Practice Questions

Q70.If the domain of the function f(x) = sin−1 ( 2x+3x−1 ) is R −(α, β), then 12αβ is equal to : (1) 32 (2) 40 (3) 24 (4) 36

Q70.If the system of linear equations 𝑥- 2𝑦+ 𝑧= - 4 2𝑥+ 𝛼𝑦+ 3𝑧= 5 3𝑥- 𝑦+ 𝛽𝑧= 3 has infinitely many solutions, then 12𝛼+ 13𝛽 is equal to (1) 60 (2) 64 (3) 54 (4) 58

Q70.Let B = [ 11 35 ] then 2β −α is equal to (1) 16 (2) 2 (3) 8 (4) 10 is equal to cot−1 √1−x1+x )dx

Q70.Let A be a square matrix such that AAT = I. Then 12 A[( A + AT)2 + (A −AT)2] (1) A2 + I (2) A3 + I (3) A2 + AT (4) A3 + AT

Q70.Let the mean and the variance of 6 observation 𝑎, 𝑏, 68, 44, 48, 60 be 55 and 194, respectively if 𝑎> 𝑏, then 𝑎+ 3𝑏 is (1) 200 (2) 190 (3) 180 (4) 210 Q71. 1 1 −1 −1 0 0 Let A be a 3 × 3 real matrix such that 𝐴 0 = 2 0 , 𝐴 0 = 4 0 , 𝐴 1 = 2 1 . Then, the system 1 1 1 1 0 0 𝑥 1 𝐴−3𝐼 𝑦 = 2 has 𝑧 3 (1) unique solution (2) exactly two solutions (3) no solution (4) infinitely many solutions

Q70.Consider the relations 𝑅1 and 𝑅2 defined as 𝑎𝑅1𝑏⇔𝑎2 + 𝑏2 = 1 for all 𝑎, 𝑏, ∈𝑅 and 𝑎, 𝑏𝑅2𝑐, 𝑑⇔𝑎+ 𝑑= 𝑏+ 𝑐 for all 𝑎, 𝑏, 𝑐, 𝑑∈𝑁× 𝑁. Then (1) Only 𝑅1 is an equivalence relation (2) Only 𝑅2 is an equivalence relation (3) 𝑅1 and 𝑅2 both are equivalence relation (4) Neither 𝑅1 nor 𝑅2 is an equivalence relation

Q70.Let a1, a2, . . . , a10 be 10 observations such that ∑10k=1 ak = 50 and ∑∀k<j ak ⋅aj = 1100. Then the standard deviation of a1, a2, … , a10 is equal to : (1) 5 (2) √5 (3) 10 (4) √115

Q71.Let f(x) = 4 cos3 x + 3√3 cos2 x −10. The number of points of local maxima of f in interval (0, 2π) is (1) 3 (2) 4 (3) 1 (4) 2

Q71.Let f, g : R →R be defined as : f(x) = |x −1| and g(x) = {ex,x + MARA1, xx ≥0≤0 Then the function f(g(x)) is (1) neither one-one nor onto. (2) one-one but not onto. (3) onto but not one-one. (4) both one-one and onto.

Q71.Let x = mn (m, n are co-prime natural numbers) be a solution of the equation cos(2 sin−1 x) = 19 and let α, β(α > β) be the roots of the equation mx2 −nx −m + n = 0. Then the point (α, β) lies on the line (1) 3x + 2y = 2 (2) 5x −8y = −9 (3) 3x −2y = −2 (4) 5x + 8y = 9 −1 < x < 1. Then at x = 12 , the value of 225(y′ −y′′) is equal to

Q71.If the domain of the function sin−1 ( 3x−222x−19 ) + loge ( 3x2−8x+5x2−3x−10 ) (1) 100 (2) 95 (3) 97 (4) 98

Q71.Let 𝑓: 𝑅→𝑅 be a function defined 𝑓𝑥= 𝑥 / 4 and 𝑔𝑥= 𝑓𝑓𝑓𝑓𝑥 then 18 ∫0√2√5 1 + 𝑥41 (1) 33 (2) 36 (3) 42 (4) 39

Q71.Let A = [ 10 21 ] and B = I + adj(A) + (adj A)2 + … + (adj A)10 . Then, the sum of all the elements of the matrix B is: (1) -124 (2) 22 (3) -88 (4) -110

Q71.Let the system of equations 𝑥+ 2𝑦+ 3𝑧= 5, 2𝑥+ 3𝑦+ 𝑧= 9, 4𝑥+ 3𝑦+ 𝜆𝑧= 𝜇 have infinite number of solutions. Then 𝜆+ 2𝜇 is equal to: (1) 28 (2) 17 (3) 22 (4) 15

Q71.Let f(x) = 7−sin1 5x be a function defined on R. Then the range of the function f(x) is equal to ; (1) [ 71 , 61 ] (2) [ 81 , 51 ] (3) [ 71 , 51 ] (4) [ 81 , 61 ]

Q71.Let f(x) = { x−a+ a ifif −a0 <≤xx ≤a≤0 g : [−a, a] →[−a, a] is (1) neither one-one nor onto. (2) onto. (3) both one-one and onto. (4) one-one. Q72. , x < 0 ⎧ tan((a+1)x)+bx tan x For a, b > 0, let f(x) = be a continous function at x = 0. Then ba is equal to : ⎨ 3, x = 0 √ax+b2x2−√ax , x > 0 ⎩ b√ax√x (1) 6 (2) 4 (3) 5 (4) 8

Q71. r 1 n22 + For α, β ∈R and a natural number n, let Ar = 2r 2 n2 −β . Then n(3n−1) 3r −2 3 2 (1) 0 (2) 4α + 2β (3) 2α + 4β (4) 2n

Q71.For 𝛼, 𝛽, 𝛾≠0. If sin−1𝛼+ sin−1𝛽+ sin−1𝛾= 𝜋 and 𝛼+ 𝛽+ 𝛾𝛼−𝛾+ 𝛽= 3𝛼𝛽, then 𝛾 equal to √3 1 (1) (2) 2 √2 (3) √3 - 1 (4) √3 2√2

Q71.If the system of equations 2𝑥+ 3𝑦−𝑧= 5 𝑥+ 𝛼𝑦+ 3𝑧= −4 3𝑥−𝑦+ 𝛽𝑧= 7 has infinitely many solutions, then 13𝛼𝛽 is equal to (1) 1110 (2) 1120 (3) 1210 (4) 1220

Q71.Let f: R - -1 →R and g: R - -5 →R be defined as fx = 2x + 3 and gx = |x | + 1 . Then the domain of the function 2 2 2x + 1 2x + 5 fog is : 5 (1) R - - (2) 𝑅 2 7 5 7 (3) R - - (4) R - - - 4 2, 4

Q71.Let f(x) = x5 + 2x3 + 3x + 1, x ∈R , and g(x) be a function such that g(f(x)) = x for all x ∈R . Then g(7) g′(7) is equal to : (1) 14 (2) 42 (3) 7 (4) 1

Q71.Consider the system of linear equation x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ2z = μ2 + 15, where λ, μ ∈R. Which one of the following statements is NOT correct? (1) The system has unique solution if λ ≠12 and (2) The system is inconsistent if λ = 12 and μ ≠1 μ ≠1, 15 (3) The system has infinite number of solutions if (4) The system is consistent if λ ≠12 λ = 21 and μ = 15 + (loge(3 −x))−1 is [−α, β) −{γ}, then α + β + γ is

Q71.Let f(x) = ax3 + bx2 + cx + 41 be such that f(1) = 40, f ′(1) = 2 and f ′(1) = 4. Then a2 + b2 + c2 is equal to: (1) 73 (2) 62 (3) 51 (4) 54

Q71.If f(x) = { 21 +−x2x,3 , 0−1≤x≤x≤3< 0 ; g(x) = { x,−x,0 <−3x ≤1≤x ≤0 , then range of (f ∘g(x)) is (1) (0, 1] (2) [0, 3) (3) [0, 1] (4) [0, 1)

Q72.If the domain of the function 𝑓𝑥= √𝑥2 −25 + + 2𝑥−15 is −∞, 𝛼∪𝛽, ∞, then 𝛼2 + 𝛽3 is equal to: 4 −𝑥2 log10𝑥2 (1) 140 (2) 175 (3) 150 (4) 125