Practice Questions

Q84.For x ∈R, x ≠0, if y(x) is a differentiable function such that x ∫x y(t)dt = (x + 1) ∫x ty(t)dt, then y(x) 1 1 equals (where C is a constant) (1) Cx3 e x1 (2) C e−1x x2 (3) C x (4) C e−1x x e−1 x3 dx, where [x] denotes the greatest integer less than or equal to x, is

Q85.The area (in sq. units) of the region {(x, y) : y2 ≥2x and x2 + y2 ≤4x, x ≥0, y ≥0} is (1) π −4√23 (2) π2 −2√23 (3) π −43 (4) π −83 JEE Main 2016 (03 Apr) JEE Main Previous Year Paper

Q85.If 2 ∫1 tan−1 xdx = ∫1 cot−1(1 −x + x2)dx, then ∫1 tan−1(1 −x + x2)dx is equal to 0 0 0 (1) π 2 + ln 2 (2) ln 2 (3) π 2 −ln 4 (4) ln 4

Q85.The value of the integral ∫10 [x2−28x+196]+[x2][x2] 4 (1) 1 (2) 6 3 (3) 7 (4) 3

Q86.If a curve y = f(x) passes through the point (1, −1) and satisfies the differential equation, y (1 + xy)dx = x dy, then f(−12 ) is equal to (1) 2 (2) 4 5 5 (3) −25 (4) −45 → → → → If b is not parallel to →c, then the b × b + = √32

Q86.The solution of the differential equation dx dy + 2y sec x = tan2y x , where 0 ≤x < π2 and y(0) = 1 , is given by (1) y2 = 1 + sec x+tanx x (2) y = 1 + sec x+tanx x (3) y = 1 − sec x+tanx x (4) y2 = 1 − sec x+tanx x y−2

Q86.The area (in sq. units) of the region described by A = {(x, y) y ≥x2 −5x + 4, x + y ≥1, y ≤0} is (1) 19 (2) 17 6 6 (3) 7 (4) 13 2 6

Q87.Let →a, b and →cbe three unit vectors such that →a × ( →c) ( →c). → angle between →a and b is (1) 2π (2) 5π 3 6 (3) 3π (4) π 4 2

Q87.The number of distinct real values of λ , for which the lines x−1 1 = = z−12 , are 2 = z+3λ2 and x−31 = y−2λ2 coplanar is (1) 2 (2) 4 (3) 3 (4) 1 JEE Main 2016 (10 Apr Online) JEE Main Previous Year Paper

Q87.In a triangle ABC , right angle at vertex A , if the position vectors of A, B and C are respectively 3ˆi + ˆj − ˆk, −ˆi + 3ˆj + pˆk and 5ˆi + qˆj −4ˆk , then the point (p, q) lies on a line: (1) Making an obtuse angle with the positive (2) Parallel to x −axis direction of x −axis (3) Parallel to y −axis (4) Making an acute angle with the positive direction of x −axis

Q88.If the line, x−3 2 = y+2−1 = z+43 lies in the plane lx + my −z = 9, then l2 + m2 is equal to (1) 5 (2) 2 (3) 26 (4) 18

Q88.The shortest distance between the lines x 2 = 2y = 1z and x+2−1 = y−48 = z−54 , lies in the interval: (1) (3, 4] (2) (2, 3] (3) [1, 2) (4) [0, 1)

Q89.Let ABC be a triangle whose circumcentre is at P . If the position vectors A, B, C and P are →a, b,→cand 4 respectively, then the position vector of the orthocentre of this triangle, is : → → (1) →a + b + →c (2) →a + b + →c 2 −( ) (3) (→a +→b+ →c) (4) →0 2

Q89.The distance of the point (1, −2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes x −y + 2z = 3 and 2x −2y + z + 12 = 0, is : (1) 2 (2) √2 (3) 2√2 (4) 1 √2

Q89.The distance of the point (1, −5, 9) from the plane x −y + z = 5 measured along the line x = y = z is (1) 10 (2) 20 √3 3 (3) 3√10 (4) 10√3

Q90.If A and B are any two events such that P(A) = 25 and P(A ∩B) = 203 , then the conditional probability, P(A|(A′ ∪B′)), where A′ denotes the complement of A , is equal to : (1) 11 (2) 5 20 17 (3) 8 (4) 1 17 4 JEE Main 2016 (09 Apr Online) JEE Main Previous Year Paper

Q90.An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is (1) 496 (2) 192 729 729 (3) 240 (4) 256 729 729 JEE Main 2016 (10 Apr Online) JEE Main Previous Year Paper

Q90.Let two fair six-faced dice A and B be thrown simultaneously. If E1 is the event that die A shows up four, E2 is the event that die B shows up two and E3 is the event that the sum of numbers on both dice is odd, then which of the following statements is not true? (1) E1 and E3 are independent (2) E1, E2 and E3 are independent (3) E1 and E2 are independent (4) E2 and E3 are independent JEE Main 2016 (03 Apr) JEE Main Previous Year Paper

Q61.Let α and β be the roots of equation x2 −6x −2 = 0 . If an = αn −βn, ∀ n ≥1, then the value of a10−2a8 is 2a9 equal to (1) −3 (2) 6 (3) −6 (4) 3 is

Q62.If 2 + 3i is one of the roots of the equation 2x3 −9x2 + kx −13 = 0, k ∈R, then the real root of this equation (where i2 = −1) : (1) Exists and is equal to 1 (2) Does not exist 2 (3) Exists and is equal to 1 (4) Exists and is equal to −12

Q62.A complex number z is said to be unimodular if |z| = 1 . Let z1 and z2 are complex numbers such that z1−2z2 2−z1 z 2 unimodular and z2 is not unimodular, then the point z1 lies on a (1) circle of radius √2 (2) straight line parallel to x-axis (3) straight line parallel to y-axis (4) circle of radius 2

Q62.If the two roots of the equation, (a −1) (x4 + x2 + 1) + (a + 1)(x2 + x + 1) 2 = 0 are real and distinct, then the set of all values of a is equal to (1) (0, 12 ) (2) (−12 , 0) ∪(0, 12 ) (3) (−∞, −2) ∪(2, ∞) (4) (−12 , 0)

Q63.The number of integers greater than 6000 that can be formed, using the digits 3, 5, 6, 7 and 8 , without repetition is (1) 72 (2) 216 (3) 192 (4) 120

Q63.If z is a non-real complex number, then the minimum value of Im z5 is (Where Im z = Imaginary part of z ) (Im z)5 JEE Main 2015 (11 Apr Online) JEE Main Previous Year Paper (1) −2 (2) −4 (3) −5 (4) −1

Q63.The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman is (1) 1960 (2) 15! (3) (15!)2 (4) 14!