Practice Questions

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.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

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.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.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.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 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

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

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

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.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.Let (2, 3) be the largest open interval in which the function f(x) = 2 loge(x −2) −x2 + ax + 1 is strictly increasing and (b, c) be the largest open interval, in which the function g(x) = (x −1)3(x + 2 −a)2 is strictly decreasing. Then 100(a + b −c) is equal to : (1) 420 (2) 360 (3) 160 (4) 280

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.The square of the distance of the point ( 157 , 327 , 7) from the line x+13 = y+35 = z+57 in the direction of the vector ^i + 4^j + 7^k is : (1) 54 (2) 44 (3) 41 (4) 66 y2

Q17.Let αθ and βθ be the distinct roots of 2x2 + (cos θ)x −1 = 0, θ ∈(0, 2π). If m and M are the minimum and the maximum values of α4θ + β4θ , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27

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

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.A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (α, β), then 3β −2α is equal to : (1) 10 (2) 15 (3) 12 (4) 14

Q18.Let α, β(α ≠β) be the values of m , for which the equations x + y + z = 1; x + 2y + 4z = m and x + 4y + 10z = m2 have infinitely many solutions. Then the value of ∑10n=1 (nα + nβ) is equal to : (1) 3080 (2) 560 (3) 3410 (4) 440

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

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.The value of (sin 70∘) (cot 10∘cot 70∘−1) is (1) 2/3 (2) 1 (3) 0 (4) 3/2 dx 1 1 1 , then 3( b + c) is equal to

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.If the midpoint of a chord of the ellipse x2 α 9 + 4 = 1 is (√2, 4/3), and the length of the chord is 2√α3 , then is : (1) 20 (2) 22 (3) 18 (4) 26 2025 (28 Jan Shift 2) JEE Main Previous Year Paper