Practice Questions

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.

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.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.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.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.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}

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.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 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.In a set of 2n observations, half of them are equal to ' a ' and the remaining hall are equal to ' −a′ '. If the standard deviation of all the observations is 2 ; then the value of |a| is : (1) 2 (2) √2 (3) 4 (4) 2√2

Q75.Statement-1: The statement A →(B →A) is equivalent to A →(A ∨B). Statement-2: The statement ∼[(A ∧B) →(∼A ∨B)] is a Tautology. (1) Statement- 1 is false; Statement- 2 is true. (2) Statement-1 is true; Statement-2 is true; Statement- 2 is not 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 the correct explanation for Statement-1.

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

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.Consider : Statement - I : (p ∧~q) ∧(~p ∧q) is a fallacy. Statement - II : (p →q) ↔(~q →~p) is a tautology. (1) Statement - I is true; statement - II is false. (2) Statement - I is false; Statement - II is true. (3) Statement - I 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.

Q76.The mean of a data set consisting of 20 observations is 40 . If one observation 53 was wrongly recorded as 33 , then the correct mean will be: (1) 41 (2) 49 (3) 40.5 (4) 42.5

Q76.Let R = {(x, y) : x, y ∈N and x2 −4xy + 3y2 = 0}, where N is the set of all natural numbers. Then the relation R is : (1) reflexive but neither symmetric nor transitive. (2) symmetric and transitive. (3) reflexive and symmetric, (4) reflexive and transitive. JEE Main 2013 (23 Apr Online) JEE Main Previous Year Paper

Q76.A common tangent to the conics x2 = 6y and 2x2 −4y2 = 9 is: (1) x −y = 32 (2) x + y = 1 (3) x + y = 92 (4) x −y = 1 Then the number of non-singular matrices in the set S is : : aij ∈{0, 1, 2}, a11 = a22}

Q76.All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ? (1) mode (2) variance (3) mean (4) median

Q76.If two vertices of an equilateral triangle are A(−a, 0) and B(a, 0), a > 0, and the third vertex C lies above x- axis then the equation of the circumcircle of △ABC is : (1) 3x2 + 3y2 −2√3ay = 3a2 (2) 3x2 + 3y2 −2ay = 3a2 (3) x2 + y2 −2ay = a2 (4) x2 + y2 −√3ay = a2

Q77.Let A , other than I or −I, be a 2 × 2 real matrix such that A2 = I, I being the unit matrix. Let Tr(A) be the sum of diagonal elements of A. Statement-1: Tr(A) = 0 Statement-2: det(A) = −1 (1) Statement-1 is true; Statement- 2 is false. (2) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (3) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1. (4) Statement-1 is false; Statement- 2 is true.

Q77.The matrix A2 + 4A −5I , where I is identity matrix and A = [14 −32 ], equals : (1) 2 1 (2) 0 −1 4 4 [2 0 ] [2 2 ] (3) 2 1 (4) 1 1 32 32 [2 0 ] [1 0 ]

Q77.Let R = {(3, 3)(5, 5), (9, 9), (12, 12), (5, 12), (3, 9) , (3, 12), (3, 5)} be a relation on the set A = {3, 5, 9, 12} . Then, R is : (1) reflexive, symmetric but not transitive. (2) symmetric, transitive but not reflexive. (3) an equivalence relation. (4) reflexive, transitive but not symmetric. Q78. ⎡3 4 1 ⎤ If p, q, r are 3 real numbers satisfying the matrix equation, [pqr] 3 2 3 = [3 0 1 ] then 2p + q −r ⎣2 0 2 ⎦ equals : (1) −3 (2) −1 (3) 4 (4) 2

Q77.Let S = {( a11a21 a12a22 ) (1) 27 (2) 24 (3) 10 (4) 20