Practice Questions

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.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.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.Which of the following is equivalent to the Boolean expression p ∧~q ? (1) ~p →~q (2) ~ ( q →p ) (3) ~ ( 𝑝→𝑞) (4) ~ ( 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.If the Boolean expression (p ∧q) ⊛(p ⊗q) is a tautology, then ⊛ and ⊗ are respectively given by (1) →, → (2) ∧, ∨ (3) ∨, → (4) ∧, →

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)

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 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

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

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

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

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 → → →

Q79.If →a = 2, →b = 5 and →a×→b = 8, then →a⋅→b is equal to: (1) 6 (2) 4 (3) 3 (4) 5

Q79.Consider the three planes P1 : 3x + 15y + 21z = 9 P2 : x −3y −z = 5, and P3 : 2x + 10y + 14z = 5 Then, which one of the following is true? (1) P2 and P3 are parallel. (2) P1, P2 and P3 all are parallel. (3) P1 and P2 are parallel. (4) P1 and P3 are parallel.

Q79.The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is: (1) x + 3z = 10 (2) x + 3z = 0 (3) 3x + z = 6 (4) 3x −z = 0

Q81.There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is

Q81.If log3 2, log3(2x −5), log3(2x −72 ) are in an arithmetic progression, then the value of x is equal to _____.

Q1. For the four sets of three measured physical quantities as given below. Which of the following options is correct? (i) A1 = 24.36, B1 = 0.0724, C1 = 256.2 (ii) A2 = 24.44, B2 = 16.082, C2 = 240.2 (iii) A3 = 25.2, B3 = 19.2812, C3 = 236.183 (iv) A4 = 25, B4 = 236.191, C4 = 19.5 (1) A4 + B4 + C4 < A1 + B1 + C1 < A3 + B3 + C3 < A2 + B2 + C2 (2) A1 + B1 + C1 = A2 + B2 + C2 = A3 + B3 + C3 = A4 + B4 + C4 (3) A1 + B1 + C1 < A2 + B2 + C2 = A3 + B3 + C3 < A4 + B4 + C4 (4) A1 + B1 + C1 < A3 + B3 + C3 < A2 + B2 + C2 < A4 + B4 + C4