Practice Questions

Q70.Let E C denote the complement of an event E . Let E1, E2 and E3 be any pairwise independent events with P(E1) > 0 and P(E1 ∩E2 ∩E3) = 0 then P((E 2C ∩E 3C )/E1) is equal to (1) P(E 2C ) + P(E3) (2) P(E 3C ) −P(E 2C ) (3) P(E3) −P(E 2C ) (4) P(E 3C ) −P(E2) 1 n

Q70.Box 1 contains 30 cards numbered 1 to 30 and Box 2 contains 20 cards numbered 31 to 50 . A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box 1 is (1) 2 (2) 8 3 17 (3) 4 (4) 2 17 5

Q71.If the letters of the word ′ MOTHER′ be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word ′ MOTHER′ is.....

Q75.If →a and b are unit vectors, then the greatest value of √3→a+ b + →a− b is JEE Main 2020 (06 Sep Shift 1) JEE Main Previous Year Paper

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 equation 375 𝑥2 - 25𝑥- 2 = 0, then 𝑛 𝛽𝑟 is equal to: ∑ 𝑟= 1 lim ∑ 𝑟=𝑛 1 𝛼𝑟+ 𝑛→∞lim 𝑛→∞ (1) 1 (2) 21 12 346 (3) 7 (4) 29 116 358

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.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.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.Let 𝛼 and 𝛽 be the roots of the equation 𝑥2 + 2𝑥+ 2 = 0, then 𝛼15 + 𝛽15 is equal to (1) -512 (2) 128 (3) 512 (4) -256

Q61.If m is chosen in the quadratic equation (m2 + 1)x2 −3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is: (1) 4√3 (2) 10√5 (3) 8√3 (4) 8√5

Q61.The number of all possible positive integral value of α for which the roots of the quadratic equation 6x2 −11x + α = 0 are rational numbers is: (1) 5 (2) 3 (3) 4 (4) 2

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 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 −

Q61.Consider the quadratic equation (c −5)x2 −2cx + (c −4) = 0, c ≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is (1) 11 (2) 12 (3) 18 (4) 10

Q61.If α, β and γ are three consecutive terms of a non-constant G.P. Such that the equations αx2 + 2βx + γ = 0 and x2 + x −1 = 0 have a common root, then α(β + γ) is equal to: (1) βγ (2) αβ (3) αγ (4) 0

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 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 value of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 −λ) x + 2 = λ has the least value is: (1) 2 (2) 49 (3) 15 (4) 1 8

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.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 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 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.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