Practice Questions

Q88.The shortest distance between the z - axis and the line x + y + 2z −3 = 0 = 2x + 3y + 4z −4, is (1) 1 (2) 2 (3) 3 (4) 4

Q89.If the shortest distance between the line x−1α = y+1−1 = 1z , (α ≠−1) , and x + y + z + 1 = 0 = 2x −y + z + 3 is 1 ,then value of α is : √3 (1) −1916 (2) 3219 (3) −1619 (4) 1932

Q90.If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is: (1) 1 (2) 1 69 26 (3) 1 (4) 1 21 15 JEE Main 2015 (11 Apr Online) JEE Main Previous Year Paper

Q90.If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is (1) 22( 13 )11 (2) 195 (3) 55( 32 )10 (4) 220( 31 )12 JEE Main 2015 (04 Apr) JEE Main Previous Year Paper

Q63.Let w(Im w≠0) be a complex number. Then, the set of all complex numbers z satisfying the equation ¯w −wz = k(1 −z), for some real number k, is (1) {z : z ≠1} (2) {z : |z| = 1, z ≠1} ¯(3) {z : z = z} (4) {z : |z| = 1}

Q64.Let f(n) = [ 13 + 1003n ]n, where [n] denotes the greatest integer less than or equal to n. Then ∑56n=1 f(n) is equal to (1) 56 (2) 1287 (3) 1399 (4) 689

Q64.If (10)9 + 2(11)1(10)8 + 3(11)2(10)7 + ...... + 10(11)9 = k(10)9, then k is equal to : JEE Main 2014 (06 Apr) JEE Main Previous Year Paper (1) 100 (2) 110 (3) 121 (4) 441 10 100

Q66.If 1 + x4 + x5 = ∑5i=0 ai (1 + xi), for all x in R, then a2 is: (1) −4 (2) 6 (3) −8 (4) 10 is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive

Q66.If the sum 3 + 5 + 7 + .... .+ up to 20 terms is equal to 21k , then k is equal to 12 12+22 12+22+32 (1) 240 (2) 120 (3) 60 (4) 180

Q66.If the coefficients of x3 and x4 in the expansion of (1 + ax + bx2)(1 −2x)18 in powers of x are both zero, then (a, b) is equal to (1) (14, 2723 ) (2) (16, 2723 ) (3) (16, 2513 ) (4) (14, 2513 )

Q71.Let a and b be any two numbers satisfying 1 + 1 = 14 . Then, the foot of perpendicular from the origin on a2 b2 the variable line x a + yb = 1 lies on : (1) A circle of radius = 2 (2) A hyperbola with each semi-axis = √2 . (3) A hyperbola with each semi-axis = 2 (4) A circle of radius = √2

Q71.The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangent to it is (1) (x2 + y2) 2 = 6x2 + 2y2 (2) (x2 + y2) 2 = 6x2 −2y2 (3) (x2 −y2) 2 = 6x2 + 2y2 (4) (x2 −y2) 2 = 6x2 −2y2

Q72.Let P(3 sec θ, 2 tan θ) and Q(3 sec ϕ, 2 tan ϕ) where θ + ϕ = π2 , be two distinct points on the hyperbola x2 . Then the ordinate of the point of intersection of the normals at P and Q is: 9 −y24 = 1 (1) 11 3 (2) −113 (3) 13 2 (4) −132 = 5, then k is equal to:

Q73.Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement p ⇒(q ∨r) is: (1) (p ∨q) ⇒r (2) (p ⇒q) ∨(p ⇒r) (3) (p ⇒∼q) ∧(p ⇒r) (4) (p ⇒q) ∧(p ⇒∼r)

Q75.Two ships A and B are sailing straight away from a fixed point O along routes such that ∠AOB is always 120∘ . At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr ): (1) 260 (2) 260 √37 37 (3) 80 (4) 80 √37 37

Q79.Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p′i(x) and p′′i(x) be the first and second order derivatives of pi(x) respectively. Let, p1(x) p′1(x) p′′1x( A(x) = ⎡ p2(x) p′2(x) p′′2( ⎤ ⎞ p3(x) p′3(x) p′′3(x ⎣ ⎦ ⎠ and B(x) = [A(x)]TA(x). Then determinant of B(x) : (1) is a polynomial of degree 6 in x. (2) is a polynomial of degree 3 in x. (3) is a polynomial of degree 2 in x. (4) does not depend on x.

Q79.If a, b, c are non - zero real numbers and if the system of equations (a −1)x = y + z JEE Main 2014 (09 Apr Online) JEE Main Previous Year Paper (b −1)y = x + z (c −1)z = x + y has a non - trivial solution, then ab + bc + ca equals : (1) −1 (2) a + b + c (3) abc (4) 1 is equal to :

Q81.Let f(x) = x|x|, g(x) = sin x and h(x) = (g ∘f)(x). Then (1) h(x) is not differentiable at x = 0. (2) h(x) is differentiable at x = 0, but h′(x) is not continuous at x = 0 (3) h′(x) is continuous at x = 0 but it is not (4) h′(x) is differentiable at x = 0 differentiable at x = 0

Q83.If m is a non-zero number and ∫x5m−1+2x4m−1 dx = f(x) + c, then f(x) is equal to (x2m+xm+1)3 (1) (x5m−x4m) (2) 1 x4m 2m(x2m+xm+1)2 2m (x2m+xm+1)2 (3) x5m (4) 2m(x5m+x4m) 2m(x2m+xm+1)2 (x2m+xm+1)2

Q85.The area (in sq. unit) of the region described by A = {(x, y) : x2 + y2 ≤1 and y2 ≤1 −x} is (1) π 2 −23 (2) π2 + 32 (3) π 2 + 34 (4) π2 −43

Q87.If ^x, ^y and ^z are three unit vectors in threedimensional space, then the minimum value of |^x + ^y|2 + |^y + ^z|2 + |^z + ^x|2 (1) 3 (2) 3 2 (3) 3√3 (4) 6

Q88.The image of the line x−1 3 = 1 = z−4−5 in the plane 2x −y + z +3=0 is the line (1) x−3 3 = y+51 = z−2−5 (2) x−3−3 = y+5−1 = z−25 (3) x+3 3 = y−51 = z−2−5 (4) x+3−3 = y−5−1 = z+25

Q88.Equation of the plane which passes through the point of intersection of lines x−1 3 = 1 = z−32 and x−3 1 = y−12 = z−23 and has the largest distance from the origin is: JEE Main 2014 (09 Apr Online) JEE Main Previous Year Paper (1) 4x + 3y + 5z = 50 (2) 3x + 4y + 5z = 49 (3) 5x + 4y + 3z = 57 (4) 7x + 2y + 4z = 54

Q89.Equation of the line of the shortest distance between the lines x 1 = −1 = 1z and x−10 = y+1−2 = 1z is JEE Main 2014 (19 Apr Online) JEE Main Previous Year Paper (1) −2 x = 1y = 2z (2) x1 = −1y = −2z y+1 (3) x−1 1 = −1 = −2z (4) x−11 = y+1−1 = 1z

Q68.Statement-1: The number of common solutions of the trigonometric equations 2 sin2 θ −cos 2θ = 0 and 2 cos2 θ −3 sin θ = 0 in the interval [0, 2π] is two. Statement-2: The number of solutions of the equation, 2 cos2 θ −3 sin θ = 0 in the interval [0, π] is two. (1) Statement-1 is true; Statement-2 is true; (2) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-2 is not a correct explanation for statement-1. statement-1. (3) Statement-1 is false; Statement-2 is true. (4) Statement-1 is true; Statement-2 is false.