Practice Questions

Q60.The statement (p →(q →p)) →(p →(p ∨q)) is : (1) equivalent to (p ∧q) ∨(~q) (2) a contradiction (3) equivalent to (p ∨q) ∧(~p) (4) a tautology

Q61. x −2 2x −3 3x −4 If Δ = 2x −3 3x −4 4x −5 = Ax3 + Bx2 + Cx + D , then B + C is equal to : 3x −5 5x −8 10x −17 (1) −1 (2) 1 (3) −3 (4) 9 Q62. 2π −(sin−1 45 + sin−1 135 + sin−1 1665 ) is equal to : (1) π (2) 5π 2 4 (3) 3π (4) 7π 2 4

Q61.Let f : (1, 3) →R, be a function defined by f(x) = x[x] , where [x], denotes the greatest integer ≤x. Then 1+x2 the range of f , is (1) ( 25 , 35 ] ∪( 34 , 45 ) (2) ( 25 , 12 ) ∪( 35 , 45 ] (3) ( 25 , 45 ] (4) ( 35 , 45 )

Q61.If the mean and the standard deviation of the data 3, 5, 7, a, b are 5and 2 respectively, then a and b are the roots of the equation: (1) x2 −10x + 18 = 0 (2) 2x2 −20x + 19 = 0 (3) x2 −10x + 19 = 0 (4) x2 −20x + 18 = 0

Q61.If the system of equations x + y + z = 2 2 x + 4 y −z = 6 3x + 2y + λz = μ has infinitely many solutions, then : (1) λ + 2μ = 14 (2) 2λ −μ = 5 (3) λ −2μ = −5 (4) 2λ + μ = 14

Q61.For which of the following ordered pairs (μ, δ), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = μ 4x + 4y + 4z = δ is inconsistent? (1) (4, 3) (2) (4, 6) (3) (1, 0) (4) (3, 4)

Q61.If for some α and β in R , the intersection of the following three planes x + 4y −2z = 1 x + 7y −5z = β x + 5y + αz = 5 is a line in R3 , then α + β is equal to: (1) 0 (2) 10 (3) 2 (4) −10 Q62. ; x < 0 ⎧ sin(a+2)x+sinxx If f(x) = is continuous at x = 0 , then a + 2b is equal to: ⎨ b ; x = 0 ; x > 0 ⎩ (x+3x2)1/3−x1/3x1/3 (1) 1 (2) −1 (3) 0 (4) −2

Q61.Let S be the set of all λ ∈R for which the system of linear equations 2x −y + 2z = 2 x −2y + λz = −4 x + λy + z = 4 has no solution. Then the set S (1) Contains more than two elements (2) Is an empty set (3) Is a singleton (4) Contains exactly two elements

Q61.A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be: (1) 63 (2) 36 (3) 54 (4) 38

Q61.Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m ) above the line AC is : (1) 20/3 (2) 5 (3) 10/3 (4) 6

Q61.Let R1 and R2 be two relations defined as follows : R1 = {(a, b) ∈R2 : a2 + b2 ∈Q} and R2 = {(a, b) ∈R2 : a2 + b2 ∉Q} , where Q is the set of all rational numbers, then (1) R1 is transitive but R2 is not transitive. (2) R2 is transitive but R1 is not transitive. (3) Neither R1 nor R2 is transitive. (4) R1 and R2 are both transitive. Q62. ⎡ 2 −1 1 ⎤ Let A be a 3 × 3 matrix such that adj A = −1 0 2 and B =adj (adjA). If |A| = λ and ⎣ 1 −2 −1 ⎦ (B−1) ⊤= μ, then the ordered pair (|λ|, μ) is equal to (1) (3, 811 ) (2) (9, 91 ) (3) (3, 81) (4) (9, 811 )

Q61.Which of the following is a tautology? (1) (~p) ∧(p ∨q) →q (2) (q →p) ∨~(p →q) (3) (~q) ∨(p ∧q) →q (4) (p →q) ∧(q →p) Q62. ⎡ 1 2 1 ⎤ Let A = where P = −2 3 −4 then the set A {X = (x, y, z)T : PX = 0 and x2 + y2 + z2 = 1} ⎣ 1 9 −1 ⎦ (1) Is a singleton. (2) Is an empty set. (3) Contains more than two elements (4) Contains exactly two elements Q63. ⎡a b c ⎤ Let a, b, c ∈R be all non-zero and satisfies a3 + b3 + c3 = 2. If the matrix A = b c a satisfies ⎣ c a b ⎦ ATA = I, then a value of abc can be (1) −13 (2) 13 (3) 3 (4) 23

Q61. cos2 x 1 + sin2 x sin 2x Let m and M be respectively the minimum and maximum value values of 1 + cos2 x sin2 x sin 2x cos2 x sin2 x 1 + sin 2x Then the ordered pair (m, M) is equal to: (1) (3, 3) (2) (−3, −1) (3) (4, 1) (4) (1, 3)

Q61.Let a −2b + c = 1. x + a x + 2 x + 1 If f(x) = x + b x + 3 x + 2 , then: x + c x + 4 x + 3 (1) f(−50) = 501 (2) f(−50) = −1 (3) f(50) = −501 (4) f(50) = 1 4 ] = A. Then the function, f(x) = [x2] sin(πx) is x

Q61.For a suitably chosen real constant a, let a function, f : R −{−a} →R be defined by f(x) = a+xa−x . Further supposed that for any real number x ≠−a,and f(x) ≠−a, (fof)(x) = x. Then f(−12 ) is equal to : (1) 3 1 (2) −13 (3) −3 (4) 3

Q61.Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j−2)aij , where i, j = 1,2, 3 . If the determinant of B is 81 , then determinant of A is (1) 1 (2) 3 3 (3) 1 (4) 1 81 9

Q61.If g(x) = x2 + x −1 and (gof)(x) = 4x2 −10x + 5, then f( 54 ) is equal to (1) 3 2 (2) −12 (3) 2 1 (4) −32 tanα+cotα 1 3π dy 5π + sin2α , α ∈( 4 , π), then dα at α = 6 is 1+tan2α )

Q62.The values of λ and μ for which the system of linear equations x + y + z = 2, x + 2 y + 3 z = 5, x + 3y + λz = μ has infinitely many solutions, are respectively (1) 6 and 8 (2) 5 and 7 (3) 5 and 8 (4) 4 and 9 ∞ Σ = 2, x, y ∈N , where N is the set of all natural numbers, then the value

Q62.A survey shows that 63% of the people in a city read newspaper A whereas 76% read news paper B. If x% of the people read both the newspapers, then a possible value of x can be: (1) 29 (2) 37 (3) 65 (4) 55 where i = √−1, then which one of the following is not (θ = 24π ) and A5 = [ ac bd ],

Q62.Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector 1 0 0 b on the right side is equal to b1, b2 and b3 respectively. If x1 = ⎡ 1 ⎤, x2 = ⎡2 ⎤, x3 = ⎡0⎤ ; 1 1 1 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 1 0 0 b1 = ⎡ 0 ⎤, b2 = ⎡ 2 ⎤, b3 = ⎡0 ⎤, then the determinant of A is equal to 0 0 2 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (1) 4 (2) 2 (3) 1 (4) 3 2 2

Q62.Let f : R →R be a function defined by f(x) = max {x, x2}.Let S denote the set of all points in R,where f is not differentiable.Then : (1) {0, 1} (2) {0} (3) ϕ (an empty set) (4) {1} π ,

Q62.If y(α) = √2( (1) 4 (2) 43 (3) −4 (4) −14

Q62.If the minimum and the maximum values of the function f : [ π4 , π2 ] →R, defined by −sin2 θ −1 −sin2 θ 1 f(θ) = −cos2 θ −1 −cos2 θ 1 are m and M respectively, then the ordered pair (m, M) is equal to : 12 10 −2 (1) (0, 2√2) (2) (−4, 0) (3) (−4, 4) (4) (0, 4)

Q62.Let [t] denote the greatest integer ≤t and x→0x[lim discontinuous, when x is equal to: (1) √A + 1 (2) √A + 5 (3) √A + 21 (4) √A

Q62.The inverse function of f(x) = 82x−8−2x , x ∈(−1, 1), is __________. 82x+8−2x (1) 4 1 loge( 1+x1−x ) (2) 14 loge( 1−x1+x ) (3) 1 4 (loge) loge( 1−x1+x ) (4) 14 log8( 1+x1−x ) = π6 , then