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1. The number of real values of a satisfying the equation a 2 2a sin x 1 0 is (a) Zero (b) One (c) Two (d) Infinite 2. For positive integers n1 , n 2 the value of the expression (1 i)n1 (1 i 3 )n1 (1 i 5 )n2 (1 i7 )n2 where i 1 is a real number if and only if 3. 4. (a) n1 n 2 1 (b) n1 n 2 1 (c) n1 n 2 (d) n1 0, n 2 0 Given that the equation z 2 ( p iq)z r i s 0, where p, q, r, s are real and non-zero has a real root, then (a) pqr r 2 p 2 s (b) prs q 2 r 2 p (c) qrs p 2 s 2q (d) pqs s 2 q 2r If x 5 2 4 , then the value of the expression x 4 9 x 3 35 x 2 x 4 is (b) 160 (d) 60 (a) 160 (c) 60 5. If (a) b d 3 i (a ib)(c id ) , then tan 1 tan 1 has the value c a 3 2n , n I (c) n 6. 3 ,n I 3 ,n I b c a 1, then cos( ) cos( ) cos( ) is equal to c a b (a) 3/2 (b) – 3/2 (c) 0 (d) 1 If (1 i)(1 2i)(1 3i).....( 1 ni) a ib , then a2 b 2 (c) 9. 6 If a cos i sin , b cos i sin , (a) a 2 b 2 8. (d) 2n ,n I c cos i sin and 7. (b) n 2.5.10.... (1 n 2 ) is equal to (b) a 2 b 2 (d) a2 b 2 If z is a complex number, then the minimum value of | z | | z 1| is (a) 1 (b) 0 (c) 1/2 (d) None of these For any two complex numbers z1 and z 2 and any real numbers a and b; | (az1 bz 2 )| 2 | (bz 1 az 2 )| 2 (a) (a 2 b 2 )(| z1 | | z 2 |) (b) (a 2 b 2 )(| z1 | 2 | z 2 | 2 ) (c) (a 2 b 2 )(| z1 | 2 | z 2 | 2 ) (d) None of these 10. 11. The locus of z satisfying the inequality log 1 / 3 | z 1| log1 / 3 | z 1| is (a) R (z) 0 (b) R (z) 0 (c) I (z) 0 (d) None of these If z1 a ib and z 2 c id are complex numbers such that | z1 | | z 2 | 1 and R(z1 z 2 ) 0, then the pair of complex numbers w1 a ic and w2 b id satisfies 12. (a) | w1 | 1 (b) | w2 | 1 (c) R(w1 w2 ) 0, (d) All the above Let z and w be two complex numbers such that | z | 1, | w | 1 and | z iw | | z iw | 2 . Then z is equal to (a) 1 or i (c) 1 or – 1 13. (b) i or i (d) i or –1 The maximum distance from the origin of coordinates to the point z satisfying the equation z (a) 1 ( a 2 1 a) 2 1 a is z 1 ( a 2 2 a) 2 1 ( a 2 4 a) (c) 2 (d) None of these (b) 14. Find the complex number z satisfying the equations (b) 6 8i (d) None of these (a) 6 (c) 6 8 i, 6 17 i 15. If z 1 , z 2 , z 3 are complex numbers such that | z1 | | z 2 | | z 3 | (a) Equal to 1 (c) Greater than 3 16. 1 1 1 1 , then | z 1 z 2 z 3 | is z1 z 2 z 3 (b) Less than 1 (d) Equal to 3 z z1 , then the value of | z 7 9i| is If z1 10 6i, z 2 4 6i and z is a complex number such that amp z z2 4 equal to (a) 17. z 12 5 z4 , 1 z 8i 3 z 8 (b) 2 2 2 (c) 3 2 (d) 2 3 If z 1 , z 2 , z 3 be three non-zero complex number, such that z 2 z 1 , a | z 1 |, b | z 2 | and c | z 3 | suppose that a b c z b c a 0 , then arg 3 z2 c a b z z1 (a) arg 2 z 3 z1 2 is equal to z z1 (b) arg 2 z 3 z1 2 18. z z1 z z1 (c) arg 3 (d) arg 3 z z 1 z 2 z1 2 Let z and w be the two non-zero complex numbers such that | z | | w | and arg z arg w . Then z is equal to (a) w 19. (b) w (c) w (d) w If | z 25 i | 15 , then | max .amp (z) min .amp (z)| 3 (a) cos 1 5 (c) 20. 21. 2 3 cos 1 5 z arg 2 z 3 Let z, w be complex numbers such that z iw 0 and arg zw . Then arg z equals (b) / 2 (d) / 4 If (1 x )n C 0 C 1 x C 2 x 2 ..... C n x n , then the value of C0 C2 C4 C6 ..... is (c) 2 n sin n 2 n n/2 (d) 2 cos 4 (b) 2 n cos (a) 2 n 23. 3 3 (d) sin 1 cos 1 5 5 z If z 1 , z 2 and z 3 , z 4 are two pairs of conjugate complex numbers, then arg 1 z4 (a) 0 (b) 2 3 (c) (d) 2 (a) 5 / 4 (c) 3 / 4 22. 3 (b) 2 cos 1 5 n 2 If x cos i sin and y cos i sin , then x m y n x m y n is equal to (a) cos(m n ) equals (b) cos(m n ) (c) 2 cos(m n ) (d) 2 cos(m n ) 8 sin 2r 2r i cos is 9 9 24. The value of 25. (a) 1 (b) 1 (c) i (d) i If a, b, c and u, v, w are complex numbers representing the vertices of two triangles such that c (1 r)a rb r 1 and w (1 r)u rv , where r is a complex number, then the two triangles (a) Have the same area (b) Are similar (c) Are congruent (d) None of these 26. Suppose z1 , z 2 , z 3 are the vertices of an equilateral triangle inscribed in the circle | z | 2 . If z1 1 i 3 , then values of z 3 and z 2 are respectively 27. 28. (a) 2, 1 i 3 (b) 2, 1 i 3 (c) 1 i 3 ,2 (d) None of these [IIT 1994] If the complex number z1 , z 2 the origin form an equilateral triangle then z12 z 22 (a) z1 z 2 (b) z1 z 2 (c) z 2 z1 (d) | z1 | 2 | z 2 | 2 If at least one value of the complex number z x iy satisfy the condition | z 2 | a 2 3a 2 and the inequality | z i 2 | a 2 , then 29. (a) a 2 (b) a 2 (c) a 2 (d) None of these If z, iz and z iz are the vertices of a triangle whose area is 2 units, then the value of | z | is (a) – 2 (c) 4 (b) 2 (d) 8 30. If z 2 z | z | | z | 2 0 , then the locus of z is 31. (a) A circle (b) A straight line (c) A pair of straight lines (d) None of these If cos cos cos sin sin sin 0 then cos 3 cos 3 cos 3 equals to 32. 33. (a) 0 (b) cos( ) (c) 3 cos( ) (d) 3 sin( ) If z r cos r n2 i sin r n2 , where r = 1, 2, 3,….,n, then lim z 1 z 2 z 3 ... z n is equal to n (a) cos i sin (b) cos(/2) i sin(/2) (c) e i / 2 (d) 3 e i If the cube roots of unity be 1, , 2 , then the roots of the equation (x 1)3 8 0 are (a) 1, 1 2 , 1 2 2 (b) 1, 1 2 , 1 2 2 (c) 1, 1, 1 (d) None of these 34. 35. If 1, , 2 , 3 ......., n 1 are the n, n th roots of unity, then (1 )(1 2 ).....( 1 n 1 ) equals (a) 0 (b) 1 (c) n (d) n 2 The value of the expression 1.(2 )(2 2 ) 2.(3 )(3 2 ) ....... .... (n 1).(n )(n 2 ), where is an imaginary cube root of unity, is 36. 37. (a) 1 (n 1)n(n 2 3 n 4 ) 2 (b) 1 (n 1)n(n 2 3 n 4 ) 4 (c) 1 (n 1)n(n 2 3 n 4 ) 2 (d) 1 (n 1)n(n 2 3 n 4 ) 4 1 i 3 If i 1 , then 4 5 2 2 334 1 i 3 3 2 2 (a) 1 i 3 (b) 1 i 3 (c) i 3 (d) i 3 365 is equal to If a cos(2 / 7) i sin(2 / 7), then the quadratic equation whose roots are a a 2 a 4 and a 3 a 5 a 6 is (a) x 2 x 2 0 (b) x 2 x 2 0 38. (c) x 2 x 2 0 (d) x 2 x 2 0 th Let z 1 and z 2 be n roots of unity which are ends of a line segment that subtend a right angle at the origin. 39. Then n must be of the form (a) 4k + 1 (b) 4k + 2 (c) 4k + 3 (d) 4k Let is an imaginary cube roots of unity then the value of 2( 1)( 2 1) 3(2 1)(2 2 1) ..... (n 1)(n 1)(n 2 1) is 2 n(n 1) (a) n 2 n(n 1) (b) 2 2 2 n(n 1) (c) n 2 40. (d) None of these is an imaginary cube root of unity. If (1 2 )m (1 4 )m , then least positive integral value of m is (a) 6 (c) 4 (b) 5 (d) 3