Practice Questions

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 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, 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) = 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 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 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.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: 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(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) = 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.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

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 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.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.Let f : [−1, 2] →R be given by f(x) = 2x2 + x + [x2] −[x], where [t] denotes the greatest integer less than or equal to t. The number of points, where f is not continuous, is : (1) 5 (2) 6 (3) 3 (4) 4

Q72.Consider the function f : [ 12 , 1] →R defined by f(x) = 4√2x3 −3√2x −1. Consider the statements (I) The curve y = f(x) intersects the x-axis exactly at one point (II) The curve y = f(x) intersects the x-axis at x = cos 12π Then (1) Only (II) is correct (2) Both (I) and (II) are incorrect (3) Only (I) is correct (4) Both (I) and (II) are correct

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 domain of the function f(x) = cos−1( 2−|x|4 ) equal to : (1) 12 (2) 9 (3) 11 (4) 8

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.Let 𝑓: 𝑅→𝑅 and 𝑔: 𝑅→𝑅 be defined as 𝑓𝑥= log𝑒𝑥, 𝑥> 0 and 𝑔𝑥= 𝑥, 𝑥≥0 . Then, 𝑔𝑜𝑓: 𝑅→𝑅 is: 𝑒−𝑥, 𝑥≤0 𝑒𝑥, 𝑥< 0 (1) one-one but not onto (2) neither one-one nor onto (3) onto but not one-one (4) both one-one and onto

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

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.A variable line L passes through the point (3, 5) and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is : (1) 30 (2) 25 (3) 40 (4) 35

Q72.Let y = loge( 1−x21+x2 ), (1) 732 (2) 746 (3) 742 (4) 736