Practice Questions

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

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

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.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 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¯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

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.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

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.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 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.Let y = y(x) be the solution of the differential equation cos x(loge(cos x))2dy + (sin x −3y sin x loge(cos x))dx = 0, x ∈(0, π2 ). If y ( π4 ) = loge−1 2 , then y ( π6 ) is equal to : (1) 1 (2) 2 loge(3)−loge(4) loge(3)−loge(4) (3) 1 (4) − 1 loge(4)−loge(3) loge(4)

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. 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.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

Q19.If in the expansion of (1 + x)p(1 −x)q , the coefficients of x and x2 are 1 and -2 , respectively, then p2 + q2 is equal to : (1) 18 (2) 13 (3) 8 (4) 20 a

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

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.If the equation of the parabola with vertex V ( 32 , 3) and the directrix x + 2y = 0 is αx2 + βy2 −γxy −30x −60y + 225 = 0, then α + β + γ is equal to : ∣ ∣ 2025 (24 Jan Shift 2) JEE Main Previous Year Paper (1) 7 (2) 9 (3) 8 (4) 6 (1+β2) (1+γ2) (1+α2) is + + +

Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121

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.Let the curve z(1 + i) + ¯z(1 −i) = 4, z ∈C, divide the region |z −3| ≤1 into two parts of areas α and β . Then |α −β| equals : (1) 1 + π2 (2) 1 + π3 (3) 1 + π6 (4) 1 + π4