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.If the circle x2 + y2 −6x −8y + (25 −a2) = 0 touches the axis of x, then a equals. (1) 0 (2) ±4 (3) ±2 (4) ±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, )

Q69.If the x-intercept of some line L is double as that of the line, 3x + 4y = 12 and the y-intercept of L is half as that of the same line, then the slope of L is : (1) −3 (2) −3/8 (3) −3/2 (4) −3/16

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.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 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.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)

Q70.Statement 1: The only circle having radius √10 and a diameter along line 2x + y = 5 is x2 + y2 −6x +2y = 0 . Statement 2 : 2x + y = 5 is a normal to the circle x2 + y2 −6x + 2y = 0 . (1) Statement 1 is false; Statement 2 is true. (2) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1. (3) Statement 1 is true; Statement 2 is false. (4) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

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.Statement-1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P. Statement-2: The system of parabolas y2 = 4ax satisfies a differential equation of degree 1 and order 1. JEE Main 2013 (09 Apr Online) JEE Main Previous Year Paper (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 true; Statement- 2 is false. (4) Statement-1 is false; Statement- 2 is true.

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

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.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

Q72.Statement-1: The line x −2y = 2 meets the parabola, y2 + 2x = 0 only at the point (−2, −2). Statement-2: The line y = mx − 2m1 (m ≠0) is tangent to the parabola, y2 = −2x at the point (− 2m21 , −1m ) JEE Main 2013 (22 Apr Online) JEE Main Previous Year Paper (1) Statement-1 is true; Statement- 2 is false. (2) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1. (3) Statement-1 is false; Statement-2 is true. (4) Statement-1 a true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.

Q72.Given : A circle, 2x2 + 2y2 = 5 and a parabola, y2 = 4√5x. Statement - I : An equation of a common tangent to these curves is y = x + √5 . Statement - II : If the line, y = mx + √5m (m ≠0) is their common tangent, then m satisfies m4 −3m2 + 2 = 0 . JEE Main 2013 (07 Apr) JEE Main Previous Year Paper (1) Statement - I is true; Statement - II is false. (2) Statement - I is false; Statement - II is true. (3) Statement - I is true; Statement - II is true; (4) Statement - I is true; Statement - II is true; Statement - II is a correct explanation for Statement - II is not a correct explanation for statement - I. statement - I.

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

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 )

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

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 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.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

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