Practice Questions

Q61.If z = 2 + 3i, then z5 + (z)5 is equal to: (1) 244 (2) 224 (3) 245 (4) 265

Q61.Let the minimum value v0 of v = |z|2 + |z −3|2 + |z −6i|2 , z ∈C is attained at z = z0 . Then ¯2z20 −z30 + 3 2 + v20 is equal to (1) 1000 (2) 1024 (3) 1105 (4) 1196

Q61.The minimum value of the sum of the squares of the roots of 𝑥2 + 3 - 𝑎𝑥= 2𝑎- 1 is (1) 6 (2) 4 (3) 5 (4) 8

Q63.The number of solutions of cos𝑥= sin𝑥, such that -4𝜋≤𝑥≤4𝜋 is (1) 4 (2) 6 (3) 8 (4) 12

Q64.The remainder when 32022 is divided by 5 is (1) 1 (2) 2 (3) 3 (4) 4

Q64.If a1, a2, a3 … and b1, b2, b3 … . are A.P. and a1 = 2, a10 = 3, a1b1 = 1 = a10b10 then a4b4 is equal to (1) 28 (2) 28 27 24 (3) 23 (4) 22 26 23 Q65. α = sin 36° is a root of which of the following equation (1) 16x4 −20x2 + 5 = 0 (2) 16x4 + 20x2 + 5 = 0 (3) 10x4 −10x2 −5 = 0 (4) 16x4 −10x2 + 5 = 0

Q65.Let p : Ramesh listens to music. q : Ramesh is out of his village r : It is Sunday s : It is Saturday Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as (1) ((~q) ∧(r ∨s)) ⇒p (2) (q ∧(r ∨s)) ⇒p (3) p ⇒(q ∧(r ∨s)) (4) p ⇒((~q) ∧(r ∨s))

Q66.Let p, q, r be three logical statements. Consider the compound statements S1 : ((~p) ∨q) ∨((~p) ∨r) and S2 : p →(q ∨r) Then, which of the following is NOT true? (1) If S2 is True, then S1 is True (2) If S2 is False, then S1 is False (3) If S2 is False, then S1 is True (4) If S1 is False, then S2 is False

Q67.The boolean expression (~(p ∧q)) ∨q is equivalent to (1) q →(p ∧q) (2) p →q (3) p →(p →q) (4) p →(p ∨q)

Q67.Which of the following statements is a tautology? (1) ~𝑝∨𝑞⇒𝑝 (2) 𝑝⇒~𝑝∨𝑞 (3) ~𝑝∨𝑞⇒𝑞 (4) 𝑞⇒~𝑝∨𝑞

Q67.If vertex of parabola is (2, −1) and equation of its directrix is 4x −3y = 21, then the length of latus rectum is (1) 2 (2) 8 (3) 12 (4) 16

Q67.Consider the following statements: A: Rishi is a judge. B: Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is (1) B →(A ∨C) (2) (~B) ∧(A ∧C) (3) B →((~A) ∨(~C)) (4) B →(A ∧C)

Q67.The statement (p ∧q) ⇒(p ∧r) is equivalent to (1) q ⇒(p ∧r) (2) p ⇒(p ∧r) (3) (p ∧r) ⇒(p ∧q) (4) (p ∧q) ⇒r JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper

Q67.Consider the following two propositions : 𝑃1: ~𝑝→~𝑞 𝑃2: 𝑝∧~𝑞∧~𝑝∨𝑞 If the proposition 𝑝→~𝑝∨𝑞 is evaluated as FALSE, then (1) 𝑃1 is TRUE and 𝑃2 is FALSE (2) 𝑃1 is FALSE and 𝑃2 is TRUE (3) Both 𝑃1 and 𝑃2 are FALSE (4) Both 𝑃1 and 𝑃2 are TRUE

Q68.If the truth value of the statement (P ∧(~R)) →((~R) ∧Q) is F , then the truth value of which of the following is F ? (1) P ∨Q →~R (2) R ∨Q →~P (3) ~(P ∨Q) →~R (4) ~(R ∨Q) →~P

Q68.Let f(x) = ax2 + bx + c be such that f(1) = 3, f(−2) = λ and f(3) = 4. If f(0) + f(1) + f(−2) + f(3) = 14 , then λ is equal to JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper (1) −4 (2) 132 (3) 23 (4) 4 2

Q68.The statement 𝑝⇒𝑞∨𝑝⇒𝑟 is NOT equivalent to: (1) 𝑝∧~𝑟⇒𝑞 (2) ~𝑞⇒~𝑟∨𝑝 (3) 𝑝⇒𝑞∨𝑟 (4) 𝑝∧~𝑞⇒𝑟

Q68. (p ∧r) ⇔(p ∧(~q)) is equivalent to (~p) when r is (1) p (2) ~p (3) q (4) ~q

Q69.Which of the following matrices can NOT be obtained from the matrix -1 2 by a single elementary row 1 -1 operation? (1) 0 1 (2) 1 -1 1 -1 -1 2 (3) -1 2 (4) -1 2 -2 7 -1 3

Q69.The mean and variance of the data 4, 5, 6, 6, 7, 8, x, y where x < y are 6 and 49 respectively. Then x4 + y2 is equal to (1) 320 (2) 420 (3) 162 (4) 674

Q69.Negation of the Boolean statement (p ∨q) ⇒((~r) ∨p) is equivalent to: (1) p ∧(~q) ∧r (2) (~p) ∧(~q) ∧r (3) (~p) ∧q ∧r (4) p ∧q ∧(~r)

Q72.The value of tan-1cos15𝜋 is equal to sin𝜋 4 𝜋 𝜋 (1) - (2) - 4 8 (3) -5𝜋 (4) -4𝜋 12 9

Q74.The value of the integral ∫2−2 (ex|x|+1)x3+x (1) 5e2 (2) 3e−2 (3) 4 (4) 6 dy ax−by+a

Q75.The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is 3643 , equal to: (1) 3 (2) 5 (3) 7 (4) 9

Q80.A random variable X has the following probability distribution: X 0 1 2 3 4 P(X) k 2k 4k 6k 8k The value of P( 1<x<4x≤2 )is equal to (1) 4 (2) 2 7 3 (3) 3 (4) 4 7 5 ¯