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

Q74.The median of 100 observations grouped in classes of equal width is 25 . If the median class interval is 20-30 and the number of observations less than 20 is 45 , then the frequency of median class is (1) 10 (2) 20 (3) 15 (4) 12 JEE Main 2012 (19 May Online) JEE Main Previous Year Paper

Q74.Let p and q denote the following statements p : The sun is shining q : I shall play tennis in the afternoon The negation of the statement "If the sun is shining then I shall play tennis in the afternoon", is (1) q ⇒∼p (2) q∧∼p (3) p∧∼q (4) ∼q ⇒∼p

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 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

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

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.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.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: 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.

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

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.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

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.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

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.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]

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.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

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.