Practice Questions

Q74.Let p and q denote the following statements p : The sun is shining q : I shall play tennis in the afternoon The negation of the statement "If the sun is shining then I shall play tennis in the afternoon", is (1) q ⇒∼p (2) q∧∼p (3) p∧∼q (4) ∼q ⇒∼p

Q75.In a ΔPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to (1) 5π (2) π 6 6 (3) π (4) 3π 4 4 JEE Main 2012 (Offline) JEE Main Previous Year Paper Q76. ⎛1 0 0⎞ ⎛1⎞ ⎛0⎞ Let A = 2 1 0 . If u1 and u2 are column matrices such that Au1 = 0 and Au2 = 1 , then ⎝3 2 1⎠ ⎝0⎠ ⎝0⎠ u1 + u2 is equal to (1) ⎛−1⎞ (2) ⎛ −1⎞ 1 1 ⎝ 0 ⎠ ⎝ −1⎠ (3) ⎛−1⎞ (4) ⎛ 1 ⎞ −1 −1 ⎝ 0 ⎠ ⎝ −1⎠

Q75.If three distinct points A, B, C are given in the 2dimensional coordinate plane such that the ratio of the distance of each one of them from the point (1, 0) to the distance from (−1, 0) is equal to 12 , then the circumcentre of the triangle ABC is at the point (1) ( 35 , 0) (2) (0, 0) (3) ( 13 , 0) (4) (3, 0) Q76. ⎡ 0 0 a ⎤ If AT denotes the transpose of the matrix A = 0 b c , where a, b, c, d, e and f are integers such that ⎣ d e f ⎦ abd ≠0 , then the number of such matrices for which A−1 = AT is (1) 2(3!) (2) 3(2!) (3) 23 (4) 32

Q75.If two vertical poles 20 m and 80 m high stand apart on a horizontal plane, then the height (in m ) of the point of intersection of the lines joining the top of each pole to the foot of other is (1) 16 (2) 18 (3) 50 (4) 15

Q77.Let P and Q be 3 × 3 matrices with P ≠Q. If P 3 = Q3 and P 2Q = Q2P , then determinant of (P 2 + Q2) is equal to (1) −2 (2) 1 (3) 0 (4) −1

Q77.If a, b, c, are non zero complex numbers satisfying a2 + b2 + c2 = 0 and b2 + c2 ab ac ab c2 + a2 bc = ka2b2c2 , then k is equal to ac bc a2 + b2 (1) 1 (2) 3 (3) 4 (4) 2 is 3

Q78.If f ′(x) = sin(log x) and y = f ( 3−2x2x+3 ), then dxdy equals (1) sin [log ( 2x+33−2x )] (2) (3−2x2)12 (3) (3−2x2) 12 sin [log ( 3−2x2x+3 )] (4) (3−2x212 cos [log ( 2x+33−2x )] JEE Main 2012 (12 May Online) JEE Main Previous Year Paper

Q78.A value of tan−1 (sin (cos−1 (√2 ))) (1) π (2) π 4 2 (3) π (4) π 3 6

Q78.If f : R →R is a function defined by f(x) = [x] cos ( 2x−12 )π, where [x] denotes the greatest integer function, then f is (1) continuous for every real x (2) discontinuous only at x = 0 (3) discontinuous only at non-zero integral values of (4) continuous only at x = 0 x

Q78.If the system of equations x + y + z = 6 x + 2y + 3z = 10 x + 2y + λz = 0 has a unique solution, then λ is not equal to (1) 1 (2) 0 (3) 2 (4) 3

Q79.If P(S) denotes the set of all subsets of a given set S , then the number of one-to-one functions from the set S = {1, 2, 3} to the set P(S) is (1) 24 (2) 8 (3) 336 (4) 320

Q79.If f(x) = a| sin x| + be|x| + c|x|3 , where a, b, c ∈R, is differentiable at x = 0, then (1) a = 0, b and c are any real numbers (2) c = 0, a = 0, b is any real number (3) b = 0, c = 0, a is any real number (4) a = 0, b = 0, c is any real number

Q79.Consider a rectangle whose length is increasing at the uniform rate of 2 m/sec, breadth is decreasing at the uniform rate of 3 m/sec and the area is decreasing at the uniform rate of 5 m2/sec. If after some time the breadth of the rectangle is 2 m then the length of the rectangle is (1) 2 m (2) 4 m (3) 1 m (4) 3 m

Q79.The range of the function f(x) = 1+|x|x , x ∈R, is (1) R (2) (−1, 1) (3) R −{0} (4) [−1, 1]

Q80.If f(x) = xex(1−x), x ∈R, then f(x) is (1) decreasing on [−1/2, 1] (2) decreasing on R (3) increasing on [−1/2, 1] (4) increasing on R

Q80.A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72 π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is (1) 9 (2) 7 7 9 (3) 2 (4) 9 9 2

Q81.If a metallic circular plate of radius 50 cm is heated so that its radius increases at the rate of 1 mm per hour, then the rate at which, the area of the plate increases (in cm2/ hour) is (1) 5π (2) 10π (3) 100π (4) 50π

Q81.The integral of x2−x w.r.t. x is x3−x2+x−1 (1) 1 2 log (x2 + 1 + c) (2) 12 log x2 −1 + c (3) log (x2 + 1 + c) (4) log x2 −1 + c

Q81.The weight W of a certain stock of fish is given by W = nw, where n is the size of stock and w is the average weight of a fish. If n and w change with time t as n = 2t2 + 3 and w = t2 −t + 2, then the rate of change of W with respect to t at t = 1 is (1) 1 (2) 8 (3) 13 (4) 5 JEE Main 2012 (19 May Online) JEE Main Previous Year Paper

Q81.If x + |y| = 2y, then y as a function of x, at x = 0 is (1) differentiable but not continuous (2) continuous but not differentiable (3) continuous as well as differentiable (4) neither continuous nor differentiable

Q82.If dx d G(x) = etanx x , x ∈(0, π/2), then ∫1/21/4 x2 ⋅etan(πx2)dx is equal to (1) G(π/4) −G(π/16) (2) 2[G(π/4) −G(π/16)] (3) π[G(1/2) −G(1/4)] (4) G(1/√2) −G(1/2)

Q82.If a circular iron sheet of radius 30 cm is heated such that its area increases at the uniform rate of 6πcm2/hr, then the rate (in mm/hr ) at which the radius of the circular sheet increases is (1) 1.0 (2) 0.1 (3) 1.1 (4) 2.0

Q82.If the integral ∫ tan5 tanx−2x dx = x + a ln | sin x −2 cos x| + k, then a is equal to JEE Main 2012 (Offline) JEE Main Previous Year Paper (1) −1 (2) −2 (3) 1 (4) 2 dt, then g(x + π) equals

Q82. f(x) = ∫ dx is a polynomial of degree sin6 x (1) 5 in cot x (2) 5 in tan x (3) 3 in tan x (4) 3 in cot x

Q82.If f(x) = ∫( x2+sin21+x2 x ) sec2 xdx and f(0) = 0 , then f(1) equals (1) tan 1 −π4 (2) tan 1 + 1 (3) π 4 (4) 1 −π4