Practice Questions

Q74.Area of the region 𝑥, 𝑦: 𝑥2 + 𝑦- 22 ≤4, 𝑥2 ≥2𝑦 is 8 16 (1) 𝜋+ (2) 2𝜋+ 3 3 (3) 𝜋- 8 (4) 2𝜋- 16 3 3

Q75.Let x = x(y) be the solution of the differential equation 2(y + 2) loge(y + 2)dx + (x + 4 −2 loge(y + 2))dy = 0 , y > −1 with x(e4 −2) = 1 . Then x(e9 −2) is equal to (1) 3 (2) 49 (3) 32 (4) 10 9 3

Q75.Let S1 and S2 be respectively the sets of all a ∈R −{0} for which the system of linear equations ax + 2ay −3az = 1 (2a + 1) x + (2a + 3) y + (a + 1)z = 2 JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper (3a + 5) x + (a + 5) y + (a + 2) z = 3 has unique solution and infinitely many solutions. Then (1) n(S1) = 2 and S2 is an infinite set (2) S1 is an infinite set an n(S2) = 2 (3) S1 = ϕ and S2 = R −{0} (4) S1 = R −{0} and S2 = ϕ

Q75.Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of A × B each having at least 3 and at most 6 elements is (1) 752 (2) 782 (3) 792 (4) 772

Q75.If A = 2 [−√3 1 ] (1) A30 −A25 = 2I (2) A30 + A25 + A = I (3) A30 + A25 −A = I (4) A30 = A25

Q75.Let 𝐼𝑥= ∫𝑥2𝑥 ( 𝑥 tan𝑥+ 1 2 𝑑𝑥 If 𝐼0 = 0, then 𝐼𝜋4 is equal to ) (1) ( 𝜋+ 4 ) 2 𝜋2 (2) ( 𝜋+ 4 ) 2 𝜋2 loge 16 + 4 ( 𝜋+ 4 ) loge 16 - 4 ( 𝜋+ 4 ) (3) ( 𝜋+ 4 ) 2 𝜋2 (4) ( 𝜋+ 4 ) 2 𝜋2 loge 32 - 4 ( 𝜋+ 4 ) loge 32 + 4 ( 𝜋+ 4 )

Q75.In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is α and the number of persons who speaks only Hindi is β, then the eccentricity of the ellipse 25(β2x2 + α2y2) = α2β2 is (1) √119 (2) √117 12 12 (3) 3√15 (4) √129 12 12

Q75.Consider the following system of questions αx + 2y + z = 1 2αx + 3y + z = 1 3x + αy + 2z = β For some α, β ∈R . Then which of the following is NOT correct. (1) It has no solution if α = −1 and β ≠2 (2) It has no solution for α = −1 and for all β ∈R (3) It has no solution for α = 3 and for all β ≠2 (4) It has a solution for all α ≠−1 and β = 2 log(x+1)(x−2) , x ∈R is

Q75.Let 𝛼∈0, 1 and 𝛽= + + … . + log𝑒1 - 𝛼. Let 𝑃𝑛𝑥= 𝑥+ 2 3 𝑛, 𝑥∈0, 1. Then the integral ∫0 1 - 𝑡𝑑𝑡 is equal to (1) 𝛽- 𝑃50𝛼 (2) -𝛽+ 𝑃50𝛼 (3) 𝑃50𝛼- 𝛽 (4) 𝛽+ 𝑃50𝛼 𝜋 2 2 + 3sin𝑥 is equal to

Q75.Let A = ⌊aˆiˆj⌋⋅aij prime number p ∈(2, 13) is _____ .

Q75.The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is (1) 225 (2) 120 (3) 150 (4) 125

Q75.For α, β ∈R, suppose the system of linear equations x −y + z = 5 2x + 2y + αz = 8 3x −y + 4z = β has infinitely many solutions. Then α and β are the roots of (1) x2 −10x + 16 = 0 (2) x2 + 18x + 56 = 0 (3) x2 −18x + 56 = 0 (4) x2 + 14x + 24 = 0 + tan−1( 1+a2a31 )

Q75.Let f be a continuous function satisfying t2 f ( x ) + x2dx = 4 ∀t > 0 . Then f π2 is equal to ∫0 3t3, 4 (1) π2 (2) π3 π21 - -π1 + 16 16 (3) π1 - π3 (4) -π21 + π2 16 16

Q75. lim 1 1 1 … + 1 is equal to :- 𝑛→∞ 1 + 𝑛+ 2 + 𝑛+ 3 + 𝑛+ 2𝑛 (1) 0 (2) loge2 3 2 (3) loge 2 (4) loge 3

Q75.Let A = {1, 2, 3, 4, 5, 6, 7} . Then the relation R = {(x, y) ∈A × A : x + y = 7} is (1) an equivalence relation (2) symmetric but neither reflexive nor transitive (3) transitive but neither symmetric nor reflexive (4) reflexive but neither symmetric nor transitive A−1 = αA + βI and α + β = −2, then 4α2 + β2 + λ2 is equal to :

Q75.If [ 𝑡 denotes the greatest integer ≤1, then the value of 𝑥2𝑒𝑥+ 𝑥3𝑑𝑥 is : 𝑒 ∫1 (1) 𝑒9 - 𝑒 (2) 𝑒8 - 𝑒 (3) 𝑒7 - 1 (4) 𝑒8 - 1

Q75.Let A = [10 5111 ] 1 2 −1 −2 equal to JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 75 (2) 125 (3) 50 (4) 100 Q76. 1 2k 2k −1 Let Dk = n n2 + n + 2 n2 . If ∑nk=1 Dk = 96, then n is equal to _________. n n2 + n n2 + n + 2 g : D →R

Q75.If S = {x ∈R sin−1( √x2+2x+2x+1 ) −sin−1( √x2+1x ) ∑x∈S(sin((x2 + x + 5) π2 ) −cos((x2 + x + 5)π)) is equal to _________.

Q75.Let |→𝑎| = 2, | →𝑏| = 3 and the angle between the vectors →𝑎 and →𝑏 be 𝜋 2 →𝑏) × (2→𝑎- 3 →𝑏)| 4. Then |( →𝑎+ equal to (1) 441 (2) 482 (3) 841 (4) 882

Q75.Let 𝑦 = 𝑦( 𝑥) be the solution of the differential equation 𝑥3 𝑑𝑦 + ( 𝑥𝑦 – 1 ) 𝑑𝑥 = 0, 𝑥 > 0, 𝑦 1 = 3 - 𝑒. Then 𝑦1 is equal to 2 (1) 1 (2) 𝑒 (3) 2 - 𝑒 (4) 3

Q75.Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10} . Let R be a relation defined on A × B such that R = {(a1, b1), (a2, b2) : a1 ≤b2 and b1 ≤a2}. Then the number of elements in the set R is (1) 160 (2) 52 (3) 26 (4) 180

Q75.Let R be a relation defined on N as a R b is 2a + 3b is a multiple of 5, a, b ∈N. Then R is (1) not reflexive (2) transitive but not symmetric (3) symmetric but not transitive (4) an equivalence relation Q76. ⎡ et e−t(sin t −2 cos t) e−t(−2 sin t −cos t) ⎤ The set of all values of t ∈R, for which the matrix et e−t(2 sin t + cos t) e−t(sin t −2 cos t) ⎣ et e−t cos t e−t sin t ⎦ is invertible, is (1) {(2k + 1) π2 , k ∈Z} (2) {kπ + π4 , k ∈Z} (3) {kπ, k ∈Z} (4) R If the sum of the diagonal elements of = 3 ]A [α β ]

Q75.Let 𝑦= 𝑦𝑥 be a solution curve of the differential equation, 1 - 𝑥2𝑦2𝑑𝑥= 𝑦𝑑𝑥+ 𝑥𝑑𝑦, If the line 𝑥= 1 intersects the curve 𝑦= 𝑦𝑥 at 𝑦= 2 and the line 𝑥= 2 intersects the curve 𝑦= 𝑦𝑥 at 𝑦= 𝛼, then a value of 𝛼 is (1) 1 - 3𝑒2 (2) 1 + 3𝑒2 23𝑒2 + 1 23𝑒2 - 1 (3) 3𝑒2 (4) 3𝑒2 23𝑒2 - 1 23𝑒2 + 1

Q75.An arc 𝑃𝑄 of a circle subtends a right angle at its centre 𝑂. The mid point of the arc 𝑃𝑄 is 𝑅. If →𝑂𝑃= →𝑢, →𝑂𝑅= →𝑣 and →𝑂𝑄= 𝛼→𝑢+ 𝛽→𝑣, then 𝛼, 𝛽2, are the roots of the equation (1) 𝑥2 + 𝑥- 2 = 0 (2) 𝑥2 - 𝑥- 2 = 0 (3) 3𝑥2 - 2𝑥- 1 = 0 (4) 3𝑥2 + 2𝑥- 1 = 0

Q76.Let S be the set of all (λ, μ) for which the vectors λˆi −ˆj + ˆk, ˆj + 2ˆj + μˆk and 3ˆi −4ˆj + 5ˆk, where λ −μ = 5, are coplanar, then ∑(λ, μ)∈S 80(λ2 + μ2) is equal to (1) 2210 (2) 2130 (3) 2290 (4) 2370