Practice Questions

Q83.Remainder when 643232 is divided by 9 is equal to _____.

Q85.Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on A × B by (a1, b1)R (a2, b2) if and only if a1 + a2 = b1 + b2 . Then the number of elements in R is _________

Q86.Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2 sin−1 x + 3 cos−1 x = 2π5 , is _______

Q87.Let f(x) = 2x −x2, x ∈R. If m and n are respectively the number of points at which the curves y = f(x) and y = f ′(x) intersects the x−axis, then the value of m + n is

Q89.The least positive integral value of α, for which the angle between the vectors αˆi −2ˆj + 2ˆk and αˆi + 2αˆj −2ˆk is acute, is _____.

Q64.If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is (1) 103 (2) 102 (3) 101 (4) 104

Q65.If 2𝑛𝐶3: 𝑛𝐶3 = 10: 1, then the ratio 𝑛2 + 3𝑛: 𝑛2 - 3𝑛+ 4 is (1) 35: 16 (2) 27: 11 (3) 65: 37 (4) 2: 1

Q65.The largest natural number n such that 3n divides 66! is _______

Q65.If tan15° + + + tan195° = 2a, then the value of 𝑎+ is : tan75° tan105° 𝑎 (1) 4 (2) 4 - 2√3 (3) 2 (4) 5 - 3 2√3

Q67.If 2n+1P n−1 : 2n−1P n = 11 : 21 , then n2 + n + 15 is equal to :

Q67.The negation of the expression 𝑞∨( ( ~𝑞) ∧𝑝) is equivalent to (1) ( ~𝑝) ∧( ~𝑞) (2) 𝑝∧( ~𝑞) (3) ( ~𝑝) ∨( ~𝑞) (4) ( ~𝑝) ∨𝑞

Q67.Statement ( 𝑃⇒𝑄) ∧( 𝑅⇒𝑄) is logically equivalent to (1) 𝑃⇒𝑅∨𝑄⇒𝑅 (2) 𝑃∧𝑅⇒𝑄 (3) 𝑃⇒𝑅∧𝑄⇒𝑅 (4) 𝑃∨𝑅⇒𝑄

Q67.The negation of the statement 𝑝∨𝑞∧𝑞∨~𝑟 is (1) 𝑝∨𝑟∧~𝑞 (2) ~𝑝) ∨𝑟∧~𝑞 (3) ~𝑝∨~𝑞∨~𝑟 (4) ~𝑝∨~𝑞∧~𝑟

Q68.Negation of p ∧(q ∧~(p ∧q)) is (1) (~(p ∧q)) ∨p (2) p ∨q (3) ~(p ∨q) (4) (~(p ∧q)) ∧q

Q69.The statement ( 𝑝∧( ~𝑞) ) ∨( ( ~𝑝) ∧ 𝑞) ∨( ( ~𝑝) ∧ ( ~𝑞) ) is equivalent to _____ (1) ~𝑝∨𝑞 (2) ~𝑝∨~𝑞 (3) 𝑝∨~𝑞 (4) 𝑝∨𝑞 Q70. 1 2 3 Let for 𝐴= 𝛼3 1 , 𝐴= 2. If |2 adj ( 2 adj ( 2𝐴) ) | = 32𝑛, then 3𝑛+ 𝛼 is equal to 1 1 2 (1) 9 (2) 11 (3) 12 (4) 10

Q69.The statement ~𝑝∨~𝑝∧𝑞 is equivalent to (1) ~𝑝∧𝑞 (2) 𝑝∧𝑞∧~𝑝 (3) ~𝑝∧𝑞∧𝑞 (4) ~𝑝∨𝑞 JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper

Q70.Let H be the hyperbola, whose foci are (1 ± √2, 0) and eccentricity is √2 . Then the length of its latus rectum is: (1) 3 (2) 52 (3) 2 (4) 32

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

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.Let p and q be two statements. Then ~(p ∧(p →~q) is equivalent to (1) p ∨(p ∧(~q)) (2) p ∨((~p) ∧q) (3) (~p) ∨q (4) p ∨(p ∧q)

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)

Q73.The negation of (p ∧(−q)) ∨(−p) is equivalent to (1) p ∧(−q) (2) p ∧q (3) p ∨(q ∨(−p)) (4) p ∧(q ∧(−p))

Q73.Negation of (p →q) →(q →p) is (1) (p~) ∨p (2) q ∧(~p) (3) (~q) ∧p (4) p ∨(~q)

Q75.Let A = {1, 2, 3, 4, 5, 6, 7} . Then the relation R = {(x, y) ∈A × A : x + y = 7} is (1) an equivalence relation (2) symmetric but neither reflexive nor transitive (3) transitive but neither symmetric nor reflexive (4) reflexive but neither symmetric nor transitive A−1 = αA + βI and α + β = −2, then 4α2 + β2 + λ2 is equal to :