Practice Questions

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.If the sum of an infinite GP, a, ar, ar2, ar3, … is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, … is: (1) 25 (2) 9 2 2 (3) 1 (4) 5 2 2

Q63.The number of solutions of sin7 x + cos7 x = 1, x ∈[0, 4π] is equal to (1) 11 (2) 7 (3) 5 (4) 9

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.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. cosec 18° is a root of the equation: (1) x2 −2x −4 = 0 (2) 4x2 + 2x −1 = 0 (3) x2 + 2x −4 = 0 (4) x2 −2x + 4 = 0

Q63.The value of ∑6r=0(6Cr ⋅6C6−r) is equal to : (1) 1124 (2) 1324 (3) 1024 (4) 924

Q63.The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is: (1) 26664 (2) 122664 (3) 122234 (4) 22264

Q63.Team ′A′ consists of 7 boys and n girls and Team ′B′ has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to: (1) 5 (2) 2 (3) 4 (4) 6

Q63.If 0 < a, b < 1 , and tan−1 a + tan−1 b = π4 , then the value of (a + b) −( a2+b22 ) ( a3+b33 ) −( a4+b44 ) is : (1) loge( 2e ) (2) e (3) e2 −1 (4) loge 2

Q63.If the greatest value of the term independent of x in the expansion of (x sin α + a cosx α )10 is (5!)210! value of a is equal to: (1) −1 (2) 1 (3) −2 (4) 2 10100 1

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

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.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 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.In an increasing geometric series, the sum of the second and the sixth term is 252 and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to: (1) 35 (2) 32 (3) 26 (4) 30 1 10 1 (1−x) 10 where x ∈(0, 1) is: 5 + t )

Q63.If for x, y ∈R, x > 0, y = log10 x + log10 x1/3 + log10 x1/9 + … upto ∞ terms and 2+4+6+…+2y3+6+9+…+3y = log104 x , then the ordered pair (x, y) is equal to (1) (106, 6) (2) (106, 9) (3) (102, 3) (4) (104, 6)

Q63. Let 𝑆𝑛= 1 · ( 𝑛- 1 ) + 2 · ( 𝑛- 2 ) + 3 · ( 𝑛- 3 ) + … + ( 𝑛- 1 ) · 1, 𝑛⩾4 . ∞ 2 Sn 1 The sum ∑n = 4 n! - ( n - 2 ) ! is equal to : 𝑒- 2 e - 1 (1) (2) 6 3 (3) e (4) e 6 3 20 1 4 = . If the sum of this 𝐴. 𝑃. is 189, then a6a16

Q63.Let A(a, 0), B(b, 2b + 1) and C(0, b), b ≠0, |b| ≠1 , be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is: (1) −2b (2) 2b2 b+1 b+1 (3) −2b2 (4) 2b b+1 b+1

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 value of 2 sin( 8π ) sin( 2π8 ) sin( 3π8 ) sin( 5π8 ) sin( 6π8 ) sin( 7π8 ) is : (1) 1 (2) 1 4√2 8 (3) 1 (4) 1 8√2 4 JEE Main 2021 (26 Aug Shift 2) JEE Main Previous Year Paper

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 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.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.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