Practice Questions

Q64.If the term independent of x in the expansion of (√ax2 + 2x31 )10 is 105 , then a2 is equal to : (1) 2 (2) 4 (3) 6 (4) 9 JEE Main 2024 (08 Apr Shift 2) JEE Main Previous Year Paper cos 36∘+5 sin 18∘

Q64.Let 2nd, 8th and 44th, terms of a non-constant 𝐴. 𝑃. be respectively the 1st, 2nd and 3rd terms of 𝐺. 𝑃. If the first term of A.P. is 1 then the sum of first 20 terms is equal to- (1) 980 (2) 960 (3) 990 (4) 970

Q64.Let 𝛼, 𝛽, 𝛾, 𝛿∈𝑍 and let 𝐴𝛼, 𝛽, 𝐵1, 0, 𝐶𝛾, 𝛿 and 𝐷1, 2 be the vertices of a parallelogram 𝐴𝐵𝐶𝐷. If 𝐴𝐵= √10 and the points 𝐴 and 𝐶 lie on the line 3𝑦= 2𝑥+ 1, then 2𝛼+ 𝛽+ 𝛾+ 𝛿 is equal to (1) 10 (2) 5 (3) 12 (4) 8

Q64.If α, −π2 < α < π2 is the solution of 4 cos θ + 5 sin θ = 1, then the value of tan α is (1) 10−√10 (2) 10−√10 6 12 (3) √10−10 (4) √10−10 12 6

Q64.If each term of a geometric progression a1, a2, a3, … with a1 = 18 and a2 ≠a1 , is the arithmetic mean of the next two terms and Sn = a1 + a2 + … + an , then S20 −S18 is equal to (1) 215 (2) −218 (3) 218 (4) −215

Q64.Let 3, 𝑎, 𝑏, 𝑐 be in 𝐴. 𝑃. and 3, 𝑎−1, 𝑏+ 1, 𝑐+ 9 be in 𝐺. 𝑃. Then, the arithmetic mean of 𝑎, 𝑏 and 𝑐 is: (1) -4 (2) -1 (3) 13 (4) 11 1 √𝑥

Q64.The sum of the coefficient of x2/3 and x−2/5 in the binomial expansion of (x2/3 + 12 x−2/5) 9 (1) 21/4 (2) 63/16 (3) 19/4 (4) 69/16

Q64.If the constant term in the expansion of 12 + , x ≠0, is α × 28 × 5√3, then 25α is equal to : ( 5√3x 2x ) 3√5 (1) 724 (2) 742 (3) 639 (4) 693

Q64.For 𝛼, 𝛽∈0, let 3sin ( 𝛼+ 𝛽) = 2sin ( 𝛼- 𝛽) and a real number 𝑘 be such that tan𝛼= tan𝛽. Then the 2 value of 𝑘 is equal to (1) -5 (2) 5 (3) 2 (4) -2 3 3

Q64.Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then : (1) P2 = 6√3Q (2) P2 = 36√3Q (3) P = 36√3Q2 (4) P2 = 72√3Q

Q64. n−1Cr = (k2 −8)nCr+1 if and only if : (1) 2√2 < k ≤3 (2) 2√3 < k ≤3√2 (3) 2√3 < k < 3√3 (4) 2√2 < k < 2√3 JEE Main 2024 (27 Jan Shift 1) JEE Main Previous Year Paper

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.If the coefficients of x4, x5 and x6 in the expansion of (1 + x)n are in the arithmetic progression, then the maximum value of n is: (1) 7 (2) 21 (3) 28 (4) 14

Q64.If 2tan2𝜃- 5sec𝜃= 1 has exactly 7 solutions in the interval 0, n𝜋 , for the least value of n ∈N then n k is 2 ∑k = 1 2k equal to : - 15 (1) 2152141 - 14 (2) 2142151 15 1 (3) 1 - (4) - 15 213 213214

Q64.Let 𝑚 and 𝑛 be the coefficients of seventh and thirteenth terms respectively in the expansion of 3 + 2 3𝑥 2𝑥 3 1 . Then 𝑛 3 is: 𝑚 (1) 4 (2) 1 9 9 1 9 (3) (4) 4 4

Q64.A line passing through the point A(9, 0) makes an angle of 30° with the positive direction of x-axis. If this line is rotated about A through an angle of 15° in the clockwise direction, then its equation in the new position is (1) y + x = 9 (2) x + y = 9 √3−2 √3−2 (3) x + y = 9 (4) y + x = 9 √3+2 √3+2

Q64.Let the first three terms 2, p and q , with q ≠2, of a G.P. be respectively the 7th , 8th and 13th terms of an A.P. If the 5th term of the G.P. is the nth term of the A.P., then n is equal to: (1) 163 (2) 151 (3) 177 (4) 169

Q64.Let |cos θ cos(60 −θ) cos(60 + θ)| ≤18 , θϵ[0, 2π]. Then, the sum of all θϵ[0, 2π], where cos 3θ attains its maximum value, is : (1) 15π (2) 18π (3) 6π (4) 9π

Q65.A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to: (1) 150 (2) 180 (3) 160 (4) 125

Q65.Let A(−1, 1) and B(2, 3) be two points and P be a variable point above the line AB such that the area of △PAB is 10 . If the locus of P is ax + by = 15, then 5a + 2 b is : (1) 6 (2) −65 (3) 4 (4) −125

Q65.If A(1, −1, 2), B(5, 7, −6), C(3, 4, −10) and D(−1, −4, −2) are the vertices of a quadrilateral ABCD , then its area is : (1) 48√7 (2) 12√29 (3) 24√7 (4) 24√29

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

Q65.If tan𝐴= 1 tan𝐵= and tan𝐶= 𝑥−3 + 𝑥−2 + 𝑥−1 2, 0 < 𝐴, 𝐵, 𝐶< 𝜋 then 𝐴+ 𝐵 is equal √𝑥𝑥2 + 𝑥+ 1, √𝑥2 + 𝑥+ 1 2, to: (1) 𝐶 (2) 𝜋−𝐶 (3) 2𝜋−𝐶 (4) 𝜋 −𝐶 2 JEE Main 2024 (01 Feb Shift 1) JEE Main Previous Year Paper

Q65.A ray of light coming from the point P(1, 2) gets reflected from the point Q on the x-axis and then passes through the point R(4, 3). If the point S(h, k) is such that PQRS is a parallelogram, then hk2 is equal to : (1) 70 (2) 80 (3) 60 (4) 90