Practice Questions

Q67.The statement among the following that is a tautology is: JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper (1) 𝐴∨𝐴∧𝐵 (2) 𝐴∧𝐴∨𝐵 (3) 𝐵→𝐴∧𝐴→𝐵 (4) 𝐴∧𝐴→𝐵→𝐵

Q67.The contrapositive of the statement "If you will work, you will earn money" is: (1) If you will not earn money, you will not work (2) To earn money, you need to work (3) You will earn money, if you will not work (4) If you will earn money, you will work AAT = I2 , then the value of α4 + β4 is :

Q67.The Boolean expression ( 𝑝⇒𝑞) ∧( 𝑞⇒~𝑝) is equivalent to : (1) ~𝑞 (2) 𝑞 (3) 𝑝 (4) ~𝑝

Q68.Negation of the statement ( 𝑝∨𝑟) ⇒( 𝑞∨𝑟) is : (1) ~𝑝∧𝑞∧~𝑟 (2) ~𝑝∧𝑞∧𝑟 (3) 𝑝∧~𝑞∧~𝑟 (4) 𝑝∧𝑞∧𝑟

Q68.Consider the two statements : (S1) : (p →q) ∨(~q →p) is a tautology. (S2) : (p ∧~q) ∧(~p ∨q) is a fallacy. Then : (1) only (S1) is true. (2) both (S1) and (S2) are false. (3) only (S2) is true. (4) both (S1) and (S2) are true. Q69. ⎡ 1 0 0⎤ Let A = 0 1 1 . Then A2025 −A2020 is equal to ⎣ 1 0 0⎦ (1) A6 −A (2) A6 (3) A5 (4) A5 −A

Q68.Let R = {(P, Q)|P and Q are at the same distance from the origin } be a relation, then the equivalence class of (1, −1) is the set JEE Main 2021 (26 Feb Shift 1) JEE Main Previous Year Paper (1) S = {(x, y) x2 + y2 = 1} (2) S = {(x, y) x2 + y2 = 2} (3) S = {(x, y) x2 + y2 = √2} (4) S = {(x, y) x2 + y2 = 4}

Q68.If P and Q are two statements, then which of the following compound statement is a tautology? (1) ((P ⇒Q) ∧~Q) ⇒Q (2) ((P ⇒Q) ∧~Q) ⇒~P (3) ((P ⇒Q) ∧~Q) ⇒P (4) ((P ⇒Q) ∧~Q) ⇒(P ∧Q)

Q68.Which of the following is equivalent to the Boolean expression p ∧~q ? (1) ~p →~q (2) ~ ( q →p ) (3) ~ ( 𝑝→𝑞) (4) ~ ( 𝑝→~𝑞)

Q68.Which of the following Boolean expression is a tautology ? (1) (p ∧q) ∨(p ∨q) (2) (p ∧q) ∨(p →q) (3) (p ∧q) ∧(p →q) (4) (p ∧q) →(p →q)

Q69.If the truth value of the Boolean expression ((p ∨q) ∧(q →r) ∧(~r)) →(p ∧q) is false, then the truth values of the statements p, q, r respectively can be: (1) FTF (2) TFF (3) TFT (4) FFT

Q69.Which of the following is the negation of the statement "for all M > 0, there exists x ∈S such that x ≥M ′′? (1) there exists M > 0, such that x < M for all (2) there exists M > 0, there exists x ∈S such that x ∈S x ≥M (3) there exists M > 0, there exists x ∈S such that (4) there exists M > 0 such that x ≥M for all x < M x ∈S

Q70.In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement? (1) P and Q (2) P and R (3) Q and R (4) None of these then a possible value of α is

Q70.The compound statement (P ∨Q) ∧(~P) ⇒Q equivalent to: (1) P ∨Q (2) P ∧~Q (3) ~(P ⇒Q) (4) ~(P ⇒Q) ⇔P ∧~Q

Q70.Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following: (1) The match will not be played and weather is not (2) If the match will not be played, then either good and ground is wet. weather is not good or ground is wet. (3) The match will be played and weather is not (4) The match will not be played or weather is good good or ground is wet. and ground is not wet.

Q70.The statement A →(B →A) is equivalent to : (1) A →(A ∧B) (2) A →(A ∨B) (3) A →(A ↔B) (4) A →(A →B)

Q70.If the Boolean expression (p ∧q) ⊛(p ⊗q) is a tautology, then ⊛ and ⊗ are respectively given by (1) →, → (2) ∧, ∨ (3) ∨, → (4) ∧, →

Q71.Let 𝑓: 𝑁→𝑁 be a function such that 𝑓𝑚+ 𝑛= 𝑓𝑚+ 𝑓𝑛 for every 𝑚, 𝑛∈𝑁. If 𝑓6 = 18 then 𝑓2 · 𝑓3 is equal to : (1) 54 (2) 6 (3) 36 (4) 18 JEE Main 2021 (31 Aug Shift 2) JEE Main Previous Year Paper + 𝑥- 1 𝑥- 1 is:

Q72.The domain of the function cosec−1 ( 1+xx ) is : (1) [−12 , ∞) −{0} (2) (−1, −12 ] ∪(0, ∞) (3) [−12 , 0) ∪[1, ∞) (4) (−12 , ∞) −{0}

Q75.The integral ∫ e4 logee3x+5e3loge 2x+5e2loge x−7e2loge 2xloge x (where c is a constant of integration) (1) loge x2 + 5x −7 + c (2) 4 loge x2 + 5x −7 + c (3) 1 4 loge x2 + 5x −7 + c (4) loge √x2 + 5x −7 + c π

Q75.The inverse of y = 5log x is: (1) x = 5log y (2) x = ylog 5 log y (3) y = x 1 1 log 5 (4) x = 5

Q76.If the functions are defined as f(x) = √x and g(x) following functions: f + g, f −g, f/g, g/f, g −f , where (f ± g)(x) = f(x) ± g(x), (f/g)(x) = f(x) g(x) (1) 0 ≤x ≤1 (2) 0 ≤x < 1 (3) 0 < x < 1 (4) 0 < x ≤1 1 ; |x| ≥1 |x| is differentiable at every point of the domain, then the values of a and b are

Q76.The value of the integral ∫1−1 log(x + √x2 + 1)dx is: (1) 2 (2) 0 (3) −1 (4) 1

Q77.If vectors →a1 = xˆi −ˆj + ˆk and →a2 = ˆi + yˆj + zˆk are collinear, then a possible unit vector parallel to the vector xˆi + yˆj + zˆk is: (1) + 1 (−ˆj √2 ˆk) (2) √31 (ˆi +ˆj −ˆk) (3) + ˆk) √2 1 (ˆi −ˆj) (4) √31 (ˆi −ˆj JEE Main 2021 (26 Feb Shift 2) JEE Main Previous Year Paper

Q77.In a triangle ABC, if BC→ = 3, CA→ = 5 and BA→ = 7, then the projection of the vector BA→ on BC→ is equal to JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper (1) 19 (2) 13 2 2 (3) 11 (4) 15 2 2

Q78.In a triangle ABC , if BC→ = 8, CA→ = 7, AB→ = 10 , then the projection of the vector AB→ on AC→ is equal to : (1) 25 (2) 85 4 14 (3) 127 (4) 115 20 16 → → →