Practice Questions

Q61.The number of real roots of the equation 5 + 2𝑥- 1 = 2𝑥2𝑥- 2 is : (1) 2 (2) 3 (3) 1 (4) 4 π

Q61.If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is : (1) -81 (2) 100 (3) 144 (4) -300 where x and y are real numbers then y −x equals

Q61.If α and β are the roots of the quadratic equation x2 + xsinθ −2sinθ = 0, θ ∈(0, 2π ) , then α12+β12 is equal to : (α−12+β−12).(α−β)24 (1) 26 (2) 212 (sinθ+8)12 (sinθ−4)12 (3) 212 (4) 212 (sinθ+8)12 (sinθ−8)6 , has magnitude , then −z is equal to:

Q61.If three distinct numbers 𝑎, 𝑏, 𝑐 are in G.P. and the equations 𝑎𝑥2 + 2𝑏𝑥+ 𝑐= 0 and 𝑑𝑥2 + 2𝑒𝑥+ 𝑓= 0 have a common root, then which one of the following statements is correct? (1) 𝑑 𝑒 𝑓 are in A.P. (2) 𝑑, 𝑒, 𝑓 are in A.P. 𝑎, 𝑏, 𝑐 (3) 𝑑, 𝑒, 𝑓 are in G.P. (4) 𝑑 𝑒 𝑓 are in G.P. 𝑎, 𝑏, 𝑐

Q61.The number of integral values of m for which the quadratic expression (1 + 2m) x2 −2(1 + 3m)x + 4(1 + m), x ∈R is always positive, is (1) 7 (2) 3 (3) 6 (4) 8

Q61.The sum of the solutions of the equation √𝑥- 2 + √𝑥√𝑥- 4 + 2 = 0, 𝑥> 0 is equal to (1) 10 (2) 9 (3) 12 (4) 4 JEE Main 2019 (08 Apr Shift 1) JEE Main Previous Year Paper

Q61.Let p, q ∈ Q . If 2 −√3 is a root of the quadratic equation x2 + px + q = 0, then (1) p2–4q + 12 = 0 (2) q2 + 4p + 14 = 0 (3) p2–4q–12 = 0 (4) q2–4p–16 = 0

Q61.If λ be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m −4)x + 2 = 0, then the least value of m for which λ + λ1 = 1, is : JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) 2 −√3 (2) −2 + √2 (3) 4 −2√3 (4) 4 −3√2 α −

Q62.If a > 0 and z = (1+i)2a−i √25 (1) −15 −35 i (2) −35 −15 i (3) 1 5 −35 i (4) −15 + 53 i

Q62.If z z α (α ∈R) is a purely imaginary number and |z| = 2, then a value of α is : + (1) 1 (2) 12 (3) √2 (4) 2

Q62.All the points in the set S = { α+iα−i , α ∈R}, i = √−1 lie on a (1) straight line whose slope is −1 (2) circle whose radius is √2 (3) circle whose radius is 1 (4) straight line whose slope is 1 JEE Main 2019 (09 Apr Shift 1) JEE Main Previous Year Paper

Q62.If both the roots of the quadratic equation x2 −mx + 4 = 0 are real and distinct and they lie in the interval (1, 5), then m lies in the interval: Note: In the actual JEE paper interval was [1, 5] (1) (−5, −4) (2) (3, 4) (3) (5, 6) (4) (4, 5)

Q62.Let z ∈C with Im(z) = 10 and it satisfies 22 z+nz−n = 2i −1 for some natural number n. Then (1) n = 20 and Re(z) = 10 (2) n = 40 and Re(z) = 10 (3) n = 20 and Re(z) = −10 (4) n = 40 and Re(z) = −10

Q62.If 𝑧 and 𝜔 are two complex numbers such that 𝑧𝜔= 1 and 𝑎𝑟𝑔𝑧- 𝑎𝑟𝑔( 𝜔) = 2, then: (1) 𝑧¯ω = 1 - 𝑖 (2) ¯𝑧𝜔= 𝑖 √2 (3) 𝑧¯ω = -1 + 𝑖 (4) ¯𝑧ω = - 𝑖 √2

Q62.The number of integral values of 𝑚 for which the equation, 1 + 𝑚2𝑥2 - 21 + 3𝑚𝑥+ 1 + 8𝑚= 0 has no real root, is (1) 2 (2) 3 (3) Infinitely many (4) 1 𝑖

Q62.Let z = 5 5 + . If R(z) and I(z) respectively denote the real and imaginary parts of z, ( √32 + 2i ) ( √32 −i2 ) then (1) I(z) = 0 (2) R(z) < 0 and I(z) > 0 (3) R(z) > 0 and I(z) > 0 (4) R(z) = −3

Q62.Let (−2 −13 i) 3 = x+iy27 (i = √−1), (1) 91 (2) -85 (3) 85 (4) -91

Q62.Let z1 and z2 be two complex numbers satisfying |z1| = 9 and |z2 −3 −4i| = 4. Then the minimum value of |z1 −z2| is : (1) 2 (2) √2 (3) 0 (4) 1

Q62.Let 𝐴= 𝜃∈- 𝜋 𝜋: 3 + 2𝑖 sin𝜃 is purely imaginary . Then the sum of the elements in 𝐴 is: 2, 1 - 2𝑖 sin𝜃 5𝜋 (1) (2) π 6 (3) 2𝜋 (4) 3𝜋 3 4

Q62.The equation |𝑧- 𝑖| = | 𝑧- 1 | , 𝑖= √-1, represents: 1 (1) a circle of radius (2) a circle of radius 1 2 (3) the line through the origin with slope 1 (4) the line through the origin with slope -1 JEE Main 2019 (12 Apr Shift 1) JEE Main Previous Year Paper

Q62.Let z be a complex number such that |z| + z = 3 + i ( where i = √−1) Then |z| is equal to : (1) √34 (2) 5 3 3 (3) √41 (4) 5 4 4

Q62.Let z1 and z2 be any two non-zero complex numbers such that 3|z1| = 4|z2|. If z = 3z1 + 2z2 then maximum 2z2 3z1 value of |z| is Note: In actual paper value of |z| was asked. Hence, none of the options given were correct. So we have modified the question as well as options. (1) 7 (2) 9 2 2 (3) 5 (4) 1 2 2 √172

Q62.If 𝛼 and 𝛽 be the roots of the equation 𝑥2 - 2𝑥+ 2 = 0, then the least value of 𝑛 for which 𝛼 𝑛= 1 is 𝛽 (1) 5 (2) 4 (3) 2 (4) 3

Q62.Let z ∈C be such that |z| < 1. If ω = 5(1−z)5+3z , then: JEE Main 2019 (09 Apr Shift 2) JEE Main Previous Year Paper (1) 5Re(ω) > 1 (2) 5Im(ω) < 1 (3) 5Re(ω) > 4 (4) 4Im(ω) > 5

Q63.The Number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is: (1) 220 (2) 221 (3) 220 + 1 (4) 220 - 1