Practice Questions

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

Q85.Statement 1: The degrees of the differential equations dy + y2 = x and + y = sin x are equal. Statement dx dx2 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined. (1) Statement 1 is true, Statement 2 is true, (2) Statement 1 is false, Statement 2 is true. Statement 2 is not a correct explanation of Statement 1. (3) Statement 1 is true, Statement 2 is false. (4) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

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

Q86.Statement 1: The vectors →a,→b and →c lie in the same plane if and only if →a ⋅(→b × →c) = 0 Statement 2: The vectors →u and →v are perpendicular if and only if →u ⋅→v = 0 where →u × →v is a vector perpendicular to the plane of →u and →v (1) Statement 1 is false, Statement 2 is true. (2) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation for Statement 1. (3) Statement 1 is true, Statement 2 is false. (4) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

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

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 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.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 −−

Q87.Statement 1: If the points (1, 2, 2), (2, 1, 2) and (2, 2, z) and (1, 1, 1) are coplanar, then z = 2. Statement 2: If the 4 points P, Q, R and S are coplanar, then the volume of the tetrahedron PQRS is 0. JEE Main 2012 (12 May Online) JEE Main Previous Year Paper (1) Statement 1 is false,, Statement 2 is true. (2) Statement 1 is true, Statement 2 is false. (3) Statement 1 is true, Statement 2 is true, (4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement Statement 2 is not a correct explanation of 1. Statement 1.

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

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

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.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

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.Statement 1: The shortest distance between the lines x 2 = −1y = 2z and x−14 = y−1−2 = z−14 is √2. Statement 2: The shortest distance between two parallel lines is the perpendicular distance from any point on one of the lines to the other line. (1) Statement 1 is true, Statement 2 is false. (2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1. (3) Statement 1 is false, Statement 2 is true. (4) Statement 1 is true, Statement 2 is true, , Statement 2 is not a correct explanation for Statement 1.

Q89.The equation of a plane containing the line x+1 −3 = y−32 = z+21 and the point (0, 7, −7) is (1) x + y + z = 0 (2) x + 2y + z = 21 (3) 3x −2y + 5z + 35 = 0 (4) 3x + 2y + 5z + 21 = 0

Q89.If the lines x−1 2 = y+13 = z−14 and x−31 = y−k2 = 1z intersect, then k is equal to (1) −1 (2) 29 (3) 9 (4) 0 2

Q89.The coordinates of the foot perpendicular from the point (1, 0, 0) to the line x −1 y + 1 z + 10 = = are 2 −3 8 (1) (2, −3, 8) (2) (1, −1, −10) (3) (5, −8, −4) (4) (3, −4, −2) ∑ni=1 i2

Q89.If →a = ^i −2^j + 3^k,→b = 2^i + 3^j −^k and →c = r^i + ^j + (2r −1^k are three vectors such that →c is parallel to the plane of →a and →b, then r is equal to (1) 1 (2) −1 (3) 0 (4) 2

Q89.The values of a for which the two points (1, a, 1) and (−3, 0, a) lie on the opposite sides of the plane 3x + 4y −12z + 13 = 0, satisfy JEE Main 2012 (07 May Online) JEE Main Previous Year Paper (1) 0 < a < 31 (2) −1 < a < 0 (3) a < −1 or a < 13 (4) a = 0

Q90.There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and there after a ball is drawn at random from the urn, then the probability that it is white is (1) 1 (2) 2 4 3 (3) 1 (4) 1 5 3 JEE Main 2012 (26 May Online) JEE Main Previous Year Paper

Q90.If six students, including two particular students A and B, stand in a row, then the probability that A and B are separated with one student in between them is (1) 8 (2) 4 15 15 (3) 2 (4) 1 15 15 JEE Main 2012 (19 May Online) JEE Main Previous Year Paper