Practice Questions

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. 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.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.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.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.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)

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.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 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 S = {1, 2, 3, … , 10}. Suppose M is the set of all the subsets of S , then the relation R = {(A, B) : A ∩B ≠ϕ; A, B ∈M} is : (1) symmetric and reflexive only (2) reflexive only (3) symmetric and transitive only (4) symmetric only Q72. ⎡cos x −sin x 0 ⎤ Consider the matrix f(x) = sin x cos x 0 . Given below are two statements : ⎣ 0 0 1 ⎦ Statement I: f(−x) is the inverse of the matrix f(x). Statement II: f(x) f(y) = f(x + y). In the light of the above statements, choose the correct answer from the options given below (1) Statement I is false but Statement II is true (2) Both Statement I and Statement II are false (3) Statement I is true but Statement II is false (4) Both Statement I and Statement II are true

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.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.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.The integral ∫3/41/4 cos (2 (1) 1/2 (2) −1/2 (3) −1/4 (4) 1/4

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 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

Q72.Let a and b be real constants such that the function 𝑓 defined by 𝑓𝑥= 𝑥2 + 3𝑥+ 𝑎, 𝑥≤1 be differentiable 𝑏𝑥+ 2, 𝑥> 1 2 on 𝑅. Then, the value of ∫-2 𝑓𝑥𝑑𝑥 equals 15 19 (1) (2) 6 6 (3) 21 (4) 17

Q72.Consider the function f: ( 0, 2 ) →R defined by f(x) = x + 2 and the function g ( x ) defined by 2 x min{f ( t ) }, 0 < t ≤x and 0 < x ≤1 gx = 3 . Then + x, 1 < x < 2 2 (1) g is continuous but not differentiable at x = 1 (2) g is not continuous for all x ∈( 0, 2 ) (3) g is neither continuous nor differentiable at x = 1(4) g is continuous and differentiable for all x ∈( 0, 2 )

Q72.The number of critical points of the function f(x) = (x −2)2/3(2x + 1) is (1) 1 (2) 2 (3) 0 (4) 3 6

Q72.If the function f(x) = sin 3x+α sin x−β cos 3x , x ∈R , is continuous at x = 0 , then f(0) is equal to : x3 (1) 2 (2) -2 (3) 4 (4) -4

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

Q72.If 𝑎= sin−1sin5 and 𝑏= cos−1cos5, then 𝑎2 + 𝑏2 is equal to (1) 4𝜋2 + 25 (2) 8𝜋2 −40𝜋+ 50 (3) 4𝜋2 −20𝜋+ 50 (4) 25

Q72.Let the range of the function f(x) = 2+sin 3x+cos1 3x , x ∈R be [a, b]. If α and β are respectively the A.M. and the G.M. of a and b, then αβ is equal to (1) π (2) √π (3) 2 (4) √2