Practice Questions

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.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.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.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

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 [t] be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and . The number of one-to-one functions from A to the + f : A →Z be the function f(x) = [log2 (x2 [ x35 ])] range of f is (1) 25 (2) 24 (3) 20 (4) 120

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.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

Q70.If the domain of the function f(x) = sin−1 ( 2x+3x−1 ) is R −(α, β), then 12αβ is equal to : (1) 32 (2) 40 (3) 24 (4) 36

Q70.Let A be a square matrix such that AAT = I. Then 12 A[( A + AT)2 + (A −AT)2] (1) A2 + I (2) A3 + I (3) A2 + AT (4) A3 + AT

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.Consider the relations 𝑅1 and 𝑅2 defined as 𝑎𝑅1𝑏⇔𝑎2 + 𝑏2 = 1 for all 𝑎, 𝑏, ∈𝑅 and 𝑎, 𝑏𝑅2𝑐, 𝑑⇔𝑎+ 𝑑= 𝑏+ 𝑐 for all 𝑎, 𝑏, 𝑐, 𝑑∈𝑁× 𝑁. Then (1) Only 𝑅1 is an equivalence relation (2) Only 𝑅2 is an equivalence relation (3) 𝑅1 and 𝑅2 both are equivalence relation (4) Neither 𝑅1 nor 𝑅2 is an equivalence relation

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.Let B = [ 11 35 ] then 2β −α is equal to (1) 16 (2) 2 (3) 8 (4) 10 is equal to cot−1 √1−x1+x )dx

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

Q70.Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps f : A →B , such that f(1) + f(3) = 14, is : (1) 480 (2) 240 (3) 120 (4) 180

Q70.If the system of equations x + 4y −z = λ, 7x + 9y + μz = −3, 5x + y + 2z = −1 has infinitely many solutions, then (2μ + 3λ) is equal to : (1) 3 (2) -3 (3) -2 (4) 2 where a > 0 and g(x) = (f(x ∣) −|f(x)|)/2. Then the function

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 the system of linear equations 𝑥- 2𝑦+ 𝑧= - 4 2𝑥+ 𝛼𝑦+ 3𝑧= 5 3𝑥- 𝑦+ 𝛽𝑧= 3 has infinitely many solutions, then 12𝛼+ 13𝛽 is equal to (1) 60 (2) 64 (3) 54 (4) 58

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.Considering only the principal values of inverse trigonometric functions, the number of positive real values of 𝑥 satisfying tan-1 (x) + tan-1 (2x) = π is : 4 (1) More than 2 (2) 1 (3) 2 (4) 0

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

Q70.Let a relation R on N × N be defined as: (x1, y1)R (x2, y2) if and only if x1 ≤x2 or y1 ≤y2 . Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true? (1) Both (I) and (II) are correct. (2) Only (II) is correct. (3) Neither (I) nor (II) is correct. (4) Only (I) is correct. JEE Main 2024 (04 Apr Shift 2) JEE Main Previous Year Paper

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