Practice Questions

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 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.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.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.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.

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.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

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.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.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

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}

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

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. ABCD is a trapezium such that AB and CD are parallel and BC ⊥CD. If ∠ADB = θ, BC = p and CD = q , then AB is equal to (1) p2+q2 (2) (p2+q2) sin θ p2 cos θ+q2 sin θ (p cos θ+q sin θ)2 (3) (p2+q2) sin θ (4) p2+q2 cos θ p cos θ+q sin θ p cos θ+q sin θ

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 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.

Q78.Statement-1: The system of linear equations x + (sin α)y + (cos α)z = 0 x + (cos α)y + (sin α)z = 0 x −(sin α)y −(cos α)z = 0 has a non-trivial solution for only one value of α lying in the interval (0, π2 ). Statement-2: The equation in α cos α sin α cos α sin α cos α sin α = 0 cos α −sin α −cos α has only one solution lying in the interval (0, π2 ) (1) Statement-1 is true, Statement-2 is true, (2) Statement-1 is true, Statement-2 is true, Statement-2 is not correct explantion for Statement-2 is a correct explantion for Statement-1. Statement-1. (3) Statement-1 is true, Statement- 2 is false. (4) Statememt-1 is false, Statement-2 is true. , then tan S is equal to :

Q78.Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A × B having 3 or more elements is : (1) 219 (2) 211 (3) 256 (4) 220 Q79. ⎡1 α 3 ⎤ If P = 1 3 3 is the adjoint of a 3 × 3 matrix A and |A| = 4 , then α is equal to ⎣2 4 4 ⎦ (1) 5 (2) 0 (3) 4 (4) 11

Q78. a b c If a, b, c are sides of a scalene triangle, then the value of b c a is : c a b (1) non - negative (2) negative (3) positive (4) non-positive JEE Main 2013 (09 Apr Online) JEE Main Previous Year Paper

Q78.Consider the function : f(x) = [x] + |1 −x|, −1 ≤x ≤3 where [x] is the greatest integer function. Statement −x, −1 ≤x < 0 1 −x, 0 ≤x < 1 1: f is not continuous at x = 0, 1, 2 and 3 Statement 2:f(x)= = 1 + x, 1 ≤x < 2 2 + x, 2 ≤x ≤3 (1) Statement 1 is true; Statement 2 is false, (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 true; (4) Statement 1 is false; Statement 2 is true. Statement It is a correct explanation for Statement 1.

Q79. S = tan−1 ( n2+n+11 ) + tan−1 ( n2+3n+31 ) + … + tan−1 ( 1+(n+19)(n+20)1 ) (1) 20 (2) n 401+20n n2+20n+1 (3) 20 (4) n n2+20n+1 401+20n

Q79.A spherical balloon is being inflated at the rate of 35cc/min . The rate of increase in the surface area (in cm2/min.) of the balloon when its diameter is 14 cm, is : JEE Main 2013 (25 Apr Online) JEE Main Previous Year Paper (1) 10 (2) √10 (3) 100 (4) 10√10