Practice Questions

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 𝐴= {𝑥∈ℝ: 𝑥+ 3 + 𝑥+ 4 ≤3}, 𝐵= 𝑥∈ℝ: 3𝑥∑𝑟= 1 10𝑟 < 3-3𝑥, where [𝑡] denotes greatest integer function. Then, (1) 𝐵⊂𝐶, 𝐴≠𝐵 (2) 𝐴∩𝐵= 𝜙 (3) 𝐴⊂𝐵, 𝐴≠𝐵 (4) 𝐴= 𝐵

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 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 𝑔𝑥= 𝑓𝑥+ 𝑓1 - 𝑥 and 𝑓"𝑥> 0, 𝑥∈0, 1. If 𝑔 is decreasing in the interval 0, 𝛼 and increasing in the interval 𝛼, 1, then tan-12𝛼+ tan-1 1 tan-1𝛼+ 1 is equal to 𝛼+ 𝛼 5π (1) π (2) 4 (3) 3π (4) 3π 4 2

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 ),

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.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

Q74.Let P(S) denote the power set of S = {1, 2, 3, … , 10} . Define the relations R1 and R2 on P(S) as AR1B if (A ∩Bc) ∪(B ∩Ac) = ϕ and AR2 B if A ∪Bc = B ∪Ac, ∀A, B ∈P(S) . Then : (1) both R1 and R2 are equivalence relations (2) only R1 is an equivalence relation (3) only R2 is an equivalence relation (4) both R1 and R2 are not equivalence relations 1 1 √3 then,

Q74.For 𝛼, 𝛽, 𝛾, 𝛿∈ℕ, if ∫ 𝑥 2𝑥+ and 𝐶 is constant of 𝑒 𝑥 𝑒 2𝑥log𝑒𝑥𝑑𝑥= 𝛼1 𝑥𝑒 𝛽𝑥- 1𝛾 𝑥𝑒 𝛿𝑥+ 𝐶, where 𝑒= ∑𝑛=∞ 0 𝑛!1 integration, then 𝛼+ 2𝛽+ 3𝛾- 4𝛿 is equal to (1) 1 (2) 4 (3) -4 (4) -8

Q75.An arc 𝑃𝑄 of a circle subtends a right angle at its centre 𝑂. The mid point of the arc 𝑃𝑄 is 𝑅. If →𝑂𝑃= →𝑢, →𝑂𝑅= →𝑣 and →𝑂𝑄= 𝛼→𝑢+ 𝛽→𝑣, then 𝛼, 𝛽2, are the roots of the equation (1) 𝑥2 + 𝑥- 2 = 0 (2) 𝑥2 - 𝑥- 2 = 0 (3) 3𝑥2 - 2𝑥- 1 = 0 (4) 3𝑥2 + 2𝑥- 1 = 0

Q75.Let 𝐼𝑥= ∫𝑥2𝑥 ( 𝑥 tan𝑥+ 1 2 𝑑𝑥 If 𝐼0 = 0, then 𝐼𝜋4 is equal to ) (1) ( 𝜋+ 4 ) 2 𝜋2 (2) ( 𝜋+ 4 ) 2 𝜋2 loge 16 + 4 ( 𝜋+ 4 ) loge 16 - 4 ( 𝜋+ 4 ) (3) ( 𝜋+ 4 ) 2 𝜋2 (4) ( 𝜋+ 4 ) 2 𝜋2 loge 32 - 4 ( 𝜋+ 4 ) loge 32 + 4 ( 𝜋+ 4 )

Q76.The area enclosed by the closed curve 𝐶 given by the differential equation 𝑑𝑦 𝑥+ 𝑎 = 0, 𝑦1 = 0 is 4𝜋. Let 𝑃 𝑑𝑥+ 𝑦- 2 and 𝑄 be the points of intersection of the curve 𝐶 and the 𝑦-axis. If normals at 𝑃 and 𝑄 on the curve 𝐶 intersect 𝑥-axis at points 𝑅 and 𝑆 respectively, then the length of the line segment 𝑅𝑆 is (1) 2√3 (2) 2√3 3 (3) 2 (4) 4√3 3 JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

Q76.Let →𝑢= ^𝑖- ^𝑗- 2 ^𝑘, →𝑣= 2 ^𝑖+ ^𝑗- ^𝑘, →𝑣· →𝑤= 2 and →𝑣× →𝑤= →𝑢+ 𝜆 →𝑣, then →𝑢· →𝑤 is equal to 3 (1) 1 (2) 2 2 (3) 2 (4) - 3

Q76.Let P be a square matrix such that P 2 = I −P . For α, β, γ, δ ∈N, if P α + P β = γl −29P and P α −P β = δl −13P , then α + β + γ −δ is equal to (1) 18 (2) 40 (3) 22 (4) 24

Q76.Let the solution curve 𝑦= 𝑦( 𝑥) of the differential equation 𝑑𝑦 3𝑥5tan-1𝑥33 𝑦= 2𝑥 exp 𝑥3 - tan-1𝑥3 pass through 𝑑𝑥- 1 + 𝑥6 2 √( 1 + 𝑥) 6 the origin. Then 𝑦( 1 ) is equal to: (1) exp4 - 𝜋 (2) exp𝜋- 4 4√2 4√2 (3) exp1 - 𝜋 (4) exp4 + 𝜋 4√2 4√2 → →

Q76.Let f : R →R be a function defined by f(x) = log√m {√2(sin −2}, for some the range of f is [0, 2]. Then the value of m is _____ . (1) 5 (2) 3 (3) 2 (4) 4

Q77. x + 1 x x If x x + λ x = 89 (103x + 81), then λ, λ3 are the roots of the equation x x x + λ2 (1) 4x2 + 24x −27 = 0 (2) 4x2 −24x −27 = 0 (3) 4x2 + 24x + 27 = 0 (4) 4x2 −24x + 27 = 0

Q77.The number of functions f : {1, 2, 3, 4} →{a ∈Z : |a| ≤8} satisfying f(n) + n1 f(n + 1) = 1, ∀ n ∈{1, 2, 3} is (1) 3 (2) 4 (3) 1 (4) 2 λ (1 + | cos x|)Q78. , 0 < x < π2 |cos x| ⎧ μ, x = π2 is continuous at x = π2 , then If the function f(x) = ⎨ cot 6x cot 4x ⎩ e , π2 < x < π 9λ + 6 logc μ + μ6 −e6λ is equal to (1) 11 (2) 8 (3) 2e4 + 8 (4) 10

Q77.Let D be the domain of the function f(x) = sin−1(log3x( 6+2−5xlog3 x )). If the range of the function defined by g(x) = x −[x], ( [x] is the greatest integer function), is (α, β), then α2 + β5 is equal to (1) 135 (2) 45 (3) 46 (4) 136

Q78.Let (a, b) ⊂(0, 2π) be the largest interval for which sin−1(sin θ) −cos−1(sin θ) > 0, θ ∈(0, 2π), holds . If αx2 + βx + sin−1(x2 −6x + 10) + cos−1(x2 −6x + 10) = 0 and α −β = b −a, then α is equal to; JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper (1) π (2) π 8 48 (3) π (4) π 16 12

Q78.The line, that is coplanar to the line 𝑥+ 3 = 𝑦- 1 = 𝑧- 5 , is -3 1 5 (1) 𝑥+ 1 = 𝑦- 2 = 𝑧- 5 (2) 𝑥+ 1 = 𝑦- 2 = 𝑧- 5 -1 2 4 -1 2 5 (3) 𝑥- 1 = 𝑦- 2 = 𝑧- 5 (4) 𝑥+ 1 = 𝑦- 2 = 𝑧- 5 -1 2 5 1 2 5

Q78.One vertex of a rectangular parallelopiped is at the origin 𝑂 and the lengths of its edges along 𝑥, 𝑦 and 𝑧 axes are 3, 4 and 5 units respectively. Let 𝑃 be the vertex ( 3, 4, 5 ) . Then the shortest distance between the diagonal 𝑂𝑃 and an edge parallel to 𝑧 axis, not passing through 𝑂 or 𝑃 is 12 (1) (2) 12√5 √5 12 12 (3) (4) 5√5 5

Q78.Let the foot of perpendicular of the point P(3, –2, –9) on the plane passing through the points (–1, –2, –3), (9, 3, 4), (9, –2, 1) be Q(α, β, γ). Then the distance Q from the origin is (1) √42 (2) √38 (3) √35 (4) √29

Q79.The set of all a ∈R for which the equation x|x −1| + |x + 2| + a = 0 has exactly one real root, is (1) (−7, ∞) (2) (−∞, ∞) (3) (−6, −3) (4) (−∞, −3) dx = Q80. ∫∞0 e3x+6e2x+11ex+66 (1) loge( 3227 ) (2) loge( 51281 ) (3) loge( 25681 ) (4) loge( 30227 )