Practice Questions

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 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 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.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 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 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 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

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

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.The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3) which are reflexive and transitive but not symmetric, is _________. . If B = , then the sum of all the elements of the matrix ∑50n=1 Bn is [−1 −1 ] A[ 1 1 ]

Q75.If S = {x ∈R sin−1( √x2+2x+2x+1 ) −sin−1( √x2+1x ) ∑x∈S(sin((x2 + x + 5) π2 ) −cos((x2 + x + 5)π)) is equal to _________.

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.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 →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.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 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 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.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

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

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

Q78.Consider a function f : N →R, satisfying f(1) + 2f(2) + 3f(3) + … + xf(x) = x(x + 1)f(x) ; x ≥2 with f(1) = 1 . Then f(2022)1 + f(2028)1 is equal to JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 8200 (2) 8000 (3) 8400 (4) 8100

Q78.Let A = {1, 2, 3, 5, 8, 9} . Then the number of possible functions f : A →A such that f(m ⋅n) = f(m) ⋅f(n) for every m, n ∈A with m ⋅n ∈A is equal to ax + bx2, a ≠2b have a common extreme point,

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