Practice Questions

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.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.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)

Q72.Let 5𝑓𝑥+ 4𝑓 𝑥= 𝑥+ 3, 𝑥> 0 . Then 18 ∫1 𝑓𝑥𝑑𝑥 is equal to (1) 5 loge2 + 3 (2) 10 loge2 + 6 (3) 10 loge2 - 6 (4) 5loge2 - 3 ∞ 3 𝑥- 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.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.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.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

Q73.Let the mean of 6 observations 1, 2, 4, 5, x and y be 5 and their variance be 10 . Then their mean deviation about the mean is equal to (1) 7 (2) 3 3 (3) 8 (4) 10 3 3

Q73.Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and α(> 0), and the mean and standard deviation of marks of class B of n students be respectively 55 and 30 −α. If the mean and variance of the marks of the combined class of 100 + n students are respectively 50 and 350 , then the sum of variances of classes A and B is (1) 500 (2) 450 (3) 650 (4) 900

Q73.Let 𝑦= 𝑓𝑥= sin3𝜋 𝜋 + 5𝑥2 + 1 2. Then, at 𝑥= 1, 3cos 3√2-4𝑥3 (1) 2𝑦' + √3𝜋2𝑦= 0 (2) 2𝑦' + 3𝜋2𝑦= 0 (3) √2𝑦' - 3𝜋2𝑦= 0 (4) 𝑦' + 3𝜋2𝑦= 0

Q73.If p, q and r are three propositions, then which of the following combination of truth values of p, q and r makes the logical expression {(p ∨q) ∧((~p) ∨r)} →((~q) ∨r) false ? (1) p = T, q = F, r = T (2) p = T, q = T, r = F (3) p = F, q = T, r = F (4) p = T, q = F, r = F

Q73.Let S be the set of all values of a1 for which the mean deviation about the mean of 100 consecutive positive integers a1, a2, a3, … . , a100 is 25 . Then S is (1) ϕ (2) {99} (3) N (4) {9}

Q73.The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12 . If the new mean of the marks is 10. 2. then their new variance is equal to: (1) 4. 04 (2) 4. 08 (3) 3. 96 (4) 3. 92 Q74. ⎡ 1 logx y logx z ⎤ Let x, y, z > 1 and A = logy x 2 logy z . Then adj (adj A2) is equal to ⎣ logz x logz y 3 ⎦ (1) 64 (2) 28 (3) 48 (4) 24

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.Let 𝑓 be a differentiable function such that 𝑥2𝑓𝑥- 𝑥= 4 𝑥𝑡 𝑓𝑡 𝑑𝑡, 𝑓1 = 2 Then 18 𝑓3 is equal to ∫0 3. (1) 210 (2) 160 (3) 150 (4) 180

Q73.Let 𝑓𝑥= 2𝑥+ tan-1𝑥 and 𝑔𝑥= log𝑒√1 + 𝑥2 + 𝑥, 𝑥∈0, 3. Then (1) There exists 𝑥∈0, 3 such that 𝑓'𝑥< 𝑔'𝑥 (2) max 𝑓𝑥> max 𝑔𝑥 (3) There exist 0 < 𝑥1 < 𝑥2 < 3 such that 𝑓𝑥< 𝑔𝑥, (4) min 𝑓'𝑥= 1 + max 𝑔'𝑥 ∀𝑥∈𝑥1, 𝑥2 Q74. 1 + sin2𝑥 cos2𝑥 sin2𝑥 𝜋 𝜋 Let 𝑓𝑥= sin2𝑥 1 + cos2𝑥 sin2𝑥 , x ∈ 6, 3 . If 𝛼 and 𝛽 respectively are the maximum and the sin2𝑥 cos2𝑥 1 + sin2𝑥 minimum values of 𝑓, then 19 19 (1) 𝛽2 - 2√𝛼= 4 (2) 𝛽2 + 2√𝛼= 4 9 (3) 𝛼2 - 𝛽2 = 4√3 (4) 𝛼2 + 𝛽2 = 2

Q73.Let △, ∇∈{∧, ∨} be such that (p →q) △(p∇q) is a tautology. Then (1) △= ∧, ∇= ∨ (2) △= ∨, ∇= ∧ (3) △= ∨, ∇= ∨ (4) △= ∧, ∇= ∧

Q73.Suppose 𝑓: 𝑅→0, ∞ be a differentiable function such that 5𝑓𝑥+ 𝑦= 𝑓𝑥· 𝑓𝑦, ∀ 𝑥, 𝑦∈𝑅, If 𝑓3 = 320, then ∑𝑛=5 0 𝑓𝑛 is equal to: (1) 6875 (2) 6575 (3) 6825 (4) 6528 JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

Q73.Among the statements (S1) : (p ⇒q) ∨((~p) ∧q) is a tautology (S2) : (q ⇒p) ⇒((~p) ∧q) is a contradiction (1) Neither (S1) and (S2) is True (2) Both (S1) and (S2) are True (3) Only (S2) is True (4) Only (S1) is True

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

Q73.The statement B ⇒((~A) ∨B) is not equivalent to : (1) B ⇒(A ⇒B) (2) A ⇒(A ⇔B) (3) A ⇒((~A) ⇒B) (4) B ⇒((~A) ⇒B) ¯¯

Q73.Let the positive numbers a1, a2, a3, a4 and a5 be in a G.P. Let their mean and variance be 1031 and mn respectively, where m and n are co-prime. If the mean of their reciprocals is 31 and a3 + a4 + a5 = 14, then 10 m + n is equal to ____________.

Q73.Let 9 = x1 < x2 < … < x7 be in an A.P. with common difference d. If the standard deviation of x1, x2 … , x7 ¯¯is 4 and the mean is x , then x + x6 is equal to : JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper + 1 ) (2) 34 (1) 18(1 √3 + 8 ) (4) 25 (3) 2(9 √7

Q73.Let [x] denote the greatest integer function and f(x) = max{1 + x + [x], 2 + x, x + 2[x]}, 0 ≤x ≤2 , where f is not continuous and n be the number of points in (0, 2), where f is not differentiable. Then (m + n)2 + 2 is equal to (1) 2 (2) 11 (3) 6 (4) 3 α, β > 0 , then α4 −β4 is equal to dx = α1 loge( α+1β ),