Practice Questions

Q70.For the function f(x) = (cos x) −x + 1, x ∈R, between the following two statements (S1) f(x) = 0 for only one value of x in [0, π]. (S2) f(x) is decreasing in [0, π2 ] and increasing in [ π2 , π]. (1) Both (S1) and (S2) are correct. (2) Both (S1) and (S2) are incorrect. (3) Only (S2) is correct. (4) Only (S1) is correct.

Q70.Let the relations R1 and R2 on the set X = {1, 2, 3, … , 20} be given by R1 = {(x, y) : 2x −3y = 2} and R2 = {(x, y) : −5x + 4y = 0}. If M and N be the minimum number of elements required to be added in R1 and R2 , respectively, in order to make the relations symmetric, then M + N equals (1) 12 (2) 16 (3) 8 (4) 10 α

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

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(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) = 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.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 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) = 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)

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.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, 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 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.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.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) = 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: 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. 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

Q72.Let the sum of the maximum and the minimum values of the function f(x) = 2x2+3x+82x2−3x+8 be mn , where gcd(m, n) = 1. Then m + n is equal to : (1) 195 (2) 201 (3) 217 (4) 182 2x , x < 0

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.Given that the inverse trigonometric function assumes principal values only. Let x, y be any two real numbers in [−1, 1] such that cos−1 x −sin−1 y = α, −π2 ≤α ≤π. Then, the minimum value of x2 + y2 + 2xy sin α is (1) 0 (2) -1 (3) 1 2 (4) −12 72x−9x−8x+1

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