Practice Questions

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 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.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 →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.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 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.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 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 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.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.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.The number of functions f : {1, 2, 3, 4} →{a ∈Z : |a| ≤8} satisfying f(n) + n1 f(n + 1) = 1, ∀ n ∈{1, 2, 3} is (1) 3 (2) 4 (3) 1 (4) 2 λ (1 + | cos x|)Q78. , 0 < x < π2 |cos x| ⎧ μ, x = π2 is continuous at x = π2 , then If the function f(x) = ⎨ cot 6x cot 4x ⎩ e , π2 < x < π 9λ + 6 logc μ + μ6 −e6λ is equal to (1) 11 (2) 8 (3) 2e4 + 8 (4) 10

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.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

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

Q78.Let ( 𝛼, 𝛽, 𝛾) be the image of point 𝑃( 2, 3, 5 ) in the plane 2𝑥+ 𝑦- 3𝑧= 6. Then 𝛼+ 𝛽+ 𝛾 is equal to (1) 5 (2) 10 (3) 12 (4) 9

Q78.Let the sets A and B denote the domain and range respectively of the function f(x) = 1 , where [x] √[x]−x denotes the smallest integer greater than or equal to x. Then among the statements (S1) : A ∩B = (1, ∞) −N and (S2) : A ∪B = (1, ∞) (1) Only (S2) is true (2) Only (S1) is true (3) Neither (S1) nor (S2) is true (4) Both (S1) and (S2) are true

Q78.If f(x) = 22x , x ∈R, then f( 20231 ) + f( 20232 ) + f( 20233 ). . . . . . . . . f( 20222023 ) is equal to 22x+2 (1) 2011 (2) 1010 (3) 2010 (4) 1011

Q78.The shortest distance between the lines 𝑥+ 2 = 𝑦 = 𝑧- 5 and 𝑥- 4 = 𝑦- 1 = 𝑧+ 3 is 1 -2 2 1 2 0 (1) 8 (2) 6 (3) 7 (4) 9 𝑥+ 3 𝑦+ 2 1 - 𝑧

Q78.Let the foot of perpendicular of the point P(3, –2, –9) on the plane passing through the points (–1, –2, –3), (9, 3, 4), (9, –2, 1) be Q(α, β, γ). Then the distance Q from the origin is (1) √42 (2) √38 (3) √35 (4) √29

Q78.Let f : R →R be a differentiable function that satisfies the relation f(x + y) = f(x) + f(y) −1, ∀ x, y ∈R. If f ′(0) = 2 , then |f(−2)| is equal to

Q78.One vertex of a rectangular parallelopiped is at the origin 𝑂 and the lengths of its edges along 𝑥, 𝑦 and 𝑧 axes are 3, 4 and 5 units respectively. Let 𝑃 be the vertex ( 3, 4, 5 ) . Then the shortest distance between the diagonal 𝑂𝑃 and an edge parallel to 𝑧 axis, not passing through 𝑂 or 𝑃 is 12 (1) (2) 12√5 √5 12 12 (3) (4) 5√5 5