Practice Questions

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

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.The domain of f(x) = e2 loge x−(2x+3) (1) R −{−1, 3} (2) (2, ∞) −{3} (3) (−1, ∞) −{3} (4) R −{3}

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 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 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 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.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 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 f : R →R be a function defined by f(x) = log√m {√2(sin −2}, for some the range of f is [0, 2]. Then the value of m is _____ . (1) 5 (2) 3 (3) 2 (4) 4

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

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 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.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.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.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 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

Q77.Let D be the domain of the function f(x) = sin−1(log3x( 6+2−5xlog3 x )). If the range of the function defined by g(x) = x −[x], ( [x] is the greatest integer function), is (α, β), then α2 + β5 is equal to (1) 135 (2) 45 (3) 46 (4) 136

Q77.Let →𝑎= 2 ^i + 3 ^j + 4 ^k, →𝑏= ^i - 2 ^j - 2 ^k and →𝑐= - ^i + 4 ^j + 3 ^k . If →𝑑 is a vector perpendicular to both →𝑏 and →𝑐, 2 is equal to and →𝑎· →𝑑= 18, then |→𝑎× →𝑑| JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper (1) 640 (2) 680 (3) 720 (4) 760

Q77.Let 𝑃𝑄𝑅 be a triangle. The points𝐴, 𝐵 and 𝐶 are on the sides 𝑄𝑅, 𝑅𝑃 and 𝑃𝑄 respectively such that 𝑄𝐴 𝑅𝐵 𝑃𝐶 1 Then Area∆𝑃𝑄𝑅 is equal to 𝐴𝑅= 𝐵𝑃= 𝐶𝑄= 2. Area∆𝐴𝐵𝐶 (1) 4 (2) 1 5 (3) 2 (4) 2

Q77. x + 1 x x If x x + λ x = 89 (103x + 81), then λ, λ3 are the roots of the equation x x x + λ2 (1) 4x2 + 24x −27 = 0 (2) 4x2 −24x −27 = 0 (3) 4x2 + 24x + 27 = 0 (4) 4x2 −24x + 27 = 0