Practice Questions

Q67.If 0 ≤x < 2π, then the number of real values of x, which satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0, is (1) 7 (2) 9 (3) 3 (4) 5 JEE Main 2016 (03 Apr) JEE Main Previous Year Paper

Q67.If m and M are the minimum and the maximum values of 4 + 12 sin22x −2cos4x, x ∈R, then M −m is equal to: (1) 15 (2) 9 4 4 (3) 7 (4) 1 4 4

Q68.Let P = {θ (1) P ⊂Q and Q −P ≠ ϕ (2) Q ⊄ P (3) P = Q (4) P ⊄ Q

Q69.A straight line through origin O meets the lines 3y = 10 −4x and 8x + 6y + 5 = 0 at points A and B respectively. Then, O divides the segment AB in the ratio (1) 2 : 3 (2) 1 : 2 (3) 4 : 1 (4) 3 : 4

Q69.The centres of those circles which touch the circle, x2 + y2 −8x −8y −4 = 0, externally and also touch the x - axis, lie on (1) A hyperbola (2) A parabola (3) A circle (4) An ellipse which is not a circle

Q69.If a variable line drawn through the intersection of the lines x 3 + 4y = 1 and x4 + 3y = 1 , meets the coordinate axes at A and B, (A ≠B),then the locus of the midpoint of AB is: (1) 7xy = 6(x + y) (2) 4(x + y)2 −28(x + y) + 49 = 0 (3) 6xy = 7(x + y) (4) 14(x + y)2 −97(x + y) + 168 = 0

Q70.The point (2, 1) is translated parallel to the line L : x −y = 4 by 2√3 units. If the new point Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is (1) x + y = 2 −√6 (2) 2x + 2y = 1 −√6 (3) x + y = 3 −3√6 (4) x + y = 3 −2√6

Q70.If one of the diameters of the circle, given by the equation, x2 + y2 −4x + 6y −12 = 0, is a chord of a circle S , whose centre is at (−3, 2), then the radius of S is (1) 5 (2) 10 (3) 5√2 (4) 5√3

Q71.A circle passes through (−2, 4) and touches the y−axis at (0, 2). Which one of the following equations can represent a diameter of this circle ? (1) 2x −3y + 10 = 0 (2) 3x + 4y −3 = 0 (3) 4x + 5y −6 = 0 (4) 5x + 2y + 4 = 0 y2

Q71.Equation of the tangent to the circle, at the point (1, −1), whose center, is the point of intersection of the straight lines x −y = 1 and 2x + y = 3 is: JEE Main 2016 (10 Apr Online) JEE Main Previous Year Paper (1) x + 4y + 3 = 0 (2) 3x −y −4 = 0 (3) x −3y −4 = 0 (4) 4x + y −3 = 0

Q72.If the tangent at a point on the ellipse x2 27 + 3 = 1 meets the coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is (1) 3√3 (2) 92 (3) 9 (4) 9√3

Q72. P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 , respectively. If the normal at P passes through Q, then the minimum value of t21 , is (1) 8 (2) 4 (3) 6 (4) 2 y2

Q73.Let a and b respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation 9e2 −18e + 5 = 0 . If S(5, 0) is a focus and 5x = 9 is the corresponding directrix of this hyperbola, then a2 −b2 is equal to (1) −7 (2) −5 (3) 5 (4) 7 t2 f(x)−x2f(t)

Q74.Let P = lim (1 + tan2 √x ) , then log P is equal to x→0+ (1) 1 (2) 1 2 4 (3) 2 (4) 1

Q74. lim 2x tan(1−cosx−x2x)2tan 2x x→0 (1) 2 (2) −12 (3) −2 (4) 12

Q75.If x→∞(1lim + (1) 2 (2) 32 (3) 1 (4) 2 2 3

Q76.The mean of 5 observations is 5 and their variance is 12. 4. If three of the observations are 1, 2 & 6; then the value of the remaining two is : (1) 1, 11 (2) 5, 5 (3) 5, 11 (4) None of these

Q76.If the standard deviation of the numbers 2, 3, a and 11 is 3. 5, then which of the following is true ? (1) 3a2 −34a + 91 = 0. (2) 3a2 −23a + 44 = 0. (3) 3a2 −26a + 55 = 0. (4) 3a2 −32a + 84 = 0.

Q77.The angle of elevation of the top of a vertical tower from a point A, due east of it is 45o . The angle of elevation of the top of the same tower from a point B, due south of A is 30o . If the distance between A and B is 54√2m , then the height of the tower (in meters), is: (1) 108 (2) 36√3 (3) 54√3 (4) 54

Q77.If the mean deviation of the numbers 1, 1 + d, … , 1 + 100d from their mean is 255 , then a value of d is : (1) 10. 1 (2) 5. 05 (3) 20. 2 (4) 10 Q78. ⎡ √32 21 ⎤ 1 1 T If P = , A = and Q = PAP T, then P Q2015 P is : √3 [0 1 ] ⎣−12 2 ⎦ (1) [00 20150 ] (2) [20151 20150 ] (3) [10 20151 ] (4) [20150 20151 ]

Q77.A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30° . After walking for 10 minutes from JEE Main 2016 (03 Apr) JEE Main Previous Year Paper A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60° . Then the time taken (in minutes) by him, from B to reach the pillar, is (1) 20 (2) 5 (3) 6 (4) 10

Q78.If A = [ 5a3 −b2 ] and A. adjA = A AT , then 5a + b is equal to (1) 4 (2) 13 (3) −1 (4) 5

Q78.Let A, be a 3 × 3 matrix, such that A2 −5A + 7I = O. Statement - I : A−1 = 71 (5I −A). Statement - II : The polynomial A3 −2A2 −3A + I ,can be reduced to 5(A −4I). Then : (1) Both the statements are true (2) Both the statements are false (3) Statement - I is true, but Statement - II is false (4) Statement - I is false, but Statement - II is true , then the determinant of the matrix (A2016 −2A2015 −A2014) is :

Q79.The system of linear equations x + λy −z = 0 λx −y −z = 0 x + y −λz = 0 has a non -trivial solution for (1) Exactly two values of λ (2) Exactly three values of λ (3) Infinitely many values of λ (4) Exactly one value of λ

Q79. cos x sin x sin x The number of distinct real roots of the equation, sin x cos x sin x = 0 in the interval [−π4 , π4 ] is : sin x sin x cos x (1) 1 (2) 4 (3) 2 (4) 3