Practice Questions

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

Q71.Let the tangents at the points A(4, −11) and B(8, −5) on the circle x2 + y2 −3x + 10y −15 = 0 , intersect at the point C . Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to (1) 3√3 (2) 2√13 4 (3) √13 (4) 2√13 3 Q72. 1−cos(x2−4px+q2+8q+16) ⎧ , x ≠2p Let x = 2 be a root of the equation x2 + px + q = 0 and f(x) = (x−2p)4 . Then ⎨ ⎩ 0, x = 2p x→2p+[f(x)]lim where [⋅] denotes greatest integer function, is (1) 2 (2) 1 (3) 0 (4) −1

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.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 𝑆 denote the set of all real values of 𝜆 such that the system of equations 𝜆𝑥+ 𝑦+ 𝑧= 1 𝑥+ 𝜆𝑦+ 𝑧= 1 𝑥+ 𝑦+ 𝜆𝑧= 1 is inconsistent, then ∑𝜆∈𝑆𝜆2 + 𝜆 is equal to (1) 2 (2) 12 (3) 4 (4) 6 - 1

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. (√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 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

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 range of 𝑓𝑥= 4sin-1 𝑥2 is 𝑥2 + 1 (1) [0, 2𝜋] (2) [0, 𝜋] (3) [0, 2𝜋) (4) [0, 𝜋) 𝜋 4 𝑒-𝑥tan 50 𝑥𝑑𝑥 Q73. 𝑒-𝜋4 + ∫0 The value of 𝜋 ∫04 𝑒-𝑥(tan49𝑥+ tan51𝑥)𝑑𝑥 (1) 51 (2) 50 (3) 25 (4) 49 JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

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

Q72.The converse of ((~p) ∧q) ⇒r is (1) ((~p) ∨q) ⇒r (2) (~r) ⇒p ∧q (3) (~r) ⇒((~p) ∧q) (4) (p ∨(~q)) ⇒(~r)

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.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.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.The number of values of r ∈{p, q, ~p, ~q} for which ((p ∧q) ⇒(r ∨q) ∧((p ∧r) ⇒q) is a tautology, is : (1) 1 (2) 2 (3) 4 (4) 3

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

Q72.If 𝐼𝑥= ∫𝑒sin2𝑥cos𝑥 sin2𝑥- sin𝑥𝑑𝑥 and 𝐼0 = 1, then 𝐼 𝜋 is equal to 3 (1) -1 34 (2) 1 34 2𝑒 2𝑒 3 (3) -𝑒 4 (4) 𝑒 34

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.Which of the following statements is a tautology? (1) p →(p ∧(p →q)) (2) (p ∧q) →(~(p) →q) (3) (p ∧(p →q)) →~q (4) p ∨(p ∧q)