Practice Questions

Q1. Let f(x) = ∫t0 (1) 253 (2) 154 (3) 125 (4) 157 →

Q2. Let f : R →R be a function defined by f(x) = (2 + 3a)x2 + ( a+2a−1 )x + b, a ≠1. If f(x + y) = f(x) + f(y) + 1 −27 xy , then the value of 28 ∑5i=1 |f(i)| is (1) 545 (2) 715 (3) 735 (4) 675

Q21.Let P be the image of the point Q(7, −2, 5) in the line L : x−12 = y+13 = 4z and R(5, p, q) be a point on Then the square of the area of △PQR is ________. x + 1 + C, where C is the

Q21.Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B , and a denotes the area of △SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is -

Q21.Let A be a square matrix of order 3 such that det(A) = −2 and det(3 adj(−6 adj(3A))) = 2m+n ⋅3mn, m > n . Then 4 m + 2n is equal to _______ , then m −n is equal to _______

Q21.Let the circle C touch the line x −y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length 4 along the line −3x + 2y = 1. Let H be the hyperbola x2 −y2 = 1, whose one of the foci is the √13 α2 β2 centre of C and the length of the transverse axis is the diameter of C . Then 2α2 + 3β2 is equal to ______

Q22.The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

Q22.If ∑5r=0 11C22r2r+2 = mn , gcd(m, n) = 1

Q22.Let M denote the set of all real matrices of order 3 × 3 and let S = {−3, −2, −1, 1, 2}. Let S1 = {A = [aij] ∈M : A = AT and aij ∈ S, ∀i, j}, S2 = {A = [aij] ∈M : A = −AT and aij ∈ S, ∀i, j}, S3 = {A = [aij] ∈M : a11 + a22 + a33 = 0 and aij ∈ S, ∀i, j}. If n ( S1 ∪2 US3) = 125α, then α equals _______

Q22.Let f : (0, ∞) →R be a twice differentiable function. If for some a ≠0, ∫10 f(λx)dλ = af(x), f(1) = 1 and f(16) = 18 , then 16 −f ′ ( 161 ) is equal to _______.

Q23.If y = y(x) is the solution of the differential equation, ( 2 ), −2 ≤x ≤2, y(2) = 4 , then y2(0) is equal to √4 −x2 dxdy = ((sin−1 x 2 x π2−8 ( 2 )) −y) sin−1

Q23.The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

Q23.Let A(6, 8), B(10 cos α, −10 sin α) and C(−10 sin α, 10 cos α), be the vertices of a triangle. If L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then (5a −3h + 6k + 100 sin 2α) is equal to ______ -. , −1 < x < 1 such that

Q24.Let y2 = 12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ) = 1474 . Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2 + 64y2 −αx −64√3y = β , then β −α is equal to ________.

Q24.Let the function, f(x) = {−3ax2a2 + bx,−2, xx <⩾11 be differentiable for all x ∈R, where a > 1, b ∈R. If the area of the region enclosed by y = f(x) and the line y = −20 is α + β√3, α, β ∈Z , then the value of α + β is ________

Q24.Let f be a differentiable function such that 2(x + 2)2f(x) −3(x + 2)2 = 10 ∫x0 (t + 2)f(t)dt, f(2) is equal to ______.

Q24.Let y = f(x) be the solution of the differential equation dydx + x2−1xy = √1−x2x6+4x f(0) = 0. If 6 ∫1/2−1/2 f(x)dx = 2π −α then α2 is equal to _______ .

Q25.Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then (QR)2 is equal to _______ -.

Q25.Let α, β be the roots of the equation x2 −ax −b = 0 with Im(α) < Im(β). Let Pn = αn −βn . If P3 = −5√7i, P4 = −3√7i, P5 = 11√7i and P6 = 45√7i , then α4 + β4 is equal to . ∣∣ 2025 (23 Jan Shift 2) JEE Main Previous Year Paper

Q25.Let f(x) = limn→∞∑nr=0 ( tan(x/2r+1)+tan3(x/2r+1)1−tan2(x/2r+1) )

Q25.Let [t] be the greatest integer less than or equal to t. Then the least value of p ∈N for which + … + ≥1 is equal to ________. ] + limx→0+ (x ([ x1 ] + [ x2 ] + … + [ xp ]) −x2 ([ x21 [ x222 ] [ x292 ])) →

Q25.If the area of the larger portion bounded between the curves x2 + y2 = 25 and y = |x −1| is 1 4 (bπ + c), b, c ∈N , then b + c is equal to

Q25.Let →a = ^i +^j + ^k, b = 2^i + 2^j + ^k and d = →a × b. If→cis a vector such that →a ⋅→c= |→c|, |→c−2→a|2 = 8 and the → → → π angle between d and→cis , then |10 −3 b ⋅→c| + |d ×→c|2 is equal to 4

Q25.Let H1 : x2a2 −y2b2 A2 B2 and e2 respectively. If the product of the lengths of 12√5 respectively. Let their ecentricities be e1 = √52 their transverse axes is 100√10, then 25e22 is equal to ________.

Q25.Let integers a, b ∈[−3, 3] be such that a + b ≠0. Then the number of all possible ordered pairs (a, b), for z + 1 ω ω2 which z−a = 1 and ω z + ω2 1 = 1, z ∈C, where ω and ω2 are the roots of x2 + x + 1 = 0, is z+b ω2 1 z + ω equal to ________.