Practice Questions

Q83.If [x] is the greatest integer ≤x, then the value of the integral ∫0.9−0.9 ([x2] + log ( 2−x2+x ))dx is (1) 0.486 (2) 0.243 (3) 1.8 (4) 0

Q83.If g(x) = ∫x0 cos 4t (1) g(x) (2) g(x) + g(π) g(π) (3) g(x) −g(π) (4) None of these

Q83.The area enclosed by the curves y = x2, y = x3 , x = 0 and x = p, where p > 1 , is 1/6 . The p equals (1) 8/3 (2) 16/3 (3) 2 (4) 4/3

Q83.The value of the integral ∫0.90 [x −2[x]]dx, where [.] denotes the greatest integer function is (1) 0.9 (2) 1.8 (3) −0.9 (4) 0

Q84.The area of the region bounded by the curve y = x3 , and the lines, y = 8 , and x = 0 , is (1) 8 (2) 12 (3) 10 (4) 16

Q84.If ∫xe tf(t)dt = sin x −x cos x −x22 , for all x ∈R −{0}, then the value of f ( π6 ) is (1) 1/2 (2) 1 (3) 0 (4) −1/2

Q84.The area bounded by the parabola y2 = 4x and the line 2x −3y + 4 = 0, in square unit, is (1) 2 (2) 1 5 3 (3) 1 (4) 1 2 JEE Main 2012 (26 May Online) JEE Main Previous Year Paper = x is

Q84.The area bounded between the parabolas x2 = 4y and x2 = 9y, and the straight line y = 2 is (1) 20√2 (2) 10√2 3 (3) 20√2 (4) 10√2 3

Q84.If a straight line y −x = 2 divides the region x2 + y2 ≤4 into two parts, then the ratio of the area of the smaller part to the area of the greater part is (1) 3π −8 : π + 8 (2) π −3 : 3π + 3 (3) 3π −4 : π + 4 (4) π −2 : 3π + 2 d2y

Q85.The general solution of the differential equation dx dy + x2 y = x2 is (1) y = cx−3 −x24 (2) y = cx3 −x24 (3) y = cx2 + x35 (4) y = cx−2 + x35

Q85.The parabola y2 = x divides the circle x2 + y2 = 2 into two parts whose areas are in the ratio (1) 9π + 2 : 3π −2 (2) 9π −2 : 3π + 2 (3) 7π −2 : 2π −3 (4) 7π + 2 : 3π + 2 x dy)

Q85.The integrating factor of the differential equation (x2 −1 dxdy + 2)xy (1) 1 (2) x2 −1 x2−1 (3) x2−1 (4) x x x2−1

Q85.The population p(t) at time t of a certain mouse species satisfies the differential equation dp(t)dt = 0.5 p(t) −450. If p(0) = 850 , then the time at which the population becomes zero is (1) 2 ln 18 (2) ln 9 (3) 1 2 ln 18 (4) ln 18

Q86.If →u = ^j + 4^k, →v = ^i + 3^k and →w = cos θ^i + sin θ^j are vectors in 3-dimensional space, then the maximum possible value of |→u × →v ⋅→w| is (1) √3 (2) 5 (3) √14 (4) 7

Q86.Let ^a and ^b be two unit vectors. If the vectors →c = ^a + 2^b and →d = 5^a −4^b are perpendicular to each other, then the angle between ^a and ^b is (1) π (2) π 6 2 (3) π (4) π 3 4 −−

Q86.Let y(x) be a solution of (2+sin dx = cos x. If y(0) = 2, then y ( π2 ) equals (1+y) (1) 5 (2) 2 2 (3) 7 (4) 3 2

Q86.If a + b + c = 0, |→a| = 3, |→b| = 5 and |→c| = 7, then the angle between →a and →b is (1) π (2) π 3 4 (3) π (4) π 6 2

Q87.Let ABCD be a parallelogram such that AB→ =→q, AD→ = →p and ∠BAD be an acute angle. If →r is the vector that coincides with the altitude directed from the vertex B to the side AD, then →r is given by (1) →r = 3→q −3(→p⋅→q) →p (2) →r = −→q+ (→p⋅→p) ( →p⋅→p→p⋅→q )→p →p⋅→q 3(→p⋅→q) (3) →r = →q (4) →r = −3→q + →p −( →p⋅→p )→p (→p⋅→p)

Q87.If the three planes x = 5, 2x −5ay + 3z −2 = 0 and 3bx + y −3z = 0 contain a common line, then (a, b) is equal to (1) ( 158 , −15 ) (2) ( 15 , −815 ) (3) (−815 , 51 ) (4) (−15 , 158 )

Q87.The distance of the point −^i + 2^j + 6^k from the straight line that passes through the point 2^i + 3^j −4^k and is parallel to the vector 6^i + 3^j −4^k is (1) 9 (2) 8 (3) 7 (4) 10

Q87. ABCD is parallelogram. The position vectors of A and C are respectively, 3^i + 3^j + 5^k and ^i −5^j −5^k. If −−→ → M is the midpoint of the diagonal DB, then the magnitude of the projection of OM on OC , where O is the origin, is (1) 7√51 (2) 7 √50 (3) 7√50 (4) 7 √51

Q88.Consider the following planes P : x + y −2z + 7 = 0 Q : x + y + 2z + 2 = 0 R : 3x + 3y −6z −11 = 0 (1) P and R are perpendicular (2) Q and R are perpendicular (3) P and Q are parallel (4) P and R are parallel

Q88.An equation of a plane parallel to the plane x −2y + 2z −5 = 0 and at a unit distance from the origin is (1) x −2y + 2z −3 = 0 (2) x −2y + 2z + 1 = 0 (3) x −2y + 2z −1 = 0 (4) x −2y + 2z + 5 = 0

Q88.A unit vector which is perpendicular to the vector 2^i −^j + 2^k and is coplanar with the vectors ^i + ^j −^k and 2^i + 2^j −^k is (1) 2^j+^k (2) 3^i+2^j−2^k √5 √17 (3) 3^i+2^j+2^k (4) 2^i+2^j−^k √17 3

Q88.If →a = ^i −2^j + 3^k,→b = 2^i + 3^j −^k and →c = λ^i + ^j + (2λ −1^k) are coplanar vectors, then λ is equal to (1) 0 (2) −1 (3) 2 (4) 1