Practice Questions

Q23.Let →c be the projection vector of →b = λ^i + 4^k, λ > 0, on the vector →a = ^i + 2^j + 2^k. If |→a + →c| = 7, then the area of the parallelogram formed by the vectors →b and →c is ________

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.The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is -

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 _______ .

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

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

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

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 _________

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 →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

Q32.Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be (1) 2.0 cm (2) 1.5 cm (3) 1.0 cm (4) 0.5 cm

Q38.A capacitor, C1 = 6μ F is charged to a potential difference of V0 = 5 V using a 5 V battery. The battery is removed and another capacitor, C2 = 12μ F is inserted in place of the battery. When the switch 'S' is closed, the charge flows between the capacitors for some time until equilibrium condition is reached. What are the 2025 (29 Jan Shift 2) JEE Main Previous Year Paper charges (q1 and q2) on the capacitors C1 and C2 when equilibrium condition is reached. (1) q1 = 10μC, q2 = 20μC (2) q1 = 30μC, q2 = 15μC (3) q1 = 20μC, q2 = 10μC (4) q1 = 15μC, q2 = 30μC