Practice Questions

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

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

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 →𝑎= 2 ^𝑖+ 7 ^𝑗- ^𝑘, ^𝑏= 3 ^𝑖+ 5 ^𝑘 and →𝑐= ^𝑖- ^𝑗+ 2 ^𝑘 Let →𝑑 be a vector which is perpendicular to both →𝑎 and → → → 𝑏, and →𝑐· 𝑑= 12. Then- ^𝑖+ ^𝑗- ^𝑘· →𝑐× 𝑑 is equal to (1) 24 (2) 44 (3) 42 (4) 48

Q76.Let for a triangle 𝐴𝐵𝐶 →𝐴𝐵= - 2 ^𝑖+ ^𝑗+ 3 ^𝑘 →𝐶𝐵= 𝛼 ^𝑖+ 𝛽 ^𝑗+ 𝛾 ^𝑘 →𝐶𝐴= 4 ^𝑖+ 3 ^𝑗+ 𝛿 ^𝑘 → → If 𝛿> 0 and the area of the triangle 𝐴𝐵𝐶 is 5√6 then 𝐶𝐵· 𝐶𝐴 is equal to (1) 60 (2) 54 (3) 108 (4) 120

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.If A = [λ1 105 ], (1) 12 (2) 19 (3) 14 (4) 10

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.If the sum of all the solutions of + cot−1( 1−x22x ) tan−1( 1−x22x ) = π3 , −1 < x < 1, x ≠0, is α − √34 , then α is equal to _____ .

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 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 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.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 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.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