Complex Number
Short Description
Complex Number...
Description
COMPLEX NUMBER LEVEL-I 1.
If z1 , z2 are two complex numbers such that arg(z1+z2) = 0 and Im(z1z2) = 0, then (A) z1 = - z2 (B) z1 = z2 (C) z 1= z 2 (D) none of these
2.
Roots of the equation xn –1 = 0, n I, (A) form a regular polygon of unit circum-radius . (C) are non-collinear.
3.
4.
(B) lie on a circle. (D) A & B
Which of the following is correct (A) 6 + i > 8 – i (C) 6 + i > 4 + 2i
(B) 6 + i > 4 - i (D) None of these
If (1+i3)1999 = a+ib, then (A) a = 21998, b = 219983 (C) a=-21998, b = -219983
(B) a = 21999, b = 219993 (D) None of these
5.
If z = 1 + i 3 , then | arg ( z) | + | arg ( z ) | equals (A) /3 (B) 2/3 (C) 0 (D) /2
6.
The equation z z i i 3 z z 1 i 3 = 0 represents a circle with 1 3 1 3 and radius 1 (A) centre , (B) centre , and radius 1 2 2 2 2
1 3 and radius 2 (C) centre , 2 2
1 3 (D) centre , and radius 2 2 2
7.
Number of solutions to the equation (1 –i)x = 2x is (A) 1 (B) 2 (C) 3 (D) no solution
8.
If arg( z ) 0, then arg( z ) arg( z ) (A)
9.
4
(C)
2
The number of solutions of the equation z 2 z (A) one
10.
(B)
(B) two
(C) three
(D) 2
2
0, where z C is (D) infinitely many
If is an imaginary cube root of unity, then (1 + –2)7 equals (A) 128 (B) –128 2 (C) 128 (D) –128 2
11.
If z1 and z2 be the nth roots of unity which subtend right angle at the origin. Then n must be of the form (A) 4k + 1 (B) 4k + 2 (C) 4k + 3 (D) 4k
12.
For any two complex numbers z1 and z2 | 7 z1 + 3z2|2 + |3z1 – 7 z2|2 is always equal to (A) 16(|z1|2 + |z2|2) (B) 4(|z1|2 + |z2|2) 2 2 (C) 8(|z1| + |z2| ) (D) none of these
13.
If is an nth root of unity other than unity itself, then the value of 1 + + 2 + ………+ n –1 is ………………………………
14.
Locus of ‘z’ in the Argand plane is z 2, then the locus of z + 1 is -
15.
16.
(A) a straight line
(B) a circle with centre (1, 0)
(C) a circle with centre (0, 0)
(D) a straight line passing through (0, 0)
Value of 1999 299 1 is (A) 1 (C) 0
(B) 2 (D) -1
Square root(s) of ‘–1’ is/ are -
1 1 i 2 1 (C) 1 i 2
1 i 1 3 1 (D) 1 i 2
(A)
17.
The real value of ‘’ for which (A) n , (C) n
18.
3 2i sin is real is 1 2i sin
n I
, nI 2
(B) n (D)
5 6 5 (C) 6
(B)
6
(D) None
Which one is not a root of the fourth root of unity (A) i (B) 1 (C)
i 2
(D) –i
, nI 3
n , n I 2
Principal argument of z 3 i is (A)
19.
(B)
20.
If z 3 2 z 2 4 z 8 0 then (A) z 1
(B) z 2
(C) z 3
(D) None
LEVEL-II 1.
If a,b, c are three complex numbers such that c =(1– ) a + b, for some non-zero real number , then points corresponding to a,b, c are (A) vertices of a triangle (B) collinear (C) lying on a circle (D) none of these
2.
If z be any complex number such that |3z –2| + |3z +2| = 4, then locus of z is (A) an ellipse (B) a circle (C) a line-segment (D) None of these
3.
If arg z1 = arg(z2), then (A) z2 = k z1-1 (k > 0) (C) |z2| = | z 1|
4.
(B) z2 = kz1 (k > 0) (D) None of these.
1 1 1 1 1 1 The value of the expression 2 1 1 2 +3 2 2 2 + 4 3 3 2 + . 1 1 . . + (n+1) n n 2 , where is an imaginary cube root of unity, is (A)
n n2 2 3
(B)
n n2 2 3
2
(C)
n 2 n 1 4n 4
(D) none of these
5.
For a complex number z , | z-1| + |z +1| =2. Then z lies on a (A) parabola (B) line segment (C) circle (D) none of these
6.
If z1 and z2 are two complex numbers such that |z1| = |z2| + |z1 – z2|, then z z (A) Im 1 = 0 (B) Re 1 = 0 z2 z2 z z (C) Re 1 Im 1 z2 z2
7.
If
(D) none of these.
z1 =1 and arg (z1 z2) = 0, then z2
(A) z1 = z2 (C) z1z2 = 1
(B) |z2|2 = z1z2 (D) none of these.
8.
Number of non-zero integral solutions to (3+ 4i)n = 25n is (A) 1 (B) 2 (C) finitely many (D) none of these.
9.
If |z| < 4, then | iz +3 – 4i| is less than (A) 4 (C) 6
10.
(B) 5 (D) 9
If z is a complex number, then z2 + z 2 = 2 represents
11.
(A) a circle (C) a hyperbola
(B) a straight line (D) an ellipse
1 i = A + iB, then A2 +B2 equals to 1 i (A) 1 (B) -1
(B) 2 (D) - 2
If
12.
A,B and C are points represented by complex numbers z1, z2 and z3. If the circumcentre of the triangle ABC is at the origin and the altitude AD of the triangle meets the circumcircle again at P, then P represents the complex number zz z z (A) – 1 2 (B) – 2 3 z3 z1 z z zz (C) – 3 1 (D) 1 2 z2 z3
13.
If |z1| = |z2| and arg(z1) +arg(z2) = /2 , then (A) arg(z1-1) + arg(z2-1) = -/2 (B) z1z2 is purely imaginary (C) (z1+z2)2 is purely imaginary (D) All the above.
14.
If z1 and z2 are two complex numbers satisfying the equation (A) purely real (C) purely imaginary
z 1 iz 2 z 1 iz 2
1, then
z1 is a z2
(B) of unit modulus (D) none of these
15.
If the complex numbers z1, z2, z3, z4, taken in that order, represent the vertices of a rhombus, then (A) z1 + z3 = z2 + z4 (B) |z1 – z2| = |z2 – z3| z z3 (C) 1 is purely imaginary (D) none of these z2 z4
16.
If
17.
z1z z 2 k, z1, z 2 0 then z1z z2 (A) for k = 1 locus of z is a straight line (B) for k {1, 0} z lies on a circle (C) for k = 0 z represents a point (D) for k = 1,z lies on the perpendicular bisector of the line segment z2 z and 2 z1 z1
joining
If the equation |z – z1|2 + | z – z2|2 = k represents the equation of a circle, where z1 2+ 3i, z2 4 + 3i are the extremities of a diameter, then the value of k is 1 (A) (B) 4 4 (C) 2 (D) None of these
18.
If z be a complex number and ai a1z b1z a2 z b2 z determinant b1z a1z b2 z a2 z b1z a1 b2 z a2
, bi , ( i= 1,2,3) are real numbers, then the value of the a3 z b3 z b3 z a3 z is equal to b3 z a3
(A) (a1 a2 a3 + b1 b2 b3 ) |z|2 (C) 0
(B) |z|2 (D) None of these
19.
If z = x + iy satisfies the equation arg (z-2) = arg(2z+3i), then 3x-4y is equal to (A) 5 (B) -3 (C) 7 (D) 6
20.
If a complex number x satisfies log1 /
2
point represented by z is (A) |z| = 5 (C) |z|> 1 21.
| z |2 2 | z | 6 1 (B) x – y < 0 (C) 4x – 2y > 3 (D) none of these
35.
Let Z1 and Z2 be the complex roots of ax2 + bx + c = 0, where a b c > 0. Then
(A) | Z1 + Z2 | 1 (C) |Z1 | = |Z2| = 1
(B) |Z1 + Z2 | > 2 (D) none of these
36.
If the roots of z3 + az2 + bz + c = 0, a, b, c C(set of complex numbers) acts as the vertices of a equilateral triangle in the argand plane, then (A) a2 + b = c (B) a2 = b 2 (C) a + b = 0 (D) none of these
37.
If |z1| = 4, |z2| = 4, then |z1 + z2 + 3 + 4i| is less than (A) 2 (B) 5 (C) 10 (D) 13
38.
If z = x + iy satisfies Re{z -|z –1| + 2i} = 0, then locus of z is 1 1 1 (A) parabola with focus , and directrix x + y = 2 2 2 1 1 1 (B) parabola with focus , and directrix x + y = 2 2 2 1 1 (C) parabola with focus 0, and directrix y = 2 2 1 1 (D) parabola with focus , 0 and directrix x = 2 2
39.
If |z +1| = z + 1 , where z is a complex number, then the locus of z is (A) a straight line (B) a ray (C) a circle (D) an arc of a circle
40.
Length of the z 1 arg , is z 1 4 (A) 2 2 (C) 2
41.
42.
the point
(B)
represented
2
(D) none of these
If 8iz 3 12 z 2 18 z 27i 0 then (A) z 3 2 (B) z 1 (C) z 2 3
(D) z 3 4
If z i 2 and z1 5 3i then the maximum value of iz z1 is (A) 2 31
43.
curved line traced by
(B)
31 2
(C)
31 2
(D) 7
1 sin 1 ( z 1), where z is not real, can be the angle of the triangle if i (A) Re( z ) 1, I m ( z ) 2 (B) Re( z ) 1,1 I m ( z ) 1 (C) Re( z ) I m ( z ) 0
(C) None of these
by z, when
44.
The value of ln(1) (A) does not exist (B) 2 ln i
(C) i
(D) 0
n
n
n
n
45.
If n1 , n 2 are positive integers then (1 i) 1 (1 i 3 ) 2 (1 i 5 ) 1 (1 i 7 ) 2 is a real Number if and only if (A) n1 n 2 1 (B) n1 1 n2 (C) n1 n 2 (D) n1 , n 2 be +ve integers
46.
Let z1 , z 2 be two nonreal complex cube roots of unity and z z1
2
2
z z1 be the
equation of a circle with z1 , z 2 as ends of a diameter then the value of is (A) 4 47.
(B) 3
The value of (A) i
(B) (1,4)
2k 2k i cos 7 7 k 1 (B) i (C) 1
(C) (2,5)
(D) (3,1)
sin
(D) –1
The complex numbers z1, z2 and z3 satisfying triangle which is (A) of area zero (C) equilateral
50.
2
3z 6 3i is 2 z 8 6i 4
6
49.
(D)
The center of the arc arg (A) (4,1)
48.
(C) 2
If |z| = 3 then the number (A) purely real (C) a mixed number
z1 z3 1 i 3 are the vertices of a z 2 z3 2
(B) right angled isosceles (D) obtuse angled isosceles z3 is z3
(B) purely imaginary (D) none of these
51.
If iz3 + z2 –z + i = 0, then |z| is equal to ………………………………………
52.
If and are different complex numbers with || = 1, then
53.
If the complex numbers z1, z2, z3 are in A.P., then they lie on a (A) circle
(B) parabola
(C) line
(D) ellipse
is equal to 1
54.
z If z1 and z2 are two nth roots of unity, then arg 1 is a multiple of …………………. z2
55.
The maximum value of |z| when z satisfies the condition z
56.
All non-zero complex numbers z satisfying z = iz2 are…………………………………….
57.
Common roots of the equation z3 + 2z2 + 2z +1 = 0 and z1985 + z100 + 1 = 0 is …………
2 = 2 is ……………… z
LEVEL-III 1.
If points corresponding to the complex numbers z1, z2, z3 and z4 are the vertices of a rhombus, taken in order, then for a non-zero real number k (A) z1 – z3 = i k( z2 –z4) (B) z1 – z2 = i k( z3 –z4) (C) z1 + z3 = k( z2 +z4) (D) z1 + z2 = k( z3 +z4)
2.
If z1 and z2 are two complex numbers such that | z1 – z2| = | |z1| - |z2| |, then argz1 – argz2 is equal to (A) - /4 (B) - /2 (C) /2 (D) 0
3.
If f(x) and g(x) are two polynomials such that the polynomial h(x) = x f(x3) + x2 g(x6) is divisible by x2 +x +1 , then (A) f(1) = g(1) (B) f(1) - g( 1) (C) f(1) = g(1) 0 (D) f(1) = -g(1) 0
4.
Consider a square OABC in the argand plane, where ’O’ is origin and A A(z0). Then the equation of the circle that can be inscribed in this square is; ( vertices of square are given in anticlockwise order) z 1 i (A) | z – z0(1+ i)| =|z0| (B) 2 z 0 z0 2 (C) z
z 0 1 i z0 2
(D) none of these .
5.
For a complex number z, the minimum value of |z| + | z - cos - isin| is (A) 0 (B) 1 (C) 2 (D) none of these
6.
The roots of equation zn = (z +1)n (A) are vertices of regular polygon (C) are collinear
7.
The vertices of a triangle in the argand plane are 3 + 4i, 4+ 3i and 2 6 + i, then distance between orthocentre and circumcentre of the triangle is equal to,
137 28 6 1 (C) 137 28 6 2 (A)
8.
(B) lie on a circle (D) none of these
137 28 6 1 (D) 137 28 6 . 3 (B)
One vertex of the triangle of maximum area that can be inscribed in the curve |z – 2 i| =2,is 2 +2i , remaining vertices is / are (A) -1+ i( 2 + 3 )
(B) –1– i( 2 + 3 )
(C) 1+ i( 2 – 3 )
(D) –1– i( 2 – 3 )
9.
3 z1 2 z 2 = k, then points A(z1) , B(z2), C(3, 0) and D(2, 0) (taken in clockwise If 2 z1 3 z 2 sense) will (A) lie on a circle only for k > 0 (B) lie on a circle only for k < 0 (C) lie on a circle k R (D) be vertices of a square k( 0, 1)
10.
Let ‘z’ be a complex number z2 + az + a2 = 0, then (A) locus of z is a pair of straight lines 2 (B) arg(z) = 3 (C) |z| =|a| . (D) All
11.
If z1, z2, z3 . . .. zn-1 are the roots of the equation zn-1 + zn-2 + zn-3 + . . .+z +1= 0, where n N, n > 2, then (A) n, 2n are also the roots of the same equation. (B) 1/n, 2/n are also the roots of the same equation. (C) z1, z2, . . . , zn-1 form a geometric series. (D) none of these. Where is the complex cube root of unity.
12.
The value of i log(x – i) + i2 +i3 log(x +i) + i4( 2 tan-1x), x> 0 ( where i = (A) 0 (B) 1 (C) 2 (D) 3
13.
If z = -2 + 2 3i , then z2n + 22n zn + 24n may be equal to (A) 22n (B) 0 4n (C) 3. 2 (D) none of these
14.
The value of 169e (A) 119 –120i (C) 119 + 120i
12 5 i sin1 cos1 13 13
and
‘a’
be
a
real
parameter
such
that
1 ) is
is (B) -i(120 +119i) (D) none of these
15.
Let z1 and z2 be the complex roots of the equation 3z2 + 3z+ b = 0. If the origin, together with the points represented by z1 and z2 form an equilateral triangle then the value of b is (A) 1 (B) 2 (C) 3 (D) None of these
16.
If|z-2| = min {|z-1|,| z-3|}, where z is a complex number, then 3 5 (A) Re(z) = (B) Re(z) = 2 2
3 5 (C) Re (z) , 2 2 17.
(D)
None of these
If x = 1 + i, then the value of the expression x4 – 4x3 + 7x2 – 6x + 3 is (A) -1 (B) (C) 2 (D)
1 None of these
18.
If z lies on the circle centred at origin. If area of the triangle whose vertices are z, z and z + z, where is the cube root of unity, is 4 3 sq. unit. Then radius of the circle is (A) 1 unit (B) 2 units (C) 3 units (D) 4 units
19.
If i [0, /6], i = 1, 2, 3, 4, 5 and sin 1z4 + sin2 z3 + sin3 z2 + sin 4 z + sin5 = 2, then z satisfies. 3 1 (A) | z | (B) | z | 4 2 1 3 (C) | z | (D) None of these 2 4
20.
If is the angle which each side of a regular polygon of n sides subtends at its centre, then 1 + cos + cos2 + cos3 … + cos(n-1) is equal to (A) n (B) 0 (C)1 (D) None of these
21.
Triangle ABC, A(z1), B(z2), C(z3) is inscribed in the circle |z| = 2. If internal bisector of the angle A meets its circumcircle again at D(zd) then (A) z 2d z 2 z3 (B) z2d z1z3 (C) z2d z2 z1
(D) none of these
ANSWERS LEVEL −I 1. 5. 9. 13. 17.
C B D 0 A
2. 6. 10. 14. 18.
D B D B A
3. 7. 11. 15. 19.
D A D C C
4. 8. 12. 16. 20.
A A A A B
LEVEL −II 1. 5. 9. 13. 17. 21. 25. 29. 33. 37. 41. 45. 49.
B B D D B D 0 B B D A C C
2. 6. 10. 14. 18. 22. 26. 30. 34. 38. 42. 46. 50.
C A C A C A A A C D D B B
3. 7. 11. 15. 19. 23. 27. 31. 35. 39. 43. 47. 51.
A B A A, B, C D 20 D D A B B A 1
4. 8. 12. 16. 20. 24. 28. 32. 36. 40. 44. 48.
C D B A, B, C, D B C A D D C A
52.
1
53.
C
54.
2 n
55.
1+
56.
3 1 , 2 2
57.
, 2
2. 6. 10. 14. 18.
D C D A, B D
3. 7. 11. 15. 19.
A B C A A
4. 8. 12. 16. 20.
B A A C B
LEVEL −III 1. 5. 9. 13. 17. 21.
A B C B, C B A
3
View more...
Comments
