Practice Questions

Q62.The sum of the series ∑∞n=1 n2+6n+10(2n+1)! is equal to (1) 41 8 e + 198 e−1 + 10 (2) 418 e + 198 e−1 −10 (3) −418 e + 198 e−1 −10 (4) 418 e −198 e−1 −10 + + …

Q62.The sum of the infinite series 1 + 32 + 327 + 1233 + 1734 + 2235 + … … is equal to: (1) 94 (2) 154 (3) 114 (4) 134

Q62.Let a complex number z, |z| ≠1, satisfy log 1 |z|+11 ≤2 . Then, the largest value of |z| is equal to √2 ( (|z|−1)2 ) _________. (1) 8 (2) 7 (3) 6 (4) 5

Q62.The number of solutions of the equation 32tan2𝑥+ 32sec2𝑥= 81, 0 ≤𝑥≤ 𝜋 is : 4 (1) 0 (2) 2 (3) 1 (4) 3 JEE Main 2021 (31 Aug Shift 2) JEE Main Previous Year Paper 𝑧- 𝑖

Q62.A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is: (1) 1050 (2) 1625 (3) 575 (4) 560

Q62.Consider a rectangle ABCD having 5, 6, 7, 9 points in the interior of the line segments AB, BC, CD, DA respectively. Let α be the number of triangles having these points from different sides as vertices and β be the number of quadrilaterals having these points from different sides as vertices. Then (β −α) is equal to (1) 795 (2) 1173 (3) 1890 (4) 717

Q62.Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 −S6 is equal to: (1) 1862 (2) 1842 (3) 1852 (4) 1872

Q62.Let C be the set of all complex numbers. Let S1 = {z ∈C |z–3–2i|2 = 8}, S2 = z ∈C| Re(z) ≥5 and ¯S3 = {z ∈C| |z–z| ≥8}. Then the number of elements in S1 ∩S2 ∩S3 is equal to (1) 1 (2) 0 (3) 2 (4) Infinite b ≠0, are equal, then the value of b is equal

Q62.Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 −S1) is 1000 , then the sum of the first 6n terms of the arithmetic progression is equal to: (1) 1000 (2) 7000 (3) 5000 (4) 3000

Q62.Let 𝑃1, 𝑃2 … , 𝑃15 be 15 points on a circle. The number of distinct triangles formed by points 𝑃𝑖, 𝑃𝑗, 𝑃𝑘 such that 𝑖+ 𝑗+ 𝑘≠15, is : (1) 455 (2) 419 (3) 12 (4) 443

Q62.If the equation a z 2 + αz + αz + d = 0 represents a circle where a, d are real constants then which of the following condition is correct? (1) |α|2 −ad ≠0 (2) |α|2 −ad > 0 and a ∈R −{0} (3) |α|2 −ad ≥0 and a ∈R (4) α = 0, a, d ∈R+

Q62.If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to: (1) 364 (2) 240 (3) 333 (4) 360

Q62.If n ⩾2 is a positive integer, then the sum of the series n+1C2 + 2(2C2 + 3C2 + 4C2 + … + nC2) is (1) n(n−1)(2n+1) (2) n(n+1)(2n+1) 6 6 (3) n(n+1)2(n+2) (4) n(2n+1)(3n+1) 12 6

Q63.The total number of positive integral solutions (x, y, z) such that xyz = 24 is : (1) 45 (2) 30 (3) 36 (4) 24

Q63.Let A(−1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0 , intersects lines AC and BC at point P and Q respectively. Let A1 and A2 be the areas of ΔABC and ΔPQC respectively, such that A1 = 3A2 , then the value of m is equal to : (1) 4 (2) 1 15 (3) 2 (4) 3 JEE Main 2021 (16 Mar Shift 2) JEE Main Previous Year Paper

Q63.If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x −1, then the co- ordinates of P are: (1) (−2, 8) (2) (1, 5) (3) (2, 8) (4) (3, 13)

Q63.If 𝑒cos2𝑥+ cos4𝑥+ cos6𝑥+ . . . . ∞log𝑒2 satisfies the equation 𝑡2 - 9𝑡+ 8 = 0, then the value of 2sin𝑥 where sin𝑥+ √3cos𝑥, 0 < 𝑥< 𝜋2, is equal to (1) 3 (2) 1 2 2 (3) √3 (4) 2√3

Q63.If z and ω are two complex numbers such that |zω| = 1 and arg(z) −arg(ω) = 3π2 , then arg( 1+3¯zω1−2¯zω ) is: (Here arg(z) denotes the principal argument of complex number z) (1) π 4 (2) −3π4 (3) −π4 (4) 3π4

Q63.If 𝑧 is a complex number such that is purely imaginary, then the minimum value of |𝑧- ( 3 + 3 𝑖) | is : 𝑧- 1 (1) 3√2 (2) 2√2 (3) 2√2 - 1 (4) 6√2

Q63.The sum of all values of 𝑥 in [0, 2𝜋], for which sin𝑥+ sin2𝑥+ sin3𝑥+ sin4𝑥= 0, is equal to : (1) 8𝜋 (2) 11𝜋 (3) 12𝜋 (4) 9𝜋

Q63.If n is the number of irrational terms in the expansion of (31/4 + 51/8) 60 , then (n −1) is divisible by : (1) 26 (2) 30 (3) 8 (4) 7

Q63.The minimum value of f(x) = aax + a1−ax , where a, x ∈R and a > 0, is equal to: (1) a + 1 (2) 2a (3) a + a1 (4) 2√a

Q63.For the natural numbers m, n, if (1 −y)m(1 + y)n = 1 + a1y + a2y2 + … . +am+nym+n and a1 = a2 = 10, then the value of m + n, is equal to: (1) 88 (2) 64 (3) 100 (4) 80

Q63.If the coefficients of x7 in (x2 + bx1 )11 and x−7 in (x − bx21 )11, to: (1) 2 (2) −1 (3) 1 (4) −2

Q63.If tan( π9 ), x, tan( 7π18 ) are in arithmetic progression and tan( π9 ), y, tan( 5π18 ) are also in arithmetic progression, then |x −2y| is equal to : (1) 4 (2) 3 (3) 0 (4) 1 Q64. 10 + 3(−18 ) log3(5x−1+1)} in A possible value of x, for which the ninth term in the expansion of {3log3 √25x−1+7 the increasing powers of 3(−18 ) log3(5x−1+1) is equal to 180, is : (1) 0 (2) −1 (3) 2 (4) 1