Practice Questions

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

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

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 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 ________.

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

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

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 ________.

Q26.An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component of velocity 106 m/s. If the magnitude of the electric field between the plates is 9.1 V/cm , then the vertical component of velocity of electron is (mass of electron = 9.1 × 10−31 kg and charge of electron = 1.6 × 10−19C ) (1) 0 (2) 1 × 106 m/s (3) 16 × 106 m/s (4) 16 × 104 m/s

Q26.An electric dipole of mass m, charge q, and length l is placed in a uniform electric field E = E0^i. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be: (1) 1 ml (2) ml 2π √ 2qE0 2π√ qE0 (3) 1 ml (4) ml 2π √2qE0 2π√ 2qE0

Q26.Which of the following figure represents the relation between Celsius and Fahrenheit temperatures ? (1) (2) (3) (4) ∣∣ 2025 (24 Jan Shift 2) JEE Main Previous Year Paper

Q26.A galvanometer having a coil of resistance 30Ω need 20 mA of current for full-scale deflection. If a maximum current of 3 A is to be measured using this galvanometer, the resistance of the shunt to be added to the galvanometer should be 30 X Ω, where X is Options (1) 596 (2) 149 (3) 298 (4) 447

Q26.During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is 2000Å and it becomes 6000Å when the electron jumps from state B to state C. Then the wavelength of the radiation emitted during the transition of electrons from state A to state B is (1) 4000Å (2) 2000Å (3) 3000Å (4) 6000Å