Practice Questions

Q68.The number of x ∈[0, 2π] for which √2 sin4 x + 18 cos2 x − √2 cos4 x + 18 sin2 x = 1 is: (1) 2 (2) 6 (3) 4 (4) 8

Q68.Two sides of a rhombus are along the lines, x −y + 1 = 0 and 7x −y −5 = 0 . If its diagonals intersect at (−1, −2) , then which one of the following is a vertex of this rhombus ? (1) ( 31 , −83 ) (2) (−103 , −73 ) (3) (−3, −9) (4) (−3, −8)

Q70.A ray of light is incident along a line which meets another line 7x −y + 1 = 0 at the point (0, 1). The ray is then reflected from this point along the line y + 2x = 1 . Then the equation of the line of incidence of the ray of light is : (1) 41x −25y + 25 = 0 (2) 41x + 25y −25 = 0 (3) 41x −38y + 38 = 0 (4) 41x + 38y −38 = 0

Q71.Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the center C of the circle x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its center at P is (1) x2 + y2 −x4 + 2y −24 = 0 (2) x2 + y2 −4x + 9y + 18 = 0 (3) x2 + y2 −4x + 8y + 12 = 0 (4) x2 + y2 −x + 4y −12 = 0

Q73.A hyperbola whose transverse axis is along the major axis of the conic x2 3 + 4 = 4 and has vertices at the foci of the conic. If the eccentricity of the hyperbola is 3 , then which of the following points does not lie on 2 the hyperbola ? (1) (√5, 2√2) (2) (0, 2) (3) (5, 2√3) (4) (√10, 2√3) is

Q73. (n+1) (n+2)….3n n1 is equal to lim n2n ) n→∞( (1) 9 (2) 3 log 3 −2 e2 (3) 18 (4) 27 e4 e2 1 2x

Q74.If f(x) is a differentiable function in the interval (0, ∞) such that f(1) = 1 and lim t−x = 1,for each t→x x > 0, then f( 23 ) is equal to JEE Main 2016 (09 Apr Online) JEE Main Previous Year Paper (1) 23 (2) 13 18 6 (3) 25 (4) 31 9 18 a − 4 ) 2x = e3 , then a is equal to x x2

Q80.Let a, b ∈R, (a ≠0). If the function f , defined as , 0 ≤x < 1 ⎧ 2x2a f(x) = a, 1 ≤x < √2 ,is continuous in the interval [0, ∞), then an ordered pair (a, b) can be ⎨ 2b2−4b ⎩ x3 , √2 ≤x < 8 1 −1 + −√3) (2) (√2, √3) (1) (−√2, 1 1 + −√3) (4) (−√2, √3) (3) (√2,

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)

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

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

Q61.The largest value of r, for which the region represented by the set {ω ∈C||ω −4 −i| ≤r} is contained in the region represented by the set {z ∈C||z −1| ≤|z + i|}, is equal to : (1) 2√2 (2) 32 √2 (3) √17 (4) 52 √2

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

Q68.The sum of coefficients of integral powers of x in the binomial expansion of (1 −2√x) 50 is (1) 2 1 (250 + 1) (2) 12 (350 + 1) (3) 1 2 (350) (4) 12 (350 −1)

Q69.Locus of the image of the point (2, 3) in the line (2x −3y + 4) + k(x −2y + 3) = 0, k∈R , is a (1) Circle of radius √3 (2) Straight line parallel to x-axis. (3) Straight line parallel to y-axis. (4) Circle of radius √2

Q71.Let the tangents drawn to the circle, x2 + y2 = 16 from the point P(0, h) meet the x -axis at points A and B . If the area of ΔAPB is minimum, then positive value of h is: (1) 4√2 (2) 3√2 (3) 4√3 (4) 3√3

Q72.If the tangent to the conic, y −6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x −2y = k (for some fixed k ) at a point (α, β); then (α, β) is (1) (−717 , 176 ) (2) (−817 , 172 ) (3) (−617 , 1017 ) (4) (−417 , 171 )

Q72.The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus ractum to the x2 y2 ellipse 9 + 5 = 1, is (1) 27 (2) 274 (3) 18 (4) 272

Q73.An ellipse passes through the foci of the hyperbola, 9x2 −4y2 = 36 and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is 1 , then which of the following points does not lie on the ellipse? 2 , (1) ( √392 √3) (2) ( √132 , √32 ) 2 , (3) (√13 (4) √6) (√13, 0) x is equal to

Q79.The least value of the product xyz (such that x, y and z are positive real numbers) for which the determinant x 1 1 1 y 1 is non-negative is 1 1 z (1) −1 (2) −16√2 (3) −8 (4) −2√2

Q81.Let k and K be the minimum and the maximum values of the function f(x) = (1+x)0.6 in [0, 1], respectively, 1+x0.6 then the ordered pair (k, K) is equal to: (1) (2−0.4, 1) (2) (2−0.6, 1) (3) (2−0.4, 20.6) (4) (1,20.6) JEE Main 2015 (11 Apr Online) JEE Main Previous Year Paper 1

Q82.Let f(x) be a polynomial of degree four and having its extreme values at x = 1 and x = 2. If f(x) lim + = 3, then f(2) is equal to [1 x2 ] x→0 (1) 4 (2) −8 (3) −4 (4) 0 JEE Main 2015 (04 Apr) JEE Main Previous Year Paper

Q83.Let f : R →R be a function such that f(2 −x) = f(2 + x) and f(4 −x) = f(4 + x), for all x ∈R and 2 50 ∫ f(x)dx = 5. Then the value of ∫ f(x)dx is 0 10 (1) 100 (2) 125 (3) 80 (4) 200

Q87.Let →a, b and →c be three non - zero vectors such that no two of them are collinear and × →c→a. If θ is the angle between vectors b and →c, then a value of sin θ is = 13 b (→a → → → b) ×→c (1) −2√3 (2) 2√2 3 3 (3) −√2 (4) 2 3 3