Practice Questions

Q68.Let f(θ) = 3(sin4( 3π2 −θ) + sin4(3π + θ)) −2(1 −sin2 2θ) and S = {θ ∈[0, π] ′(θ) = −√32 }. If 4β = ∑θ∈S θ then f(β) is equal to (1) 11 (2) 5 8 4 (3) 9 (4) 3 8 2

Q68.The set of all values of a2 for which the line x + y = 0 bisects two distinct chords drawn from a point P( 1+a2 , 1−a2 ) on the circle 2x2 + 2y2 −(1 + a)x −(1 −a)y = 0 , is equal to : (1) (8, ∞) (2) (0, 4] (3) (4, ∞) (4) (2, 12] JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q69.Let A(0, 1), B(1, 1) and C(1, 0) be the mid-points of the sides of a triangle with incentre at the point D. If the α and β are rational numbers, then focus of the parabola y2 = 4ax passing through D is (α + β√2, 0), where α is equal to β2 (1) 8 (2) 12 (3) 6 (4) 29 JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper

Q69.Let C(α, β) be the circumcentre of the triangle formed by the lines 4x + 3y = 69 , 4y −3x = 17 , and x + 7y = 61 . Then (α −β)2 + α + β is equal to (1) 18 (2) 17 (3) 15 (4) 16

Q69.For a triangle 𝐴𝐵𝐶, the value of cos2𝐴+ cos2𝐵+ cos2𝐶 is least. If its inradius is 3 and incentre is 𝑀, then which of the following is NOT correct? (1) Perimeter of ∆𝐴𝐵𝐶 is 18√3 (2) sin2𝐴+ sin2𝐵+ sin2𝐶= sin𝐴+ sin𝐵+ sin𝐶 (3) →MA · →MB = - 18 (4) area of ∆𝐴𝐵𝐶 is 27√3 2

Q69.For the system of linear equations 𝑥+ 𝑦+ 𝑧= 6 𝛼𝑥+ 𝛽𝑦+ 7𝑧= 3 𝑥+ 2𝑦+ 3𝑧= 14 which of the following is NOT true ? (1) If 𝛼= 𝛽= 7, then the system has no solution (2) If 𝛼= 𝛽 and 𝛼≠7 then the system has a unique solution. (3) There is a unique point ( 𝛼, 𝛽) on the line (4) For every point ( 𝛼, 𝛽) ≠( 7, 7 ) on the line 𝑥+ 2𝑦+ 18 = 0 for which the system has x - 2y + 7 = 0, the system has infinitely many infinitely many solutions solutions.

Q70.Let the determinant of a square matrix A of order m be m −n , where m and n satisfy 4m + n = 22 and 17m + 4n = 93 . If det(n adj(adj(mA))) = 3a5b6c , then a + b + c is equal to (1) 84 (2) 96 (3) 101 (4) 109

Q70.The equations of sides AB and AC of a triangle ABC are (λ + 1)x + λy = 4 and λx + (1 −λ)y + λ = 0 respectively. Its vertex A is on the y−axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola y2 = 6x in the first quadrant is (1) √6 (2) 2√2 (3) 2 (4) 4 JEE Main 2023 (24 Jan Shift 2) JEE Main Previous Year Paper

Q70.If the radius of the largest circle with centre (2, 0) inscribed in the ellipse x2 + 4y2 = 36 is r, then 12 r2 is equal to (1) 115 (2) 92 (3) 69 (4) 72

Q70.Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2 + y2 = 8 and y2 = 16x . If one of these tangents touches the two curves at Q and R, then (QR)2 is equal to (1) 64 (2) 76 (3) 81 (4) 72

Q70.Let 𝛼 be a root of the equation 𝑎- 𝑐𝑥2 + 𝑏- 𝑎𝑥+ 𝑐- 𝑏= 0 where 𝑎, 𝑏, 𝑐 are distinct real numbers such that 𝛼2 𝛼1 𝑎- 𝑐2 𝑏- 𝑎2 𝑐- 𝑏2 the matrix 1 1 1 is singular. Then the value of is 𝑏- 𝑎𝑐- 𝑏+ 𝑎- 𝑐𝑐- 𝑏+ 𝑎- 𝑐𝑏- 𝑎 𝑎 𝑏 𝑐 (1) 6 (2) 3 (3) 9 (4) 12

Q70.Let B and C be the two points on the line y + x =0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y −2x = 2 such that ΔABC is an equilateral triangle. Then, the area of the ΔABC is (1) 3√3 (2) 2√3 (3) 8 (4) 10 √3 √3

Q70.Let 𝐴 be a 2 × 2 matrix with real entries such that 𝐴' = 𝛼𝐴+ 1, where 𝛼∈ℝ- -1, 1., If det 𝐴2 - 𝐴= 4, the sum of all possible values of 𝛼 is equal to 3 (1) 0 (2) 2 (3) 2 (4) 5 2

Q71. (√3x+1+√3x−1) 6 +(√3x+1−√3x−1) 6 lim 6 6 x3 x→∞ (x+√x2−1) +(x−√x2−1) (1) is equal to 272 (2) is equal to 9 (3) does not exist (4) is equal to 27

Q71.Let R be the focus of the parabola y2 = 20x and the line y = mx + c intersect the parabola at two points P and Q. Let the points G(10, 10) be the centroid of the triangle PQR . If c −m = 6 , then PQ2 is (1) 296 (2) 325 (3) 317 (4) 346

Q71.Let P( 2√3√7 √7 perpendicular and pass through the origin. If 1 + 1 = pq , where p and q are coprime, then p + q is (PQ)2 (RS)2 equal to (1) 147 (2) 143 (3) 137 (4) 157

Q71.Let the system of linear equations –x + 2y −9z = 7 −x + 3y + 7z = 9 −2x + y + 5z = 8 −3x + y + 13z = λ has a unique solution x = α, y = β, z = γ . Then the distance of the point (α, β, γ) from the plane 2x −2y + z = λ is (1) 11 (2) 7 (3) 9 (4) 13

Q71.Let f, g and h be the real valued functions defined on R as x , x ≠0 sin(x+1) |x| (x+1) , x ≠−1 f(x) = , g(x) = and h(x) = 2[x] −f(x), where [x] is the greatest integer { 1, x = 0 { 1, x = −1 ≤x. Then the value of lim g(h(x −1)) is x→1 (1) 1 (2) sin(1) (3) −1 (4) 0

Q72.If the tangent at a point P on the parabola y2 = 3x is parallel to the line x + 2y = 1 and the tangents at the x2 y2 points Q and R on the ellipse 4 + 1 = 1 are perpendicular to the line x −y = 2, then the area of the triangle PQR is: (1) 9 (2) 5√3 √5 (3) 3 2 √5 (4) 3√5

Q72.The equations of two sides of a variable triangle are x = 0 and y = 3 , and its third side is a tangent to the parabola y2 = 6x . The locus of its circumcentre is : (1) 4y2 −18y −3x −18 = 0 (2) 4y2 + 18y + 3x + 18 = 0 (3) 4y2 −18y + 3x + 18 = 0 (4) 4y2 −18y −3x + 18 = 0 JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper

Q72. n→∞{(2 1 1 1 1 1 1 (1) 1 (2) 0 (3) √2 (4) 1 √2

Q72.Let 5𝑓𝑥+ 4𝑓 𝑥= 𝑥+ 3, 𝑥> 0 . Then 18 ∫1 𝑓𝑥𝑑𝑥 is equal to (1) 5 loge2 + 3 (2) 10 loge2 + 6 (3) 10 loge2 - 6 (4) 5loge2 - 3 ∞ 3 𝑥- 3

Q72.The equation 𝑥2 – 4𝑥+ [𝑥] + 3 = 𝑥[𝑥], where [𝑥] denotes the greatest integer function, has: (1) exactly two solutions in ( - ∞, ∞) (2) no solution (3) a unique solution in ( - ∞, 1 ) (4) a unique solution in ( - ∞, ∞) Q73. 𝑥2sin1 𝑥≠0 Let 𝑓𝑥= 𝑥; , then at 𝑥= 0 0; 𝑥= 0 (1) 𝑓 is continuous but not differentiable (2) 𝑓 is continuous but 𝑓' is not continuous (3) both 𝑓 and 𝑓' are continuous (4) 𝑓' is continuous but not differentiable

Q72.If α > β > 0 are the roots of the equation ax2 + bx + 1 = 0 , and 1 1−cos(x2+bx+a) 2 1 1 k is equal to lim ( 2(1−αx)2 ) = k ( β −1α ), then x→1α (1) 2β (2) α (3) 2α (4) β

Q72.Let 𝑓: 2, 4 →ℝ be a differentiable function such that 𝑥log𝑒𝑥𝑓'𝑥+ log𝑒𝑥𝑓𝑥+ 𝑓𝑥≥1, 𝑥∈2, 4 with 𝑓2 = 2 and 1 𝑓4 = 2. Consider the following two statements: (A) 𝑓𝑥≤1, for all 𝑥∈2, 4 (B) 𝑓𝑥≥1 / 8, for all 𝑥∈2, 4 Then, (1) Neither statement ( 𝐴) nor statement ( 𝐵) is (2) Only statement ( 𝐵) is true true (3) Both the statements ( 𝐴) and ( 𝐵) are true (4) Only statement ( 𝐴) is true √1 + 𝑒2𝑥𝑑𝑥 is equal to