Practice Questions

Q62.The sum of the series 1 + 2 + + … + 2100 when x = 2 is: x+1 x2+1 x4+1 x2100+1 (1) 1 − 2101 (2) 1 + 2101 4101−1 4101−1 (3) 1 + 2100 (4) 1 − 2100 4101−1 4201−1

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.Let C be the set of all complex numbers. Let S1 = {z ∈C : |z −2| ≤1} and 2 ¯S2 = {z ∈C : z(1 + i) + z(1 −i) ≥4}. Then, the maximum value of z −52 for z ∈S1 ∩S2 is equal to : (1) 3+2√2 (2) 5+2√2 4 2 (3) 3+2√2 (4) 5+2√2 2 4

Q62.A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = √5 inches and ∠PCB = tan−1(2). The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is : (1) tan−1( 43 ) (2) tan−1( 21 ) (3) tan−1( 43 ) (4) tan−1(1)

Q62.If α, β ∈R are such that 1 −2i (here i2 = −1) is a root of z2 + αz + β = 0, then (α −β) is equal to: (1) −7 (2) 7 (3) −3 (4) 3

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.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.If 0 < x < 1 and y = 21 x2 + 32 x3 + 43 x4 + … … , then the value of e1+y at x = 21 is: (1) 1 e2 (2) 2e 2 (3) 2e2 (4) 21 √e

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.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.If 𝑏 is very small as compared to the value of 𝑎, so that the cube and other higher powers of 𝑏 can be neglected 𝑎 in the identity 1 1 1 1 … . + 𝛼𝑛+ 𝛽𝑛2 + 𝛾𝑛3 𝑎- 𝑏+ 𝑎- 2𝑏+ 𝑎- 3𝑏+ 𝑎- 𝑛𝑏= then the value of 𝛾 is : (1) 𝑎2 + 𝑏 (2) 𝑎+ 𝑏 3𝑎3 3𝑎2 (3) 𝑏2 (4) 𝑎+ 𝑏2 3𝑎3 3𝑎3

Q62.If S = {z ∈C : z+2iz−i ∈R}, then (1) S is a circle in the complex plane (2) S contains exactly two elements (3) S contains only one element (4) S is a straight line in the complex plane

Q62.Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3r2, then r2 −d is equal to : (1) 7 −√3 (2) 7 + 3√3 (3) 7 −7√3 (4) 7 + √3

Q62.The area of the triangle with vertices P(z), Q(iz) and R(z + iz) is (1) 1 (2) 12 z 2 (3) 1 (4) 1 z + iz 2 2 2

Q62.The probability of selecting integers a ∈[−5, 30] such that x2 + 2 (a + 4) x −5a + 64 > 0 , for all x ∈R, is: (1) 7 (2) 2 36 9 (3) 1 (4) 1 6 4 JEE Main 2021 (20 Jul Shift 1) JEE Main Previous Year Paper

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.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.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 15 sin4 α + 10 cos4 α = 6, for some α ∈R, then the value of 27 sec6 α + 8 cosec6 α is equal to : (1) 350 (2) 500 (3) 400 (4) 250

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. 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.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.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 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