Practice Questions

Q52.Let z = x + iy be a non-zero complex number such that z2 = i|z|2 , where i = √−1, then z lies on the : (1) line, y = −x (2) imaginary axis (3) line, y = x (4) real axis

Q52.If Re( 2z+iz−1 ) = 1, where z = x + iy, then the point (x, y) lies on a (1) circle whose centre is at (−12 , −32 ) (2) straight line whose slope is −23 (3) straight line whose slope is 3 2 (4) circle whose diameter is √52

Q52.The imaginary part of (3 2√−54) −(3 −2√−54) ,can be (1) −√6 (2) −2√6 (3) 6 (4) √6

Q52.Let z be a complex number such that z+2i z−i = 1 and |z| = 52 . Then, the value of |z + 3i| is (1) √10 (2) 72 (3) 15 (4) 2√3 4

Q52.Let α = −1+i√32 . If a = (1 + α) ∑100k=0 α2k and b = ∑100k=0 α3k , then a and b, are the roots of the quadratic equation. (1) x2 + 101x + 100 = 0 (2) x2 −102x + 101 = 0 (3) x2 −101x + 100 = 0 (4) x2 + 102x + 101 = 0

Q52.If 3+isinθ , θ ∈[0 ,2 π], is a real number, then an argument of sinθ + icosθ is 4−icosθ (1) π −tan−1( 34 ) (2) π −tan−1( 43 ) (3) −tan−1( 43 ) (4) tan−1( 43 )

Q52.If the four complex numbers z, z, z −2 Re (z) and z −2 Re (z) represent the vertices of a square of side 4 units in the Argand plane, then |z| is equal to : (1) 4√2 (2) 4 (3) 2√2 (4) 2

Q52.The value of 2π 2π 3 2π 2π ( 1+sin 9 −i cos 99 ) is (1) 1 (2) 1 2 (1 −i√3) 2 (√3 −i) (3) −12 (√3 −i) (4) −12 (1 −i√3)

Q52.The region represented by {z = x + iy ∈C : |z|−Re (z) ≤1} is also given by the inequality (1) y2 ≥2(x + 1) (2) y2 ≤2(x + 12 ) (3) y2 ≤(x + 12 ) (4) y2 ≥x + 1

Q52.Let a, b ∈R, a ≠0 be such that the equation, ax2 −2bx + 5 = 0 has a repeated root α, which is also a root of the equation, x2 −2bx −10 = 0. If β is the other root of this equation, then α2 + β2 is equal to: (1) 25 (2) 26 (3) 28 (4) 24

Q52.If a and b are real numbers such that (2 + α)4 = a + bα, where α = −1+i√32 , then (1) 9 (2) 24 (3) 33 (4) 57

Q52.Let f : R →R be such that for all x ∈R(21+x + 21−x), f(x) and (3x + 3−x) are in A.P., then the minimum value of f(x) is (1) 2 (2) 3 (3) 0 (4) 4

Q52.If z1, z2 are complex numbers such that Re (z1) = |z1 −1| and Re (z2) = |z2 −1| and arg(z1 −z2) = π6 , then Im(z1 + z2) is equal to : (1) 2√3 (2) √3 2 (3) 1 (4) 2 √3 √3

Q52.Let α and β be the roots of x2 −3x + p = 0 and γ and δ be the roots of x2 −6x + q = 0. If α, β, γ, δ from a geometric progression. Then ratio (2 q + p) : (2 q −p) is (1) 3 : 1 (2) 9 : 7 (3) 5 : 3 (4) 33 : 31

Q53.If the sum of the series 20 + 19 35 + 19 51 + 18 54 +. . . . . . . . . . up to nth term is 488 and the nth term is negative, then : (1) nth term is −4 52 (2) n = 41 (3) nth term is −4 (4) n = 60 k, then 18k is equal to:

Q53.If a, b and c are the greatest values of 19Cp, 20Cq and 21Cr respectively, then: (1) 11 a = 22b = 21c (2) 10a = 11b = 21c (3) 11 a = 22b = 42c (4) 10a = 11b = 42c

Q53.Let a1, a2, … , an be a given A.P. whose common difference is an integer and Sn = a1 + a2 + … + an . If a1 = 1, an = 300 and 15 ≤n ≤50, then the ordered pair (Sn−4, an−4) is equal to: (1) (2490, 249) (2) (2480, 249) (3) (2480, 248) (4) (2490, 248)

Q53.If z is a complex number satisfying |Re(z)| + |Im(z)| = 4, then |z| cannot be (1) √172 (2) √10 (3) √7 (4) √8

Q53.Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated ? (1) 2! 3! 4! (2) (3!)3 ⋅(4!) (3) (3!)2. (4!) (4) 3! (4!)3

Q53.The sum of the first three terms of G. P is S and their products is 27 . Then all such S lie in (1) (−∞, −9] ∪[3, ∞) (2) [−3, ∞) (3) (−∞, −3] ∪[9, ∞) (4) (−∞, 9]

Q53.There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is: (1) 3000 (2) 1500 (3) 2255 (4) 2250

Q53.If 210 + 29 ⋅31 + 28 ⋅32 + … … + 2 ⋅39 + 310 = S −211 , then S is equal to (1) 311 −212 (2) 311 (3) 3112 + 210 (4) 2. 311

Q53.Let a1, a2, a3, … , be a G. P. such that a1 < 0, a1 + a2 = 4 and a3 + a4 = 16. If ∑9i=1 ai = 4λ, then λ, is equal to. (1) −513 (2) −171 (3) 171 (4) 5113

Q53.If the 10th , term of an A.P. is 201 , and its 20th , term is 101 , then the sum of its first 200 , terms is. (1) 50 (2) 50 14 (3) 100 (4) 100 12

Q53.Let n > 2 be an integer. Suppose that there are n Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of n is (1) 201 (2) 200 (3) 101 (4) 199