Practice Questions

Q68.If lim 3 + 𝛼sin𝑥+ 𝛽cos𝑥+ log𝑒( 1 - 𝑥) = 1 then 2𝛼- 𝛽 is equal to : 𝑥→0 3tan2𝑥 3, (1) 2 (2) 7 (3) 5 (4) 1 Q69. 1 3 𝛼+ 3 2 2 The values of 𝛼, for which 1 1 = 0, lie in the interval 1 𝛼+ 3 3 2𝛼+ 3 3𝛼+ 1 0 (1) ( - 2, 1 ) (2) ( - 3, 0 ) (3) -3 3 (4) ( 0, 3 ) 2, 2

Q68.The frequency distribution of the age of students in a class of 40 students is given below. Age 15 16 17 18 19 20 If the mean deviation about the median is 1.25, then 4x + 5y No of Students 5 8 5 12 x y is equal to : (1) 46 (2) 43 (3) 44 (4) 47 Q69. 3x + 5y + λz = 3 Let λ, μ ∈R. If the system of equations 7x + 11y −9z = 2 has infinitely many solutions, then μ + 2λ is 97x + 155y −189z = μ equal to : (1) 24 (2) 25 (3) 22 (4) 27

Q68.Let α, β ∈R. Let the mean and the variance of 6 observations −3, 4, 7, −6, α, β be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is : (1) 13 (2) 16 3 3 (3) 11 (4) 14 3 3 Q69. ⎡ 1 2 α⎤ Let α ∈(0, ∞) and A = 1 0 1 . If det (adj (2A −AT) ⋅adj (A −2AT)) = 28 , then (det(A))2 is equal ⎣ 0 1 2 ⎦ to: (1) 36 (2) 16 (3) 1 (4) 49

Q68.For 0 < 𝜃< 𝜋/ 2, if the eccentricity of the hyperbola 𝑥2 −𝑦2cosec2𝜃= 5 is √7 times eccentricity of the ellipse 𝑥2cosec2𝜃+ 𝑦2 = 5, then the value of 𝜃 is: (1) 𝜋 (2) 5𝜋 6 12 𝜋 𝜋 (3) (4) 3 4

Q68.Let f : [−π2 , 2 ] →R be a differentiable function such that f(0) = 2 , If ex2−1 x→0 to : (1) 16 (2) 2 (3) 1 (4) 4

Q68.Let R be a relation on Z × Z defined by (a, b)R(c, d) if and only if ad −bc is divisible by 5 . Then R is (1) Reflexive and symmetric but not transitive (2) Reflexive but neither symmetric not transitive (3) Reflexive, symmetric and transitive (4) Reflexive and transitive but not symmetric Q69. ⎡ 1 0 0 ⎤ 3 Let A = 0 α β and 2A = 221 where α, β ∈Z , Then a value of α is ⎣ 0 β α⎦ (1) 3 (2) 5 (3) 17 (4) 9 is equal to

Q68.If the mean and variance of five observations are 24 and 194 respectively and the mean of first four 5 25 observations is 7 , then the variance of the first four observations in equal to 2 (1) 4 (2) 77 5 12 (3) 5 (4) 105 4 4

Q68. e−(1+2x) 2x1 limx→0 x is equal to (1) 0 (2) −2 e (3) e (4) e −e2

Q68. is equal to : limn→∞ (13+23+⋯⋯+n3)−(12+22+⋯⋯+n2) (1) 2 (2) 1 3 3 (3) 3 (4) 1 4 2

Q69.If the variance of the frequency distribution x c 2c 3c 4c 5c 6c is 160, then the value of c ∈N is f 2 1 1 1 1 1 (1) 7 (2) 8 (3) 5 (4) 6 and A be a 2 × 2 matrix such that AB−1 = A−1 . If BCB−1 = A and C 4 + αC 2 + βI = O,

Q69.The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is (1) 1.8 (2) 1.94 (3) √3.96 (4) √3.86

Q69.If a = lim √1+√1+x4−√2 and b = lim sin2 x , then the value of ab3 is : x→0 x4 x→0 √2−√1+cos x (1) 36 (2) 32 (3) 25 (4) 30

Q69.Consider 10 observation 𝑥1, 𝑥2, . .. 𝑥10, such that ∑𝑖=10 1 𝑥𝑖−𝛼= 2 and ∑𝑖=10 1 𝑥𝑖−𝛽2 = 40, where 𝛼, 𝛽 are 6 84 𝛽 positive integers. Let the mean and the variance of the observations be and respectively. The is equal to: 5 25 𝛼 (1) 2 (2) 3 2 (3) 5 (4) 1 2

Q69.Let 𝑓: →𝑅→0, ∞ be strictly increasing function such that lim 𝑓7𝑥 1. Then, the value of lim 𝑓5𝑥 is 𝑥→∞ 𝑓𝑥= 𝑥→∞ 𝑓𝑥−1 equal to (1) 4 (2) 0 (3) 7 (4) 1 5

Q69.If R is the smallest equivalence relation on the set {1, 2, 3, 4} such that {(1, 2), (1, 3)} ⊂R, then the number of elements in R is ______. (1) 10 (2) 12 (3) 8 (4) 15 Q70. ⎡ 2 1 2 ⎤ ⎡ 1 2 0⎤ Let A = 6 2 11 and P = 5 0 2 . The sum of the prime factors of P−1AP −2I is equal to ⎣ 3 3 2 ⎦ ⎣ 7 1 5⎦ (1) 26 (2) 27 (3) 66 (4) 23

Q69.If the mean of the following probability distribution of a random variable X : X 0 2 4 6 8 46 is , then the variance of the distribution is P(X) a 2a a + b 2b 3b 9 (1) 173 (2) 566 27 81 (3) 151 (4) 581 27 81

Q69.Consider the system of linear equations 𝑥+ 𝑦+ 𝑧= 5, 𝑥+ 2𝑦+ 𝜆2𝑧= 9 and 𝑥+ 3𝑦+ 𝜆𝑧= 𝜇, where 𝜆, 𝜇∈𝑅. Then, which of the following statement is NOT correct ? (1) System has infinite number of solution if 𝜆= 1 (2) System is inconsistent if 𝜆= 1 and 𝜇≠13 and 𝜇= 13 (3) System has unique solution if 𝜆≠1 and 𝜇≠13 (4) System is consistent if 𝜆≠1 and 𝜇= 13

Q69.Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, 𝑎, 𝑏 be 170 205 and respectively. Then the mean deviation about the mean of these 7 observations is: 7 (1) 31 (2) 28 (3) 30 (4) 32 0

Q69.Let A = {1, 2, 3, 4, 5}. Let R be a relation on A defined by xRy if and only if 4x ≤5y. Let m be the number of elements in R and n be the minimum number of elements from A × A that are required to be added to R to make it a symmetric relation. Then m + n is equal to : (1) 25 (2) 24 (3) 26 (4) 23

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.If the domain of the function 𝑓𝑥= 2𝑥+ 3 + cos-12𝑥- 1 is ( 𝛼, 𝛽], then the value of 5𝛽- 4𝛼 is equal to log𝑒 4𝑥2 + 𝑥- 3 𝑥+ 2 (1) 10 (2) 12 (3) 11 (4) 9 𝑥2𝑔𝑥𝑑𝑥

Q70. x + (√2 sin α)y + (√2 cos α)z = 0 If the system of equations x + (cos α)y + (sin α)z = 0 has a non-trivial solution, then α ∈(0, π2 ) is x + (sin α)y −(cos α)z = 0 equal to : (1) 11π (2) 5π 24 24 (3) 7π (4) 3π 24 4 is (α, β], then 3α + 10β is equal to:

Q70.If 𝐴= √2 1 , 𝐵1 , 𝐶= 𝐴𝐵𝐴𝑇 and 𝑋= 𝐴𝑇𝐶2𝐴, then det 𝑋 is equal to: −1 √2 1 1 (1) 243 (2) 729 (3) 27 (4) 891

Q70. x + y + z = 4, The values of m, n, for which the system of equations 2x + 5y + 5z = 17, has infinitely many solutions, x + 2y + mz = n satisfy the equation: (1) m2 + n2 −mn = 39 (2) m2 + n2 −m −n = 46 (3) m2 + n2 + m + n = 64 (4) m2 + n2 + mn = 68

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