Practice Questions

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.If the system of equations 2𝑥+ 𝑦- 𝑧= 5 2𝑥- 5𝑦+ 𝜆𝑧= 𝜇 𝑥+ 2𝑦- 5𝑧= 7 has infinitely many solutions, then ( 𝜆+ 𝜇) 2 + ( 𝜆- 𝜇) 2 is equal to (1) 904 (2) 916 (3) 912 (4) 920

Q71.Let the tangent to the parabola y2 = 12x at the point (3, α) be perpendicular to the line 2x + 2y = 3 . Then the square of distance of the point (6, −4) from the normal to the hyperbola α2x2 −9y2 = 9α2 at its point (α −1, α + 2) is equal to .............

Q71.Let 𝐴= 2, 3, 4 and 𝐵= 8, 9, 12. Then the number of elements in the relation 𝑅= 𝑎1, 𝑏1, 𝑎2, 𝑏2 ∈𝐴× 𝐵, 𝐴× 𝐵: 𝑎1 divides 𝑏2 and 𝑎2 divides 𝑏1 is (1) 36 (2) 24 (3) 18 (4) 12 Q72. 5! 6! 7! 1 If 𝐴= 6! 7! 8! , then adj adj 2𝐴 is equal to 5!6!7! 7! 8! 9! (1) 220 (2) 28 (3) 212 (4) 216

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.If the system of equations 𝑥+ 𝑦+ 𝑎𝑧= 𝑏 2𝑥+ 5𝑦+ 2𝑧= 6 𝑥+ 2𝑦+ 3𝑧= 3 has infinitely many solutions, then 2𝑎+ 3𝑏 is equal to (1) 25 (2) 20 (3) 23 (4) 28 1 1 2

Q71.Let the eccentricity of an ellipse x2 + y2 = 1 is reciprocal to that of the hyperbola 2x2 −2y2 = 1 . If the a2 b2 ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is _____. JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper lim 2 −2 3 2 −2 5 . . . 2 −2 2n+1 )(2 ). (2 )} is equal to

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

Q71.The set of values of a for which x→a([xlim −5] −[2x + 2]) = 0 , where, [ζ] denotes the greatest integer less than or equal to ζ is equal to (1) (−7. 5, −6. 5) (2) (−7. 5, −6. 5] (3) [−7. 5, −6. 5] (4) [−7. 5, −6. 5)

Q71.The value of 1+2−3+4+5−6+…+(3n−2)+(3n−1)−3n lim is n→∞ √2n4+4n+3−√n4+5n+4 (1) √2+1 + 2 (2) 3(√2 1) (3) 3 + 2 (√2 1) (4) 2√23

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

Q72. x→0((lim 1−cos2(3x)cos3(4x) )( (loge(2x+1))5sin3(4x) )) is equal to (1) 15 (2) 9 (3) 18 (4) 24

Q72.The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is (1) 610 (2) 106 (3) 910 (4) 109 Q73. ⎡ 1 3 α⎤ ⎡ α ⎤ Let B = 1 2 3 , α > 2 be the adjoint of a matrix A and |A| = 2. Then [α −2α α ]B −2α is equal to ⎣ α α 4 ⎦ ⎣ α ⎦ (1) 0 (2) 16 (3) −16 (4) 32

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.Among the two statements (S1) : (p ⇒q) ∧(p ∧(~q)) is a contradiction and (S2) : (p ∧q) ∨((~p) ∧q) ∨(p ∧(~q)) ∨((~p) ∧(~q)) is a tautology (1) only (S2) is true (2) only (S1) is true (3) both are false (4) both are true

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

Q72.Let 𝑆 be the set of all solutions of the equation cos-12𝑥- 2cos-1√1 - 𝑥2 = 𝜋, 𝑥∈-1 2, 12. Then ∑𝑥∈𝑆2sin-1𝑥2 is equal to -2𝜋 (1) 0 (2) 3 (3) 𝜋- sin-1√3 (4) 𝜋- 2sin-1√3 4 4

Q72.If the domain of the function 𝑓𝑥= where 𝑥 is greatest integer ≤𝑥, is [2, 6 ) , then its range is 1 + 𝑥2, 5 2 9 27 18 9 5 2 (1) 26, 5 - 29, 109, 89, 53 (2) 26, 5 (3) 5 2 - 9 27 18 9 (4) 5 2 37, 5 29, 109, 89, 53 37, 5 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 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.Consider the following statements: P : I have fever Q : I will not take medicine R : I will take rest The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to: JEE Main 2023 (30 Jan Shift 2) JEE Main Previous Year Paper (1) ((~P) ∨~Q) ∧((~P) ∨R) (2) ((~P) ∨−Q) ∧((~P) ∨~R) (3) (P ∨Q) ∧((~P) ∨R) (4) (P ∨~Q) ∧(P ∨~R)

Q72.If the domain of the function f(x) = loge(4x2 + 11x + 6) + sin−1(4x + 3) + cos−1( 10x+63 ) is (α, β] , then 36|α + β| is equal to (1) 54 (2) 72 (3) 63 (4) 45

Q72.Let the system of linear equations 𝑥+ 𝑦+ 𝑘𝑧= 2 2𝑥+ 3𝑦- 𝑧= 1 3𝑥+ 4𝑦+ 2𝑧= 𝑘 have infinitely many solutions. Then the system 𝑘+ 1 𝑥+ 2𝑘- 1 𝑦= 7 2𝑘+ 1𝑥+ 𝑘+ 5𝑦= 10 has : (1) infinitely many solutions (2) unique solution satisfying 𝑥- 𝑦= 1 (3) no solution (4) unique solution satisfying 𝑥+ 𝑦= 1

Q72.The statement (p ∧(~q)) ⇒(p ⇒(~q)) is (1) equivalent to (~p) ∨(~q) (2) a tautology (3) equivalent to p ∨q (4) a contradiction