Practice Questions

Q24.The sum of all rational terms in the expansion of (1 + 21/2 + 31/2) 6 is equal to

Q24.Let E1 : x29 + y24 = 1 same as that of E1 , and the length of minor axis of Ei is the length of major axis of Ei+1(i ≥1). If Ai is the area of the ellipse Ei , then π5 (∑∞i=1 Ai), is equal to → → →

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 S = ∈Z : Am2 + Am = 3I , where A = 2 −1 . Then n(S) is equal to ______. {m −A−6} [ 1 0 ]

Q24.The focus of the parabola y2 = 4x + 16 is the centre of the circle C of radius 5 . If the values of λ, for which C passes through the point of intersection of the lines 3x −y = 0 and x + λy = 4, are λ1 and λ2, λ1 < λ2 , then 12λ1 + 29λ2 is equal to

Q24.Number of functions f : {1, 2, … , 100} →{0, 1}, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________. y2 + = 1 be two hyperbolas having length of latus rectums 15√2 and = 1 and H2 : −x2

Q24.The interior angles of a polygon with n sides, are in an A.P. with common difference 6∘ . If the largest interior angle of the polygon is 219∘ , then n is equal to . Then limx→0 (x−f(x))ex−ef(x) is equal to

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

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 →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 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 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 ________.

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.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.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.The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is______.

Q25.Let L1 : x−13 = y−1−1 = z+10 and L2 : x−22 = 0y = z+4α , α ∈R, be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1, 1, −1) on L2 , then the value of 26α( PB)2 is _________

Q48.Two particles are located at equal distance from origin. The position vectors of those are represented by ¯A = 2^i + 3n^j + 2^k and ¯B = 2^i −2^j + 4p^k, respectively. If both the vectors are at right angle to each other, the value of n−1 is _____ .

Q50.A force f = x2y^i + y2^j acts on a particle in a plane x + y = 10. The work done by this force during a displacement from (0, 0) to (4 m, 2 m) is Joule (round off to the nearest integer)

Q63. Given below are two statements : In the light of the above statements, choose the correct answer from the options given below : (1) Both Statement I and Statement II are true (2) Statement I is false but Statement II is true (3) Statement I is true but Statement II is false (4) Both Statement I and Statement II are false

Q64. Choose the correct answer from the options given below : (1) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) (2) (A)-(I), (B)-(III), (C)-(II), (D)-(IV) (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) (4) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)

Q68. Choose the correct answer from the options given below : (1) (A)-(IV), (B)-(III), (C)-(II), (D)-(I) (2) (A)-(III), (B)-(IV), (C)-(II), (D)-(I) (3) (A)-(III), (B)-(IV), (C)-(I), (D)-(II) (4) (A)-(III), (B)-(I), (C)-(IV), (D)-(II)