Practice Questions

Q81.Let C be a curve given by y(x) = 1 + √4x −3 , x > 43 . If P is a point on C, such that the tangent at P has slope 2 , then a point through which the normal at P passes, is : 3 (1) (1, 7) (2) (3, −4) (3) (4, −3) (4) (2, 3)

Q82.Consider f(x) = tan−1(√1+sin ), point (1) ( π6 , 0) (2) ( π4 , 0) (3) (0, 0) (4) (0, 2π3 )

Q82.Let f(x) = sin4x + cos4x. Then, f is an increasing function in the interval: (1) ] 5π8 , 3π4 [ (2) ] π2 , 5π8 [ (3) ] π4 , π2 [ (4) ]0, π4 [

Q82.The minimum distance of a point on the curve y = x2 −4 from the origin is (1) √15 units 2 units (2) √192 (4) √19 units units 2 (3) √152 JEE Main 2016 (09 Apr Online) JEE Main Previous Year Paper

Q83.The integral ∫ dx is equal to (1+√x)√x−x2 (1) (2) + c + c −2√1+√x1−√x −√1−√x1+√x (3) (4) −2 + c + c √1−√x1+√x √1+√x1−√x

Q83.If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3 −1, t ∈R, meets the curve again at a point Q, then the coordinates of Q are : (1) (16t2 + 3, −64t3 −1) (2) (4t2 + 3, −8t3 −1) (3) (t2 + 3, t3 −1) (4) (t2 + 3, −t3 −1)

Q83.A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then (1) x = 2r (2) 2x = r (3) 2x = (π + 4)r (4) (4 −π)x = πr

Q84.The integral ∫ 2x12+5x9 dx, is equal to (x5+x3+1)3 (1) x5 + c (2) −x10 + c 2(x5+x3+1)2 2(x5+x3+1)2 (3) −x5 + c (4) x10 + c (x5+x3+1)2 2(x5+x3+1)2

Q84.If ∫ dx = (tan x)A + C(tan x)B + k, where k is a constant of integration, then A + B + C equals cos3 x √2 sin 2x (1) 16 (2) 27 5 10 (3) 7 (4) 21 10 5

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

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

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

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

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

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)

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. ABC is a triangle in a plane with vertices A(2, 3, 5), B(−1, 3, 2) and C(λ, 5, μ) . If the median through A is equally inclined to the coordinate axes, then the value of (λ3 + μ3 + 5) is (1) 1130 (2) 1348 (3) 1077 (4) 676 → →a+→b+→c

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

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.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