Practice Questions

Q74.The frequency distribution of daily working expenditure of families in a locality is as follows: If the mode of the distribution is Rs. 140, then the value of b is (1) 34 (2) 31 (3) 26 (4) 36

Q75.In a ΔPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to (1) 5π (2) π 6 6 (3) π (4) 3π 4 4 JEE Main 2012 (Offline) JEE Main Previous Year Paper Q76. ⎛1 0 0⎞ ⎛1⎞ ⎛0⎞ Let A = 2 1 0 . If u1 and u2 are column matrices such that Au1 = 0 and Au2 = 1 , then ⎝3 2 1⎠ ⎝0⎠ ⎝0⎠ u1 + u2 is equal to (1) ⎛−1⎞ (2) ⎛ −1⎞ 1 1 ⎝ 0 ⎠ ⎝ −1⎠ (3) ⎛−1⎞ (4) ⎛ 1 ⎞ −1 −1 ⎝ 0 ⎠ ⎝ −1⎠

Q75.Statement 1: The variance of first n odd natural numbers is n2−1 Statement 2: The sum of first n odd natural 3 n(4n2+1) number is n2 and the sum of square of first n odd natural numbers is . 3 (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 false, Statement 2 is true. (4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1. Q76. ⎡ 1 0 0⎤ ⎡ 1 0 0 ⎤ If A = 2 1 0 and B = −2 1 0 then AB equals ⎣−3 2 1⎦ ⎣ 7 −2 1 ⎦ (1) I (2) A (3) B (4) 0

Q75.If three distinct points A, B, C are given in the 2dimensional coordinate plane such that the ratio of the distance of each one of them from the point (1, 0) to the distance from (−1, 0) is equal to 12 , then the circumcentre of the triangle ABC is at the point (1) ( 35 , 0) (2) (0, 0) (3) ( 13 , 0) (4) (3, 0) Q76. ⎡ 0 0 a ⎤ If AT denotes the transpose of the matrix A = 0 b c , where a, b, c, d, e and f are integers such that ⎣ d e f ⎦ abd ≠0 , then the number of such matrices for which A−1 = AT is (1) 2(3!) (2) 3(2!) (3) 23 (4) 32

Q75.If two vertical poles 20 m and 80 m high stand apart on a horizontal plane, then the height (in m ) of the point of intersection of the lines joining the top of each pole to the foot of other is (1) 16 (2) 18 (3) 50 (4) 15

Q76.Let X and Y are two events such that P(X ∪Y =)PX ∩(Y . ) Statement 1: P ∩Y ′ = ˙PX ′ ∩(Y = 0 ) Statement 2: P(X)PY ∈2)PX ∩Y ( ) (X (1) Statement 1 is false, Statement 2 is true. (2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1. (3) Statement 1 is true, Statement 2 is false. (4) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1. Q77. ⎛ α −1⎞ ⎛ α + 1⎞ If A = 0 , B = 0 be two matrices, then ABT is a non-zero matrix for |α| not equal to ⎝ 0 ⎠ ⎝ 0 ⎠ (1) 2 (2) 0 (3) 1 (4) 3

Q76.Statement 1: If A and B be two sets having p and q elements respectively, where q > p. Then the total number of functions from set A to set B is qp Statement 2: The total number of selections of p different objects out of q objects is qCp . (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 of Statement 1. (3) Statement 1 is false, Statement 2 is true (4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.

Q77.If a, b, c, are non zero complex numbers satisfying a2 + b2 + c2 = 0 and b2 + c2 ab ac ab c2 + a2 bc = ka2b2c2 , then k is equal to ac bc a2 + b2 (1) 1 (2) 3 (3) 4 (4) 2 is 3

Q77.Let P and Q be 3 × 3 matrices with P ≠Q. If P 3 = Q3 and P 2Q = Q2P , then determinant of (P 2 + Q2) is equal to (1) −2 (2) 1 (3) 0 (4) −1

Q77.Statement 1: A function f : R →R is continuous at x0 if and only if limx→x0 f(x) exists and limx→x0 f(x) = f (x0⋅) Statement 2: A function f : R →R is discontinuous at x0 if and only if, limx→x0 f(x) exists and limx→x0 f(x) ≠f (x0. ) (1) Statement 1 is true, Statement 2 is true, (2) Statement 1 is false, Statement 2 is true. Statement 2 is not a correct explanation of Statement 1. (3) Statement 1 is true, Statement 2 is true, (4) Statement 1 is true, Statement 2 is false. Statement 2 is a correct explanation of Statement 1.

Q77.Statement 1: If the system of equations x + ky+ 3z = 0, 3x + ky −2z = 0, 2x + 3y −4z = 0 has a nontrivial solution, then the value of k is 31 . Statement 2: A system of three homogeneous equations in three variables 2 has a non trivial solution if the determinant of the coefficient matrix is zero. JEE Main 2012 (26 May Online) JEE Main Previous Year Paper (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 true,, (4) Statement 1 is true, Statement 2 is false. Statement 2 is not a correct explanation for Statement 1.

Q78.If f ′(x) = sin(log x) and y = f ( 3−2x2x+3 ), then dxdy equals (1) sin [log ( 2x+33−2x )] (2) (3−2x2)12 (3) (3−2x2) 12 sin [log ( 3−2x2x+3 )] (4) (3−2x212 cos [log ( 2x+33−2x )] JEE Main 2012 (12 May Online) JEE Main Previous Year Paper

Q78.Let A and B be non empty sets in R and f : A →B is a bijective function. Statement 1: f is an onto function. Statement 2: There exists a function g : B →A such that fog = IB . (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 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.

Q78.A value of tan−1 (sin (cos−1 (√2 ))) (1) π (2) π 4 2 (3) π (4) π 3 6

Q78.If the system of equations x + y + z = 6 x + 2y + 3z = 10 x + 2y + λz = 0 has a unique solution, then λ is not equal to (1) 1 (2) 0 (3) 2 (4) 3

Q78.If f : R →R is a function defined by f(x) = [x] cos ( 2x−12 )π, where [x] denotes the greatest integer function, then f is (1) continuous for every real x (2) discontinuous only at x = 0 (3) discontinuous only at non-zero integral values of (4) continuous only at x = 0 x

Q79.Consider the function f(x) = |x −2| + |x −5|, x ∈R. Statement 1: f ′(4) = 0 Statement 2 : f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5). (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 true; statement (4) Statement 1 is true, statement 2 is false 2 is not a correct explanation for statement 1

Q79.If P(S) denotes the set of all subsets of a given set S , then the number of one-to-one functions from the set S = {1, 2, 3} to the set P(S) is (1) 24 (2) 8 (3) 336 (4) 320

Q79.If f(x) = a| sin x| + be|x| + c|x|3 , where a, b, c ∈R, is differentiable at x = 0, then (1) a = 0, b and c are any real numbers (2) c = 0, a = 0, b is any real number (3) b = 0, c = 0, a is any real number (4) a = 0, b = 0, c is any real number

Q79.Consider a rectangle whose length is increasing at the uniform rate of 2 m/sec, breadth is decreasing at the uniform rate of 3 m/sec and the area is decreasing at the uniform rate of 5 m2/sec. If after some time the breadth of the rectangle is 2 m then the length of the rectangle is (1) 2 m (2) 4 m (3) 1 m (4) 3 m

Q79.The range of the function f(x) = 1+|x|x , x ∈R, is (1) R (2) (−1, 1) (3) R −{0} (4) [−1, 1]

Q80.A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72 π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is (1) 9 (2) 7 7 9 (3) 2 (4) 9 9 2

Q80.Let f : (−∞, ∞) →(−∞, ∞) be defined by f(x) = x3 + 1 Statement 1: The function fhas a local extremum at x = 0 Statement 2: The function f is continuous and differentiable on (−∞, ∞) and f ′(0) = 0 (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 true, Statement 2 is true, (4) Statement 1 is false, Statement 2 is true. Statement 2 is not the correct explanation for Statement 1.

Q80.Let f : [1, 3] →R be a function satisfying x ≤f(x) ≤√6 −x, for all x ≠2 and f(2) = 1, where R is the [x] set of all real numbers and [x] denotes the largest integer less than or equal to x. Statement 1: limx→2−f(x) exists. Statement 2: f is continuous at x = 2. (1) Statement 1 is true, Statement 2 is true, (2) Statement 1 is false, Statement 2 is true. Statement 2 is a correct explanation for Statement 1. (3) Statement 1 is true, Statement 2 is true, (4) Statement 1 is true, Statement 2 is false. Statement 2 is not a correct explanation for Statement 1.

Q80.Let f(x) = sin x, g(x) = x. Statement 1: f(x) ⩽gx( for )x in (0, ∞) Statement 2: f(x) ≤1 for x in (0, ∞) but g(x) →∞ as x →∞. JEE Main 2012 (07 May 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 true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1. (4) Statement 1 is false, Statement 2 is true.