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

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.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 = {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 :

Q76.Let the solution curve 𝑦= 𝑦( 𝑥) of the differential equation 𝑑𝑦 3𝑥5tan-1𝑥33 𝑦= 2𝑥 exp 𝑥3 - tan-1𝑥3 pass through 𝑑𝑥- 1 + 𝑥6 2 √( 1 + 𝑥) 6 the origin. Then 𝑦( 1 ) is equal to: (1) exp4 - 𝜋 (2) exp𝜋- 4 4√2 4√2 (3) exp1 - 𝜋 (4) exp4 + 𝜋 4√2 4√2 → →

Q76.For x ∈R, two real valued functions f(x) and g(x) are such that, g(x) = √x + 1 and fog(x) = x + 3 −√x. Then f(0) is equal to (1) 1 (2) 5 (3) 0 (4) −3

Q76.If A = [λ1 105 ], (1) 12 (2) 19 (3) 14 (4) 10

Q76.The area enclosed by the closed curve 𝐶 given by the differential equation 𝑑𝑦 𝑥+ 𝑎 = 0, 𝑦1 = 0 is 4𝜋. Let 𝑃 𝑑𝑥+ 𝑦- 2 and 𝑄 be the points of intersection of the curve 𝐶 and the 𝑦-axis. If normals at 𝑃 and 𝑄 on the curve 𝐶 intersect 𝑥-axis at points 𝑅 and 𝑆 respectively, then the length of the line segment 𝑅𝑆 is (1) 2√3 (2) 2√3 3 (3) 2 (4) 4√3 3 JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

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 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.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.Let P be a square matrix such that P 2 = I −P . For α, β, γ, δ ∈N, if P α + P β = γl −29P and P α −P β = δl −13P , then α + β + γ −δ is equal to (1) 18 (2) 40 (3) 22 (4) 24

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.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 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 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.Let →𝑢= ^𝑖- ^𝑗- 2 ^𝑘, →𝑣= 2 ^𝑖+ ^𝑗- ^𝑘, →𝑣· →𝑤= 2 and →𝑣× →𝑤= →𝑢+ 𝜆 →𝑣, then →𝑢· →𝑤 is equal to 3 (1) 1 (2) 2 2 (3) 2 (4) - 3