Practice Questions

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.The value of 30 ( −1+i√31−i ) is : (1) 65 (2) 215 i (3) −215 (4) −215 i

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

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.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.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

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.Let u = z−ki2z+i , z = x + iy and k > 0. If the curve represented by Re (u)+ Im (u) = 1 intersects the y-axis at points P and Q where PQ = 5 then the value of k is (1) 3 (2) 1 2 2 (3) 4 (4) 2

Q53.If the number of five digit numbers with distinct digits and 2 at the 10th place is 336k , then k is equal to: (1) 4 (2) 6 (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.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

Q53.If the first term of an A. P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A. P.is (1) 1 (2) 1 6 5 (3) 1 (4) 1 4 7

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.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.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.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.Total number of 6− digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appears, is (1) 1 2 (6!) (2) 6! (3) 56 (4) 25 (6!)

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.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.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.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