Practice Questions

Q70.Let A and B be two 3 × 3 matrices such that AB = I and |A| = 18 then |adj(Badj(2A))| is equal to (1) 128 (2) 32 (3) 64 (4) 102

Q70.The number of values of α for which the system of equations x + y + z = α αx + 2αy + 3z = −1 x + 3αy + 5z = 4 is inconsistent, is (1) 0 (2) 1 (3) 2 (4) 3

Q70.The negation of the Boolean expression ~𝑞∧𝑝⇒~𝑝∨𝑞 is logically equivalent to (1) 𝑝⇒𝑞 (2) 𝑞⇒𝑝 (3) ~𝑝⇒𝑞 (4) ~𝑞⇒𝑝

Q70.If the system of equations 𝑥+ 𝑦+ 𝑧= 6 2𝑥+ 5𝑦+ 𝛼𝑧= 𝛽 𝑥+ 2𝑦+ 3𝑧= 14 has infinitely many solutions, then 𝛼+ 𝛽 is equal to (1) 8 (2) 36 (3) 44 (4) 48

Q70.The value of n→∞6lim tan{∑nr=1 tan−1( r2+3r+31 )} is equal to (1) 1 (2) 2 (3) 3 (4) 6

Q70.The total number of functions, 𝑓: 1, 2, 3, 4 →1, 2, 3, 4, 5, 6 such that 𝑓1 + 𝑓2 = 𝑓3, is equal to (1) 60 (2) 90 (3) 108 (4) 126

Q70.If the inverse trigonometric functions take principal values, then cos−1( 103 cos(tan−1( 43 )) + 25 sin(tan−1( 43 ))) is equal to (1) 0 (2) π4 (3) π (4) π 3 6

Q70.Let A = (α4 −2β ) (1) −18 (2) 18 (3) −50 (4) 50 1 [t] is the greatest

Q70.If cos−1( 2y ) = loge ( x5 ) 5, |y| < 2, then (1) x2y′′ + xy′ −25y = 0 (2) x2y′′ −xy′ −25y = 0 (3) x2y′′ −xy′ + 25y = 0 (4) x2y′′ + xy′ + 25y = 0

Q70.Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then (1) The system of linear equations AX = 0 has a (2) The system of linear equations AX = 0 has unique solution infinitely many solutions (3) B is an invertible matrix (4) adj(A) is an invertible matrix

Q70.The ordered pair (a, b), for which the system of linear equations 3x −2y + z = b 5x −8y + 9z = 3 2x + y + az = −1 has no solution, is (1) (3, 13 ) (2) (−3, 31 ) (3) (−3, −13 ) (4) (3, −13 )

Q71.Let A = (−21 −52 ). Let α, β ∈R be such that αA2 + βA = 2I . Then α + β is equal to (1) −10 (2) −6 (3) 6 (4) 10

Q71.If the function f(x) = loge(1−x+x2)+loge(1+x+x2) −π π sec x−cos x , x ∈( 2 , 2 ) −{0} is continuous at x = 0 , then k is equal { k , x = 0 to: (1) 1 (2) −1 (3) e (4) 0 are continuous on R, then and g(x) =

Q71.Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos-1𝑥- 2sin-1𝑥= cos-12𝑥 is equal to (1) 0 (2) 1 (3) 1 (4) -1 2 2

Q71.The number of distinct real roots of x4 −4x + 1 = 0 is (1) 0 (2) 1 (3) 2 (4) 4

Q71.If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is (1) 11. 5 (2) 10. 5 (3) 12 (4) 11

Q71.Let A = [aij] be a square matrix of order 3 such that aij = 2j−i , for all i, j = 1, 2, 3 . Then, the matrix A2 + A3 + … + A10 is equal to (1) ( 310−12 )A (2) ( 310+12 )A (3) ( 310+32 )A (4) ( 310−32 )A

Q71.If the absolute maximum value of the function 𝑓𝑥= x2 - 2x + 7e4x3 - 12x2 - 180x + 31in the interval -3, 0 is 𝑓𝛼, then (1) 𝛼= 0 (2) 𝛼= - 3 (3) 𝛼∈-1, 0 (4) 𝛼∈-3, - 1

Q71.If y = tan−1(sec x3 −tan x3), π2 < x3 < 3π2 , then (1) xy′′ + 2y′ = 0 (2) x2y′′ −6y + 3π2 = 0 (3) x2y′′ −6y + 3π = 0 (4) xy′′ −4y′ = 0

Q71.From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60°. The pole subtends an angle 30° at the top of the tower. Then the height of the tower is: (1) 15√3 (2) 20√3 (3) 20 + 10√3 (4) 30 Q72. 2 −1 Let A = − . . . − 5C5(adj A)5 , then the sum of . If B = I −5C1(adj A) + 5C2(adj A)2 (0 2 ) all elements of the matrix B is: (1) −5 (2) −6 (3) −7 (4) −8

Q71.The system of equations -𝑘𝑥+ 3𝑦- 14𝑧= 25 -15𝑥+ 4𝑦- 𝑘𝑧= 3 -4𝑥+ 𝑦+ 3𝑧= 4 Question: is consistent for all 𝑘 in the set (1) 𝑅 (2) 𝑅- -11, 13 (3) 𝑅- -13 (4) 𝑅- -11, 11 - 1 4

Q71. a −1 0 Let f(x) = ax a −1 , a ∈R. Then the sum of the squares of all the values of a for ax2 ax a 2f ′(10) −f ′(5) + 100 = 0 is (1) 117 (2) 106 (3) 125 (4) 136 is

Q71.The domain of the function 𝑓𝑥= sin-1 𝑥2 - 3𝑥+ 2 is 𝑥2 + 2𝑥+ 7 (1) [1, ∞) (2) ( - 1, 2] (3) [ - 1, ∞) (4) ( - ∞, 2]

Q71.The domain of the function f(x) = sin−1[2x2 −3] + log2(log (x2 −5x + 5)), where 2 integer function, is 2 , 5+√52 ) 2 , 5−√52 (1) (−√5 ) (2) ( 5−√5 (3) (1, 5−√52 ) (4) [1, 5+√52 )

Q71.Let f : R →R be defined as f(x) = x −1 and g : R →{1, −1} →R be defined as g(x) = x2 . Then the x2−1 function fog is: (1) One-one but not onto (2) onto but not one-one (3) Both one-one and onto (4) Neither one-one nor onto