Practice Questions

Q69.A ray of light along x + √3y = √3 gets reflected upon reaching X−axis, the equation of the reflected ray is (1) y = √3x −√3 (2) √3y = x −1 (3) y = x + √3 (4) √3y = x −√3

Q69.Let x ∈(0, 1). The set of all x such that sin−1 x > cos−1 x, is the interval: 1 (1) (2) 1 , ( 2 , √21 ) ( √2 1) (3) (0, 1) (4) √3 2 (0, )

Q70.The x−coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as (0, 1), (1, 1) and (1, 0) is (1) 1 + √2 (2) 1 −√2 (3) 2 + √2 (4) 2 −√2

Q70.If each of the lines 5x + 8y = 13 and 4x −y = 3 contains a diameter of the circle x2 + y2 −2 (a2 −7a + 11) x −2 (a2 −6a + 6)y + b3 + 1 = 0, then : (1) a = 5 and b ∉(−1, 1) (2) a = 1 and b ∉(−1, 1) (3) a = 2 and b ∉(−∞, 1) (4) a = 5 and b ∈(−∞, 1)

Q70.The acute angle between two lines such that the direction cosines l, m, n, of each of them satisfy the equations l + m + n = 0 and l2 + m2 −n2 = 0 is : (1) 15∘ (2) 30∘ (3) 60∘ (4) 45∘

Q70.The point of intersection of the normals to the parabola y2 = 4x at the ends of its latus rectum is : (1) (0, 2) (2) (3, 0) (3) (0, 3) (4) (2, 0)

Q71.A tangent to the hyperbola x2 meets x-axis at P and y-axis at Q. Lines PR and QR are drawn such 4 −y22 = 1 that OPRQ is a rectangle (where O is the origin). Then R lies on : (1) 4 + 2 = 1 (2) 2 − 4 = 1 x2 y2 x2 y2 (3) 2 + 4 = 1 (4) 4 − 2 = 1 x2 y2 x2 y2

Q71.The circle passing through (1, −2) and touching the axis of x at (3, 0) also passes through the point (1) (5, −2) (2) (−2, 5) (3) (−5, 2) (4) (2, −5)

Q71.If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle 60∘ on the circumference of the first circle, then the radius of the arc is: JEE Main 2013 (25 Apr Online) JEE Main Previous Year Paper (1) √3 (2) 12 (3) 1 (4) None of these

Q71.If a circle C passing through (4, 0) touches the circle x2 + y2 + 4x −6y −12 = 0 externally at a point (1, −1) , then the radius of the circle C is : (1) 5 (2) 2√5 (3) 4 (4) √57

Q72.For integers m and n, both greater than 1, consider the following three statements : P : m divides n Q : m divides n2 R : m is prime, then (1) Q ∧R →P (2) P ∧Q →R (3) Q →R (4) Q →P

Q72.A point on the ellipse, 4x2 + 9y2 = 36 , where the normal is parallel to the line, 4x −2y −5 = 0 , is : (1) ( 95 , 85 ) (2) ( 85 , −95 ) (3) (−95 , 85 ) (4) ( 85 , 95 )

Q72.Equation of the line passing through the points of intersection of the parabola x2 = 8y and the ellipse x2 3 + y2 = 1 is : (1) y −3 = 0 (2) y + 3 = 0 (3) 3y + 1 = 0 (4) 3y −1 = 0

Q73.If the median and the range of four numbers {x, y, 2x + y, x −y}, where 0 < y < x < 2y, are 10 and 28 respectively, then the mean of the numbers is : (1) 18 (2) 10 (3) 5 (4) 14

Q73.Consider the system of equations : x + ay = 0, y + az = 0 and z + ax = 0 . Then the set of all real values of ' a ' for which the system has a unique solution is: (1) R −{1} (2) R −{−1} (3) {1, −1} (4) {1, 0, −1}

Q73.If a and c are positive real numbers and the ellipse x2 + y2 = 1 has four distinct points ir common with the 4c2 c2 circle x2 + y2 = 9a2 , then (1) 9ac −9a2 −2c2 < 0 (2) 6ac + 9a2 −2c2 < 0 (3) 9ac −9a2 −2c2 > 0 (4) 6ac + 9a2 −2c2 > 0

Q73.The equation of the circle passing through the foci of the ellipse x216 + y29 = 1 , and having centre at (0, 3) is (1) x2 + y2 −6y −5 = 0 (2) x2 + y2 −6y + 5 = 0 (3) x2 + y2 −6y −7 = 0 (4) x2 + y2 −6y + 7 = 0

Q73.Let the equations of two ellipses be x2 y2 x2 y2 E1 : + = 1 and E2 : + = 1, 3 2 16 b2 If the product of their eccentricities is 1 , then the length of the minor axis of ellipse E2 is : 2 (1) 8 (2) 9 (3) 4 (4) 2

Q74.If the extremities of the base of an isosceles triangle are the points (2a, 0) and (0, a) and the equation of one of the sides is x = 2a, then the area of the triangle, in square units, is : (1) 5 a2 (2) 5 a2 4 2 (3) 25a2 (4) 5a2 4

Q74.The value of limx→0 x1 [tan−1 ( 2x+1x+1 ) −π4 ] is : (1) 1 (2) −12 (3) 2 (4) 0

Q74.The value of lim (1−cosx2x)(3+costan 4x x) is equal to x→0 (1) 1 (2) 2 (3) −14 (4) 21

Q74.The statement p →(q →p) is equivalent to : (1) p →q (2) p →(p ∨q) (3) p →(p →q) (4) p →(p ∧q)

Q74.Let p and q be any two logical statements and r : p →(∼p ∨q). If r has a truth value F , then the truth values of p and q are respectively: (1) F, F (2) T, T (3) T, F (4) F, T

Q75.On the sides AB, BC, CA of a △ABC, 3, 4, 5 distinct points (excluding vertices A, B, C ) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are : (1) 210 (2) 205 (3) 215 (4) 220

Q75.Mean of 5 observations is 7 . If four of these observations are 6, 7, 8, 10 and one is missing then the variance of all the five observations is : (1) 4 (2) 6 (3) 8 (4) 2