2) xy

(3) x/y

(4) y/x

33. If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are two diameters of a circle of area 49π sq unit, the equation of the circle is

(1) x^{2} + y^{2} + 2x – 2y – 62 = 0

(2) x^{2} + y^{2} – 2x + 2y – 62 = 0

(3) x^{2} + y^{2} – 2x + 2y – 47 = 0

(4) x^{2} + y^{2} + 2x – 2y – 47 = 0

34. The image of the point (−1, 3, 4) in the plane x – 2y = 0 is

(1) (15, 11, 4)

(2)

(3) (8, 4, 4)

(4)

35. The differential equation whose solution is Ax^{2} + By^{2} = 1, where A and B are arbitrary constant, is of

(1) first order and second degree

(2) first order and first degree

(3) second order and first degree

(4) second order and second degree

36. The value of where [x] denotes the greatest integer not exceeding x, is

(1) [a] f(a) – {f(1) + f(2) + …..+ f([a])}

(2) [a] f([a]) – {f(1) + f(2) + ….+f(a)}

(3) a f([a]) – {f(1) + f(2) + …+f(a)}

(4) a f(a) – {f(1) + f(2) + …..+f([a])}

37. Let C be the circle with centre (0, 0) and radius 3 unit. The equation of the locus of the mid points of the chords of the circle that subtend an angle of 2π/3 at its centre, is

(1) x^{2} + y^{2} = 1

(2) x^{2} + y^{2} = 27/4

(3) x^{2} + y^{2} = 9/4

(4) x^{2} + y^{2} = 3/2

38. If a_{1}, a_{2}, …., a_{n} are in HP, then the expression a_{1}a_{2} + a_{2}a_{3} + ……+a_{n−1} a_{n} is equal to

(1) (n – 1) (a_{1} – a_{n})

(2) na_{1}a_{n}

(3) (n – 1) a_{1}a_{n}

(4) n(a_{1} – a_{n})

39. If z^{2} + z + 1 = 0, where z is complex number, then the value of is

(1) 54

(2) 6

(3) 12

(4) 18

40. If 0 < x < π and cos x + sin x = 1/2, then tan x is

(1)

(2)

(3)

(4)

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