Practice Questions

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

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

Q76.If the system of linear equations 7x + 11y + αz = 13 5x + 4y + 7z = β 175x + 194y + 57z = 361 has infinitely many solutions, then α + β + 2 is equal to (1) 4 (2) 3 (3) 5 (4) 6

Q76.Let A be a n × n matrix such that |A| = 2 . If the determinant of the matrix Adj (2. Adj (2 A−1)) is 284 , then n is equal to _____ . Q77. ⎛ 2 10 8⎞ If a point P(α, β, γ) satisfying (α β γ ) 9 3 8 = (0 0 0) lies on the plane 2x + 4y + 3z = 5, then ⎝ 8 4 8⎠ 6α + 9β + 7γ is equal to (1) 5 (2) −1 4 (3) 11 (4) 115

Q76.For any vector →𝑎= 𝑎1 ^𝑖+ 𝑎2 ^𝑗+ 𝑎3 ^𝑘, with 10𝑎𝑖< 1, 𝑖= 1, 2, 3, consider the following statements: 𝐴 : max𝑎1, 𝑎2, 𝑎3 ≤ →𝑎 𝐵 : | →𝑎| ≤3max𝑎1, 𝑎2, 𝑎3 JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper (1) Only 𝐵 is true (2) Only 𝐴 is true (3) Both 𝐴 and 𝐵 are true (4) Neither 𝐴 nor 𝐵 is true

Q76.Let a1 = 1, a2, a3, a4, … .. be consecutive natural numbers. Then tan−1( 1+a1a21 ) + … . . + tan−1( 1+a2021a20221 ) is equal to (1) π 4 −cot−1(2022) (2) cot−1(2022) −π4 (3) tan−1(2022) −π4 (4) π4 −tan−1(2022)

Q76.The value of ∫𝜋 sin𝑥1 + cos𝑥𝑑𝑥 3 (1) 7 - √3 - log𝑒√3 (2) -2 + 3√3 + log𝑒√3 2 10 10 (3) 3 - √3 + log𝑒√3 (4) 3 - √3 - log𝑒√3 𝑥𝑓𝑡

Q76.For the system of linear equations ax + y + z = 1 , x + ay + z = 1, x + y + az = β, which one of the following statements is NOT correct? (1) It has infinitely many solutions if α = 2 and (2) It has no solution if α = −2 and β = 1 β = −1 (3) x + y + z = 34 if α = 2 and β = 1 (4) It has infinitely many solutions if α = 1 and β = 1 n(S) denotes the number of elements ∈R : 0 < x < 1 and 2 tan−1( 1+x1−x ) = cos−1( 1+x21−x2 )} . If

Q76.Let A be a 3 × 3 matrix such that |adj(adj(adj. A))| = 124 . Then A−1adj A is equal to (1) 2√3 (2) √6 (3) 12 (4) 1

Q76.Let the position vectors of the points 𝐴, 𝐵, 𝐶 and 𝐷 be 5 ^i + 5 ^j + 2λ ^k, ^i + 2 ^j + 3 ^k, - 2 ^i + λ ^j + 4 ^k and - ^i + 5 ^j + 6 ^k . Let the set 𝑆= {𝜆∈ℝ: the points 𝐴, 𝐵, 𝐶 and 𝐷 are coplanar } . The 2 ∑𝜆∈𝑆(𝜆+ ) 2 is equal to 37 (1) 25 (2) 2 (3) 14 (4) 41

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

Q76.Let 𝑂 be the origin and the position vector of the point 𝑃 be - ^𝑖- 2 ^𝑗+ 3𝑘. If the position vectors of the points 𝐴, 𝐵 and 𝐶 are -2 ^𝑖+ ^𝑗- 3𝑘, 2 ^𝑖+ 4 ^𝑗- 2𝑘 and -4 ^𝑖^ + 2 ^𝑗- 𝑘 respectively, then the projection of the vector → → → 𝑂𝑃 on a vector perpendicular to the vectors 𝐴𝐵 and 𝐴𝐶 is 8 (1) 3 (2) 3 7 10 (3) (4) 3 3