Jmetquant Practice Paper-2

JMET PAPER - 2 π

1.

∫ x f (sin x ) dx =

8.

Area of the region bounded by x = [y] where [y] represents the greatest integer less than or equal to y. The y-axis and the lines y = 3 and y = 4 is (1) 4 (2) 3 (3) 7 (4) None of these

9.

A sector with a central angle α is cut out of a circle to make a cone. Find the value of the angle α, which will yield the greatest possible volume.

0

(1)

ππ ∫ f (sin x ) dx 20 π

(3) 2.

π ∫ f (sin x ) dx 2 0 2

π

(2) π ∫ f (sin x ) dx 0

(4)

ππ ∫ f (sin x ) dx 40

If A, B and C are the angles of a triangle, then the value of the determinant cos(A + B ) tan(B + C) 0 sin(A + B + C ) sin B sin(π / 2 − C ) is sin(− B ) 0 tan A

(1) 2 (3) 1 3.

4.

(2) –1 (4) None of these

Let A and B be two finite sets have 4 and 6 elements respectively. A mapping is selected at random from the set of all mappings from A to B. The probability that the mapping selected is an injection is (1) 7/18 (2) 1/3 (3) 5/18 (4) 2/9 Let A = {1, 2, 3} and let R and S be relations on A defined as R = {(1, 1), (2, 2), (3, 1), (3, 3)}, S = {(1, 2), (2, 3), (3, 1)}. Then (1) (S o R )−1 ⊆ R −1 o S−1

(2) (S o R )−1 ⊇ R −1 o S−1 (3) (S o R )−1 = R −1 o S−1

(4) (S o R )−1 and R −1 o S −1 are not comparable.

5.

6.

7.

If 5 sin θ + 12 cos θ = ±13, then find the number of solutions for the given expression, given 0 ≤ θ ≤ 360° (1) 0 (2) 1 (3) 2 (4) 3 x – 1), then the If f(x) = minimum (4 – x, 2 maximum possible value of f(x) is 2 (1) 0 (2) 3 (3) 1.5 (4) none of these Identify the true statement: (1) If a, b, c are position vectors of points which are collinear, then la + mb + nc = 0 for all values of l, m and n. (2) Volume of the tetrahedron with co-terminus edges as a, b, c is [a, b, c ] . (3) a, b and c are coplanar if c = x a + y b for unique scalars x and y. (4) If a and b are parallel vectors and c is any vector then [a, b, c ] ≠ 0.

(1) π

2 3

(2) π

4 3

(3) π

8 3

(4) None of these

10. For two positive real numbers x and y if xy > x + y, then which of the following must be true? (1) xy > 4 (2) xy ≥ 4 (3) xy ≤ 4 (4) None of these 11. The equation a sin x + b cos x = c where

c > a2 + b2 has (1) (2) (3) (4)

one solution two solutions no solution infinite number of solutions

12. AB is a vertical pole. The end A is on the level ground, C is the middle point of AB and P is a point on the level ground. The portion CB subtends an angle β at P. If AP = n.AB, then tan β = n2 n (1) (2) 2 (2n + 1) (2n + 1)

(3)

n

(4)

(2n − 1) 2

n2 (2n2 − 1)

13. The quadrilateral whose vertices are represented by the complex numbers 0, z, iz and z(1 + i) is (1) a trapezium. (2) a rectangle but not square. (3) a square. (4) a rhombus. 14. If x and y are the sides of a rectangle such that 2x + y = 3, the maximum possible area is (1) 3/4 sq.units (2) 3/2 sq.units (3) 9/8 sq.units (4) 9/4 sq.units

r r

r

r

15. Let OA = 6a + b , OB = b and OC = 12a , where O is origin. Let A1 denote the area of the triangle OAB and A2 denote the area of the parallelogram →

→

with OA and OC as adjacent A1 = kA2, then k = 1 (3) 4 (1) 8 (2) 8

sides. (4)

If

1 4

1

16. If cos-1x + cos-1y xy + yz + xz = (1) 0 (2) 1

+

cos-1z

=

(3) 2

3π,

then

(4) 3

17. At a point on the curve f(x) = x3 – 5x defined on the interval [0, 2], the tangent is parallel to the chord joining end points on the curve. Find the abcissa of this point. 2 −2 (2) (1) 3 3 (3) 2/3 (4) Both (1) and (2) ∞

18. If S1 = ∑ an and S 2 = n=0

then (1) S1 = 5S2 (3) S1 = 4S2

1 ⎛a+4⎞ ⎟ , for 0 < a < 1, ∑⎜ 5 n=0 ⎝ 5 ⎠

(4) None of these

24. If f(a) = 3, f '(a) = –3, g(a) = 2 and g '(a) = 3, then f (a )g(x ) − f (x )g(a ) = a−x (1) –15 (2) 12 (3) –12 lim

(2) S1 = S2 (4) S1 = 3S2

20. If x 1, x 2, x 3, ….., x n are in G.P.,

log x n + 2 log x n +1 log x n log x n + 5 log x n + 4 log x n + 3 equals log x n + 7

⎧ x tan 2x ⎪ sin3x sin5x , x ≠ 0 ⎪ (3) f(x) = ⎨ ⎪2 ,x = 0 ⎪⎩ 15

n

∞

19. What can be said about the nature of the real valued function f(x) = ⏐x⏐? (1) only injective. (2) only surjective. (3) bijective. (4) neither surjective nor injective.

log x n + 8

1 ⎧ ⎪x sin 2 , x ≠ 0 (2) f(x) = ⎨ x ⎪3 ,x = 0 ⎩

log x n + 6

(1) log x n x n + 8 – 2log x n (2) log x 1 x 3 – 2 log x 2 (3) log (x 1 x 2 …….. x n) (4) 0

+4

x →a

25. Betting in a casino on a roulette wheel is as follows: The markings on the wheel are from 101 to 400. If the wheel stops at a multiple of 7, the player wins Rs. 100/–. If it stops at a multiple of 13, the player wins Rs. 150/–. If it stops at a multiple of 7 and 13, the player wins Rs. 250/–. The entry fee for the play is Rs. 25/–. What is the expected gain or loss for the player? (1) Gain of Rs.0.83 (2) Loss of Rs.2.53 (3) Gain of Rs.5.63 (4) Loss of Rs.5.62 26. An equilateral triangle is inscribed in the parabola y2 = 8x, whose one vertex is at the vertex of the parabola. The length of the side of the triangle is (1) 8

3 units

22. Anand and Bimal are first semester B−Tech students of IIT in city X. They had their schooling in two different schools located in cities Y and Z. Anand has x friends in city Y. (who studied with him in the same school). Bimal has (x + 3) friends in city Z (who studied in the same school with him). In the IIT where they study now, they have common friends. Altogether there are 12 who are friends of Anand or of Bimal. One day Anand and Bimal arranged a party for their friends and organized some games. Assume that no game is played between two friends who have studied / studying in the same institution. The maximum number of games that could be played, is (1) 23 (2) 66 (3) 46 (4) 45 23. Which of the following function is not continuous at x = 0? 1 ⎧ ⎪x cos , x ≠ 0 (1) f ( x ) = ⎨ x ⎪0 ,x = 0 ⎩

(2) 4

3 units

(3) 32

21. The number of non-negative integral solutions of 10 ≤ x1 + x2 + x3 + x4 ≤ 20, is (1) 9191 (2) 9911 (3) 9119 (4) 9991

(4) 15

3 units

(4) 16

3 units

27. One of the solution of the equation 1 − cos 10 x + cos2 10 x + .... + ( −1)n cos 10 x + .... 1 + cos 10 x + cos2 10 x + cos3 10 x + ... + cosn 10 x + ... 1 − sin 5 x is 1 + sin 5 x π (1) 15 2π (3) 15 =

π 60 π (4) 30

(2)

28. Equation of an ellipse with the centre at origin the 4 foci at (0, ± 4) and eccentricity as , is 5 (1)

x2 y2 + =1 4 25

(2)

x2 y2 + =1 25 4

(3)

x2 y2 + =1 9 25

(4)

x2 y2 + =1 25 9

(

29. The inverse of f ( x ) = 5 − (x − 8 )5 (1) 5 − (x − 8 )

5

(

(3) 8 − 5 − x 3

)

1/ 5

)

(

1/ 3

is

(2) 8 + 5 − x 3

(

)

1/ 5

(4) 5 − (x − 8 )

)

1/ 5 3

2

35. In the X-Y plane find the area of the region which satisfies x + 1 + y = 4 and x − 1 + y = 4.

30. What is the value of 1 1. 3 1. 3 . 5 1+ + + + ......... ∞ ? 7 7 . 14 7 . 14 . 21

(1) (3)

5 7 3

(1) 24 sq.units (3) 10 sq.units

7 5

(2)

(4) Either (1) or (3)

7 31. The number of 2 |x − 3x + 1| = x − 3 is (1) 0 (2) 2

distinct

solutions

(3) 3

(4) 4

2

of

2logk

32. If the equation x – 3kx + 2e – 1 = 0 has real roots such that the product of roots is 7, then the value of k is (1) 1 (2) 2 (3) 3 (4) 4 33. The curve represented by the equation y =

x 2 − 35 − 16 meets the x-axis (1) (2) (3) (4)

at only one point at exactly two point at exactly three points at exactly four points

36. The number of ordered triplets of positive integers which are solutions of the equation 3x + 2y + z = 100, is (1) 784 (2) 800 (3) 864 (4) None of these 37. Two points A and B are chosen at random on a line segment PQ of length 120 cm. Find the probability that AB ≤ 20 cm. (1) 2/9 (2) 1/6 (3) 2/7 (4) 11/36 38. A cylindrical can is to be constructed so that it can hold 1m3 of milk. The cost of constructing the top and the bottom of the can is four times the cost of constructing the sides. what should be the height (h) of the can, such that the cost of construction can be minimized? 4 2 (1) m m (2) 3 π π

(3) 34. Two towers T1 and T2 stand on either sides of a highway. The towers T1 and T2 are 15 m and 21 m high respectively and the distance between their feet is 48 m. A point P is selected on the line joining the foot of the towers T1 and T2 such that the sum of the distances (S) of the tip of the towers from the point P is minimum. Find the value of S. (1) 48 m (2) 52 m (3) 60 m (4) 54 m

(2) 16 sq.units (4) 18 sq.units

2 3

π

3

m

(4)

π m 4

39. What is the sum of the roots of the equation 4 3 2 x – 4x + 5x + 3 = 0? (1) 4 (2) –5 (3) –4 (4) None of these 40. Find the sum of the coefficients in the expansion 3 of (x + 2y + 3z) (1) 125 (2) 216 (3) 128 (4) 256

3