Practice Questions

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

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= 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

Q77.Let →𝑎 be a non-zero vector parallel to the line of intersection of the two planes described by ^𝑖+ ^𝑗, ^𝑖+ ^𝑘 and ^𝑖- ^𝑗, ^𝑗- ^𝑘. If 𝜃 is the angle between the vector →𝑎 and the vector →𝑏= 2 ^𝑖- 2 ^𝑗+ ^𝑘 and →𝑎· →𝑏= 6, then the ordered pair 𝜃, | →𝑎× →𝑏| is equal to 𝜋 𝜋 (1) 3, 3√6 (2) 4, 3√6 (3) 𝜋 6 (4) 𝜋 6 3, 4,

Q77.The range of the function f(x) = √3 −x + √2 + x is (1) [√5, √10] (2) [2√2, √11] (3) [√5, √13] (4) [√2, √7]

Q77.Let S = {x in S then : (1) n(S) = 2 and only one element in S is less then (2) n(S) = 1 and the element in S is more than 21 1 2 (3) n(S) = 1 and the element in S is less then 12 (4) n(S) = 0

Q77.Let two vertices of a triangle 𝐴𝐵𝐶 be 2, 4, 6 and 0, - 2, - 5, and its centroid be 2, 1, - 1. If the image of the third vertex in the plane 𝑥+ 2𝑦+ 4𝑧= 11 is 𝛼, 𝛽, 𝛾, then 𝛼𝛽+ 𝛽𝛾+ 𝛾𝛼 is equal to (1) 70 (2) 76 (3) 74 (4) 72

Q77.Let S be the set of all values of θ ∈[−π, π] for which the system of linear equations x + y + √3z = 0 −x + + √7z = 0 (tan θ)y x + y + (tan θ)z = 0 has non-trivial solution.Then 120 π ∑0∈S θ is equal to (1) 20 (2) 40 (3) 30 (4) 10 + (α, β) ∪(γ, δ), then 18(α2 + β2 + γ 2 + δ2)

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 f : (0, 1) →R be a function defined by f(x) = 1−e−x1 , and g(x) = (f(−x) −f(x)). Consider two statements (I) g is an increasing function in (0, 1) (II) g is one-one in (0, 1) Then, (1) Only (I) is true (2) Only (II) is true (3) Neither (I) nor (II) is true (4) Both (I) and (II) are true x−7

Q77.If the system of equations x + 2y + 3z = 3, 4x + 3y −4z = 4 and 8x + 4y −λz = 9 + μ has infinitely many solutions, then the ordered pair (λ, μ) is equal to (1) ( 725 , 215 ) (2) ( −725 , −215 ) (3) ( 725 , −215 ) (4) ( −725 , 215 )

Q77.The plane, passing through the points ( 0, – 1, 2 ) and ( – 1, 2, 1 ) and parallel to the line passing through ( 5, 1, – 7 ) and ( 1, – 1, – 1 ) , also passes through the point (1) -2, 5, 0 (2) 1, - 2, 1 (3) 2, 0, 1 (4) 0, 5, - 2

Q77.Let f : R →R be a function such that f(x) = x2+2x+1 . Then x2+1 (1) f(x) is many-one in (−∞, −1) (2) f(x) is many-one in (1, ∞) (3) f(x) is one-one in [1, ∞) but not in (−∞, ∞) (4) f(x) is one-one in (−∞, ∞) JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q77.Let A be a symmetric matrix such that |A| = 2 and [23 1 1 2 2 A is s , then βs is equal to _________. α2

Q77.For the differentiable function f : R −{0} −R, let 3f(x) + 2f( x1 ) = x1 −10, then f(3) + f ′( 41 ) is equal to (1) 33 (2) 8 5 (3) 29 (4) 13 5 1 sin 3x} = 3 Q78. 0≤x≤π{xmax −2 sin x cos x + (1) π+2−3√3 (2) π 6 (3) 0 (4) 5π+2+3√3 6

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.Let a differentiable function 𝑓 satisfy 𝑓𝑥+ ∫3 𝑡𝑑𝑡= √𝑥+ 1, 𝑥≥3. Then 12𝑓8 is equal to: (1) 34 (2) 19 (3) 17 (4) 1

Q77.Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and −−−−−→ → → → → − + = k FE , then k is equal to (AB BC) (AD −DC) (1) 4 (2) −2 (3) 2 (4) −4

Q77.If the points 𝑃 and 𝑄 are respectively the circumcenter and the orthocentre of a ∆𝐴𝐵𝐶, then →𝑃𝐴+ →𝑃𝐵+ →𝑃𝐶 is equal to _______ (1) 2→𝑄𝑃 (2) 2→𝑃𝑄 (3) →𝑃𝑄 (4) →𝑄𝑃

Q77.If 𝑦= 𝑦𝑥 is the solution curve of the differential equation 𝑑𝑦 𝑦tan𝑥= 𝑥sec𝑥, 0 ≤𝑥≤ 𝜋 𝑦0 = 1, then 𝑑𝑥+ 3, 𝜋 𝑦 is equal to 6 (1) 𝜋 - √3 2 (2) 𝜋 + √3 2√3 12 2 log𝑒 𝑒√3 12 2 loge e (3) 𝜋 - √3 2√3 (4) 𝜋 + √3 2 12 2 loge e 12 2 loge e√3

Q77.If →𝑎, 𝑏, →𝑐 are three non-zero vectors and ^𝑛 is a unit vector perpendicular to →𝑐 such that →𝑎= 𝛼 𝑏- ^𝑛, 𝛼≠0 and →𝑏· →𝑐= 12, then →𝑐× →𝑎× →𝑏 is equal to: (1) 15 (2) 9 (3) 12 (4) 6

Q77.For the system of equations x + y + z = 6 x + 2y + αz = 10 x + 3y + 5z = β, which one of the following is NOT true? (1) System has no solution for α = 3, β = 24 (2) System has a unique solution for α = −3, β = 14 (3) System has infinitely many solutions for (4) System has a unique solution for α = 3, β ≠14 α = 3, β = 14

Q78.The line 𝑙1 passes through the point 2, 6, 2 and is perpendicular to the plane 2𝑥+ 𝑦- 2𝑧= 10. Then the 𝑥+ 1 𝑦+ 4 𝑧 shortest distance between the line 𝑙1 and the line 2 = -3 = 2 is: (1) 7 (2) 19 3 19 (3) (4) 9 2

Q78.For some a, b, c ∈N, let f(x) = ax −3 and g(x) = xb + c, x ∈R. If (fog)−1 (x) = ( 1 2 ) 3 , then (f ∘g)(ac) + (g ∘f)(b) is equal to _____ .