Practice Questions

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

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 _______ -.

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)

Q61.Let 𝑆 be the set of positive integral values of 𝑎 for which 𝑎𝑥2 + 2𝑎+ 1𝑥+ 9𝑎+ 4 < 0, ∀𝑥∈ℝ. Then, the number 𝑥2 - 8𝑥+ 32 of elements in 𝑆 is: (1) 1 (2) 0 (3) ∞ (4) 3

Q61.Let S1 = {z ∈C : |z| ≤5}, S2 = {z ( z+1−√3i1−√3i ) ≥0} area of the region S1 ∩S2 ∩S3 is : (1) 125π (2) 125π 12 4 (3) 125π (4) 125π 24 6 Q62.60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is : (1) JBBOH (2) OBBJH (3) OBBHJ (4) HBBJO

Q62.Let 𝑎 and 𝑏 be two distinct positive real numbers. Let 11th term of a GP, whose first term is 𝑎 and third term is 𝑏, is equal to 𝑝th term of another GP, whose first term is 𝑎 and fifth term is 𝑏. Then 𝑝 is equal to (1) 20 (2) 25 (3) 21 (4) 24

Q62.For 0 < 𝑐< 𝑏< 𝑎, let ( 𝑎+ 𝑏– 2𝑐) 𝑥2 + ( 𝑏+ 𝑐– 2𝑎) 𝑥+ ( 𝑐+ 𝑎– 2𝑏) = 0 and 𝛼≠1 be one of its root. Then, among the two statements (I) If 𝛼∈-1, 0, then 𝑏 cannot be the geometric mean of 𝑎 and 𝑐. (II) If 𝛼∈0, 1, then 𝑏 may be the geometric mean of 𝑎 and 𝑐. (1) Both (I) and (II) are true (2) Neither (I) nor (II) is true (3) Only (II) is true (4) Only (I) is true 1 2 3

Q62.In an A.P., the sixth term a6 = 2. If the a1a4a5 is the greatest, then the common difference of the A.P., is equal to (1) 3 (2) 8 2 5 (3) 2 (4) 5 3 8

Q62.Let 𝑆= 𝑧∈𝐶: 𝑧−1 = 1 and √2 −1𝑧+ ¯𝑧- 𝑖𝑧- ¯𝑧= 2√2. Let 𝑧1, 𝑧2 ∈𝑆 be such that 𝑧1 = max𝑧∈𝑠𝑧 and 2 𝑧2 = min𝑧∈𝑠𝑧. Then √2𝑧1 −𝑧2 equals: (1) 1 (2) 4 (3) 3 (4) 2

Q62.Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to (1) 18 (2) 16 (3) 12 (4) 15

Q63.The coefficient of x70 in x2(1 + x)98 + x3(1 + x)97 + x4(1 + x)96 + … + x54(1 + x)46 is 99Cp −46Cq . Then a possible value of p + q is : (1) 55 (2) 83 (3) 61 (4) 68

Q64.Let a variable line of slope m > 0 passing through the point (4, −9) intersect the coordinate axes at the points A and B. The minimum value of the sum of the distances of A and B from the origin is JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper (1) 30 (2) 25 (3) 15 (4) 10

Q64.Let two straight lines drawn from the origin O intersect the line 3x + 4y = 12 at the points P and Q such that △OPQ is an isosceles triangle and ∠POQ = 90∘ . If l = OP2 + PQ2 + QO2 , then the greatest integer less than or equal to l is : (1) 42 (2) 46 (3) 44 (4) 48

Q65.The sum of the solutions x ∈R of the equation 3 cos 2x+cos3 2x = x3 −x2 + 6 is cos6 x−sin6 x (1) 0 (2) 1 (3) −1 (4) 3

Q65.If the value of 3 is a√5−b , where a, b, c are natural numbers and gcd(a, c) = 1, then a + b + c is c 5 cos 36∘−3 sin 18∘ equal to : (1) 40 (2) 52 (3) 50 (4) 54

Q65.Two vertices of a triangle ABC are A(3, −1) and B(−2, 3), and its orthocentre is P(1, 1). If the coordinates of the point C are (α, β) and the centre of the of the circle circumscribing the triangle PAB is (h, k), then the value of (α + β) + 2( h + k) equals (1) 5 (2) 81 (3) 15 (4) 51 and the eccentricity

Q65.Let C be a circle with radius √10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord MN is (1) 3 −√2 (2) √2 + 1 (3) √2 −1 (4) 2 −√3

Q66.The vertices of a triangle are A(−1, 3), B(−2, 2) and C(3, −1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is : (1) x + y + (2 −√2) = 0 (2) −x + y −(2 −√2) = 0 (3) x + y −(2 −√2) = 0 (4) x −y −(2 + √2) = 0

Q66.Let 𝐴𝑎, 𝑏, 𝐵3, 4 and −6, −8 respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point 𝑃2𝑎+ 3, 7𝑏+ 5 from the line 2𝑥+ 3𝑦−4 = 0 measured parallel to the line 𝑥−2𝑦−1 = 0 is (1) 15√5 (2) 17√5 7 6 (3) 17√5 (4) √5 7 17

Q66.Let 𝐴( 𝛼, 0 ) and 𝐵( 0, 𝛽) be the points on the line 5𝑥+ 7𝑦= 50. Let the point 𝑃 divide the line segment 𝐴𝐵 𝑥2 𝑦2 internally in the ratio 7: 3. Let 3𝑥- 25 = 0 be a directrix of the ellipse 𝐸: + = 1 and the corresponding 𝑎2 𝑏2 focus be 𝑆. If from 𝑆, the perpendicular on the 𝑥- axis passes through 𝑃, then the length of the latus rectum of 𝐸 is equal to 25 32 (1) (2) 3 9 (3) 25 (4) 32 9 5