Practice Questions

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.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.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.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.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.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.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 )

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.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

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.The domain of the function f(x) = sin−1( |x|+5x2+1 ) is (−∞, −a] ∪[a, ∞), then a is equal to (1) √17 (2) √17−1 2 2 (3) 1+√17 (4) √17 2 2 + 1 Q63. ⎧ aex + be−x, −1 ≤x < 1 If a function f(x) defined by f(x) = 1 ≤x ≤3 be continuous for some a, b, c ∈R and ⎨ cx2, ⎩ ax2 + 2cx, 3 < x ≤4 f ′(0) + f ′(2) = e, then the value of a is (1) 1 (2) e e2−3e+13 e2−3e−13 (3) e (4) e e2+3e+13 e2−3e+13

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

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

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 y = y(x) be a function of x satisfying y√1 −x2 = k −x√1 −y2 where k is a constant and y( 21 ) = −14 .Then dx dy at x = 12 , is equal to (1) −√54 (2) −√52 (3) 2 (4) √5 √5 2

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.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.If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for some k ∈R, then x + ( yz ) is equal to : (1) −3 (2) 9 (3) 3 (4) −9

Q63.Let xk + yk = ak, (a, k > 0) and dx 1 dy + ( xy ) 3 = 0, then k is (1) 3 (2) 4 2 3 (3) 32 (4) 13

Q63.The minimum value of 2sin x + 2cos x is : −1+ (1) √2 1 (2) 2−1+√2 2 (3) 21−√2 (4) 2 1−1√2 4 + tan−1 x, |x| ≤1 is :

Q63.If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then x a + y x + a y b + y y + b is equal to : z c + y z + c (1) y(b – a) (2) y (a – b) (3) 0 (4) y(a – c) at x = 21 is :

Q63.Let λ ∈R. The system of linear equations 2x1 −4x2 + λx3 = 1 x1 −6x2 + x3 = 2 λx1 −10x2 + 4x3 = 3 is inconsistent for : (1) exactly one positive value of λ (2) exactly one negative value of λ (3) every value of λ (4) exactly two values of λ

Q63.If y2 + loge(cos2 x) = y, x ∈(−π2 , π2 ) then : (1) y′′(0) = 0 (2) |y′(0)| + |y′′(0)| = 1 (3) |y′′(0)| = 2 (4) |y′(0)| + |y′′(0)| = 3

Q63.If A = [ cosisinθθ cosisinθθ ], true? (1) 0 ≤a2 + b2 ≤1 (2) a2 −d2 = 0 (3) a2 −c2 = 1 (4) a2 −b2 = 12 cos = a2 −b2 , where a > b > 0, then dxdy at ( π4 , π4 ) is:

Q63.If x = 2 sin θ −sin 2θ and y = 2 cos θ −cos 2θ , θ ∈[0, 2π], then d2y at θ = π is: dx2 (1) 4 3 (2) −38 (3) 2 3 (4) −34