- 1. GRAViitYCOMPLEX NUMBERMATHEMATICS-PI SECTION IStraight
Objective Type This section contains 8 multiple choice questions.
Each question has 4 choices (A), (B), (C) and (D), out of which
ONLY ONE is correct.21.If z and are two non-zero complex numbers
such that |z| = 1 and Arg z Arg = z (A) 1 (C) i22.23.24. , then
2(B) 1 (D) iLet z and be complex numbers such that z i 0 and arg z
= , then arg z = (A) (B) 4 2 3 5 (C) (D) 4 4 i z 1 e If the
imaginary part of the expression i be zero, then locus of z is z 1
e (A) straight line (B) parabola (C) unit circle (D) ellipse If is
a complex number such that || = r 1 then z between the foci is (A)
2 (C) 31 describes a conic. The distance (B) 2( 2 1) (D) 4z lie on
1 z2 (A) a line not passing through the origin (B) | z | 2 (C) the
x-axis (D) the y-axis25.If |z| = 1 and z 1, then all the values
of26.The number of solutions of the system of equations given by
|z| = 3 and | z 1 i | 2 is equal to (A) 4 (B) 2 (C) 1 (D) no
solution27.Let z = cos + isin. Then the value of151 Im(z2m1 ) m1
sin 2 1 (C) 2sin 2(A)28.1 3sin 2 1 (D) 4sin 2(B)In geometrical
progression first term and common ratio are both value of the nth
term of the progression is (A) 2n (C) 1BYat = 2 isRAJESH SIR(B) 4n
(D) 3n1 ( 3 i). Then the absolute 2
2. GRAViitYCOMPLEX NUMBERSECTION IIMultiple Correct Answer Type
This section contains 4 multiple correct answer(s) type questions.
Each question has 4 choices (A), (B), (C) and (D), out of which ONE
OR MORE is/are correct.29.If z 20i 21 21 20i , then the principal
value of arg z can be (A) 4(C) 3 4 3 (D) 4 (B) 430.If z1 = a + ib
and z2 = c + id are complex numbers such that |z1| = |z2| = 1 and
Re(z1z2) = 0 then the pair of complex numbers 1 = a + ic and 2 = b
+ id satisfies. (A) |1| = 1 (B) |2| = 1 (C) Re (1 2) = 0 (D) |1| =
231.If z1 = 5 + 12i and |z2| = 4 then (A) maximum (|z1 + iz2|) = 17
(C) minimum32.z1 4 z2 z2(B) minimum (|z1 + (1 + i)z2|) = 13 9 213
4(D) maximumz1 4 z2 z213 3If z is a complex number satisfying |z i
Re(z)| = |z Im (z)| then z lies on (A) y = x (B) y = x (C) y = x +
1 (D) y = x + 1 SECTION IIILinked Comprehension Type This section
contains 2 paragraphs. Based upon each paragraph, 3 multiple choice
questions have to be answered. Each question has 4 choices (A),
(B), (C) and (D) out of which ONLY ONE is correct.Paragraph for
Question Nos. 33 to 35 Let A, B, C be three sets of complex numbers
as defined below A = {z : Im z 1} B = {z : |z 2 i| = 3} C = {z :
Re((1 i)z) = 2 } 33.The number of elements in the set ABC is (A) 0
(C) 2(B) 1 (D) 34.Let z be any point in ABC. Then, |z + 1 i|2 + |z
5 i|2 lies between (A) 25 and 29 (B) 30 and 34 (C) 35 and 39 (D) 40
and 4435.Let z be any point in ABC and let w be any point
satisfying | 2 i| < 3. Then, |z| |w| + 3 lies between (A) 6 and
3 (B) 4 and 6 (C) 6 and 6 (D) 3 and 9BYRAJESH SIR 3.
GRAViitYCOMPLEX NUMBER Paragraph for Question Nos. 36 to 38Suppose
z and w be two complex numbers such that |z| 1, |w| 1 and |z + iw|
= |z i w | = 2. Use the result | z |2 zz and |z + w| |z| + |w|,
answer the following 36.Which of the following is true about |z|
and ||1 2 3 (C) | z | | w | 4 (A) | z || w |37.38.(B) | z |1 3 , |
w | 2 4(D) |z| = |w| = 1Which of the following is true for z and
(A) Re(z) = Re(w) (C) Re(z) = Im(w)(B) Im(z) = Im(w) (D) Im(z) =
Re(w)Number of complex numbers satisfying the above conditions is
(A) 1 (B) 2 (C) 4 (D) indeterminate SECTION IV39.40.Matrix Match
Type Match the statements/expressions in Column I with the open
intervals in Column II Column I Column II 10 (A) (P) 2 sin 0 (r )(r
) 900 r 1 (B) If roots of t2 + t + 1 = 0 be , then 4 + 4 + (Q) 4 1
1 = 4 (C) 1 cos isin If cos n isin n, then n (R) i sin i(1 cos ) =
(D) (S) If z r cos r isin r , r = 1,2,3,., then value 1 3 3 of
z1z2z3 = Number of solutions of Column I (A) (B) (C) (D)BY2z |z| 0
22z z 0 z 2 8z 0 | z 2 | 1 and | z 1| 2RAJESH SIRColumn II
(P)1(Q)3(R) (S)4 Infinite 4. GRAViitYCOMPLEX NUMBERMATHEMATICS-PII
SECTION IStraight Objective Type This section contains 4 multiple
choice questions. Each question has 4 choices (A), (B), (C) and
(D), out of which ONLY ONE is correct.20.The roots of 1 + z + z3 +
z4 = 0 are represented by the vertices of (A) a square (B) an
equilateral triangle (C) a rhombus (D) a rectangle21.If |z 1| + |z
+ 3| 8, then the range of values of |z 4| is, (A) (0, 8) (B) [1, 9]
(C) [0, 8] (D) [5, 9]22.If z1, z2 and z3 be the vertices of ABC,
taken in anti-clock wise direction and z0 be the z 0 z1 sin 2A z 0
z 3 sin 2C is equal to z 0 z 2 sin 2B z 0 z 2 sin 2B circumcentre,
then (A) 0 (C) 1 23.(B) 1 (D) 2If a, b, c, a1,b1,c1 are non zero
complex numbers satisfyinga b c 1 i and a1 b1 c1a1 b1 c1 a2 b2 c 2
0 , then 2 2 2 is equal to a b c a1 b1 c1 (A) 2i (C) 2(B) 2 + 2i
(D) 2 2i SECTION IIMultiple Correct Answer Type This section
contains 5 multiple correct answer(s) type questions. Each question
has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE
is/are correct.24.A, B, C are the points representing the complex
numbers z1, z2, z3 respectively on the complex plane and the
circumcentre of the triangle ABC lies at the origin. If the
altitude AD of the triangle ABC meets the circumcircle again at P,
then P represents the complex number zz zz z z (A) z1z 2 z 3 (B) 1
2 (C) 1 3 (D) 2 3 z3 z2 z125.If points A and B are represented by
the non-zero complex numbers z1 and z2 on the Argand plane such
that |z1 + z2| = |z1 z2| and 0 is the origin, then z z (A)
orthocentre of OAB lies at 0 (B) circumcentre of AOB is 1 2 2 z (C)
arg 1 (D) OAB is isosceles 2 z2 26.If f(x) and g(x) are two
polynomials such that the polynomial h(x) = xf(x3) + x2g(x6) is
divisible by x2 + x + 1, then (A) f(1) = g(1) (B) f(1) = g(1) (C)
h(1) = 0 (D) none of theseBYRAJESH SIR 5. GRAViitY27.28.COMPLEX
NUMBERIf ( 1) is the fifth root of unity then (A) |1 2 3 4 | 0 (C)
|1 2 | 2cos 5(B) |1 2 3 | 1 (D) |1 | 2cos 10If the lines az az b 0
and cz cz d 0 are mutually perpendicular, where a and c are
non-zero complex numbers and b and d are real numbers, then (A) aa
cc 0 (B) ac is purely imaginary a c a (C) arg (D) 2 a c cSECTION
IIIMatrix Match Type 29.Match the statements/expressions in Column
I with the open intervals in Column II Column I (A) (P) z 3 z2 Let
z1, z2 be complex numbers such that 1 1 and |z2| 1, thenColumn II3
z1 z26|z1| is equal to (B) (C) (D)30.Number of non-zero complex
number satisfying z iz 2 Let a, b (0, 1) and z1 = a + i, z2 = 1 +
bi and z3 = 0 be the vertices of an equilateral triangle then value
of a b 2 3 is equal to Consider a circle having OP as diameter
where O being origin and P be z1. Take two points Q(z2) and R(z3)
on the circle and also on the same4(R)3(S)3 2 side of OP. If POQ
=/2k, QOR = /k and 8 z2 (5 3 3) z1 z3 then k is equal to Let the
complex numbers z1, z2 and z3 represent the vertices A, B and C of
triangle ABC respectively, which is inscribed in the circle of
radius unity and centre at origin. The internal bisector of the
angle A meets the circumcircle again at the point D, which is
represent by the complex number z4, and altitude from A to BC meets
the circumcircle at E, given by z5. Now match the entries from the
following columns Column I Column II (A) (P) z z arg 2 2 3 is equal
to z4 (B) (C) (D) z4 arg is equal to z 2 z3 zz arg 1 3 is equal to
z 2 z5 (Q) z2 arg 4 z1z 5 (S)(R)RAJESH SIR 2 40 (T)BY(Q) /25 6.
GRAViitYCOMPLEX NUMBERSECTION IVInteger Answer Type This section
contains 8 questions. The answer to each of the question is a
single digit integer, ranging from 0 to 9. The appropriate bubbles
below the respective question numbers in the ORS have to be
darkened. For example, if the correct answers to question numbers
X, Y, Z and W (say) are 6, 0, 9 and 2, respectively, then the
correct darkening of bubbles will look like the following:a b c 0
where a, b, c are three distinct complex numbers, then the value of
bc ca a b a2 b2 c2 is equal to (b c) 2 (c a) 2 (a b) 231.If32.If
|z1| = 1, |z2| = 2, |z3| = 3 and |9 z1z2 + 4z1z3 + z2z3| = 12 then
|z1 + z2 + z3| is equal to33. 3z 6 3i If the complex numbers z for
which arg and |z 3 + i| = 3, are 2z 8 6i 4 4 2 4 2 k i 1 and k i 1
then k must be equal to 5 5 5 5 2034.If e 2 i/7 and f x A 0 A k x k
then the value of f(x) + f(x) + f(2x) + . + f(6x) k 1is k(A0 + A7x7
+ A14x14), then k must be equal to 35.If magnitude of a complex
number 4 3i is tripled and rotated by an angle anticlockwise about
origin then resulting complex number would 12 + i then must be
equal to36.The maximum value of |z| when z satisfies the condition
z 37.Let z1, z2 be the roots of the equation z2 + az + b = 0 where
a and b may be complex. Let A and B represent z1 and z2 in the
Argands plane. If AOB 0 and OA = OB. Then 2 = b cos2 . where value
is 238.z1, z2 are roots of the equation z2 + az + b = 0. If AOB (0
is origin), A and B represent z1 and a2 z2 is equilateral, then is
equal to .. bBYRAJESH SIR2 2 is 1 z 7. GRAViitYCOMPLEX NUMBERPAPER
I KEYMATHS 21. (D) 22. (C) 23. (C) 24. (D) 25. (D) 26. (D) 27. (D)
28. (C) 29. (ABCD) 30. (ABC) 31. (AD) 32. (AB) 33. (B) 34. (C) 35.
(D) 36. (D) 37. (D) 38. (B) 39. (A - S), (B - P), (C - Q), (D - R)
40. (A- Q), (B - S), (C - R), (D P) PAPER II KEYMATHS 20. (B) 21.
(B) 22. (C) 23. (A) 24. (BCD) 25. (ABC) 26. (ABC) 27. (ABC) 28.
(BC) 29. (A-R), (B-R), (C-Q), (D-P) 30. (A S), (B QT), (C S), (D P)
31. 2 32. 2 33. 4 34. 7 35. 9 36. 3 37. 4 38. 3BYRAJESH SIR
