Practice Questions

Q16.Suppose A and B are the coefficients of 30th and 12th terms respectively in the binomial expansion of (1 + x)2n−1 . If 2 A = 5 B , then n is equal to : (1) 22 (2) 20 (3) 21 (4) 19

Q16.A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ2 denote the mean and variance of X , then the value of 64 (μ + σ2) is : (1) 51 (2) 64 (3) 32 (4) 48

Q16.If x = f(y) is the solution of the differential equation (1 + y2) + (x −2etan−1 y) dydx is equal to : f(0) = 1, then f ( √31 ) (1) eπ/12 (2) eπ/4 (3) eπ/3 (4) eπ/6

Q16.Let for some function y = f(x), ∫x0 tf(t)dt = x2f(x), (1) 1 (2) 3 (3) 6 (4) 2 π dx = π (απ2 + β), α, β ∈Z , then (α + β)2 equals

Q16.The area of the region bounded by the curves x (1 + y2) = 1 and y2 = 2x is: (1) 2 ( π2 −13 ) (2) π2 −13 (3) π 4 −13 (4) 12 ( π2 −13 )

Q16.Let f(x) = 2x+2+16 . Then the value of 8 (f ( 151 ) + f ( 152 ) + … + f ( 5915 )) is equal to 22x+1+2x+4+32 (1) 92 (2) 118 (3) 102 (4) 108 + + (1 + x2)dy = 0, y(0) = 0.

Q16.The value of ∫e4e2 x ( e((loge x)2+1)−1 +e((6−loge x)2+1)−1 )dx (1) 2 (2) loge 2 (3) 1 (4) e2 2025 (23 Jan Shift 1) JEE Main Previous Year Paper

Q16.If I = ∫ 0π 3 dx, then ∫210 sin4x sinx+cos4x cos xx 2 2 x sin x+cos (1) π2 (2) π2 12 4 (3) π2 (4) π2 16 8 ∣∣ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2025 (23 Jan Shift 2) JEE Main Previous Year Paper

Q16.Let a straight line L pass through the point P(2, −1, 3) and be perpendicular to the lines x−12 = y+11 = z−3−2 and x−3 1 = y−23 = z+24 . If the line L intersects the yz -plane at the point Q , then the distance between the points P and Q is : (1) √10 (2) 2√3 (3) 2 (4) 3

Q17.The least value of n for which the number of integral terms in the Binomial expansion of (3√7 + 12√11)n is 183, is : (1) 2184 (2) 2196 (3) 2148 (4) 2172 ∣ ∣ 2025 (29 Jan Shift 1) JEE Main Previous Year Paper

Q17.The number of non-empty equivalence relations on the set {1, 2, 3} is : (1) 6 (2) 5 (3) 7 (4) 4

Q17.A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is : (1) 7/10 (2) 4/5 (3) 23/30 (4) 3/5

Q17.Let y = y(x) be the solution of the differential equation (xy −5x2√1 x2)dx Then y(√3) is equal to (1) √152 (2) 5√32 (3) 2√2 (4) √143 is:

Q17.Let 2¯z+i ¯z−i = 13 , z ∈C , be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points (0, 0), C and (α, 0) is 11 square units, then α2 equals: (1) 50 (2) 100 (3) 81 (4) 121 25 25

Q17.Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 29/45, then n is equal to : (1) 6 (2) 3 (3) 5 (4) 4 ∣∣ 2025 (29 Jan Shift 2) JEE Main Previous Year Paper

Q17.If ∫ −π2 2 96x2(1+ex)cos2 x (1) 64 (2) 196 (3) 144 (4) 100

Q18. a + sinx x 1 b For some a, b, let f(x) = a 1 + sinx x b , x ≠0, limx→0 f(x) = λ + μa + νb. Then a 1 b + sinx x (λ + μ + ν)2 is equal to : (1) 16 (2) 25 (3) 9 (4) 36

Q18. limx→0 cosec x (√2 cos2 x + 3 cos x −√cos2 x + sin x + 4) 2025 (24 Jan Shift 1) JEE Main Previous Year Paper (1) 0 (2) 1 √15 (3) 1 (4) − 1 2√5 2√5

Q18.Let ⟨an⟩ be a sequence such that a0 = 0, a1 = 12 and 2an+2 = 5an+1 −3an, n = 0, 1, 2, 3, … . Then ∑100k=1 ak is equal to 2025 (28 Jan Shift 1) JEE Main Previous Year Paper (1) 3a99 −100 (2) 3a100 −100 (3) 3a99 + 100 (4) 3a100 + 100

Q18.The sum of all values of θ ∈[0, 2π] satisfying 2 sin2 θ = cos 2θ and 2 cos2 θ = 3 sin θ is 2025 (22 Jan Shift 2) JEE Main Previous Year Paper (1) 4π (2) 5π6 (3) π (4) π 2

Q18.Let the shortest distance from (a, 0), a > 0, to the parabola y2 = 4x be 4 . Then the equation of the circle passing through the point (a, 0) and the focus of the parabola, and having its centre on the axis of the parabola is : (1) x2 + y2 −10x + 9 = 0 (2) x2 + y2 −6x + 5 = 0 (3) x2 + y2 −4x + 3 = 0 (4) x2 + y2 −8x + 7 = 0

Q19.Let A = {1, 2, 3, … , 10} and B = { mn : m, n ∈A, m < n and gcd(m, n) = 1}. Then n(B) is equal to : (1) 36 (2) 31 (3) 37 (4) 29 2025 (22 Jan Shift 1) JEE Main Previous Year Paper

Q19.If α + iβ and γ + iδ are the roots of x2 −(3 −2i)x −(2i −2) = 0, i = √−1, then αγ + βδ is equal to : (1) −2 (2) 6 (3) −6 (4) 2

Q19.Let S = N ∪{0}. Define a relation R from S to R by : R = {(x, y) : loge y = x loge ( 25 ), x ∈ S, y ∈R} Then, the sum of all the elements in the range of R is equal to : (1) 10 (2) 3 9 2 (3) 5 (4) 5 2 3

Q19.The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0 , 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8 , is (1) 4608 (2) 5720 (3) 5719 (4) 4607