complex numbers

CLASS WORK SHEET No-1 IIT JEE2013-VCC Topic: Complex Number

MATHEMATICS

SINGLE CORRECT CHOICE TYPE QUESTIONS 1. If

(a)

3i ai  and a is a real number, then a is 2 a i

1  3 2

(b)

1 4 3 2

1  3 2

(c) 2  3

(d)

(c) x > 2, y > 3

(d) x > 3, and y is any real number

(c) 4x + 3

(d) 3x + 4

2. If z = x + iy is such that z  4  z  2 , then (a) x > 0, y > 0 3. If x  iy 

(b) x < 0, y > 0

3 then x2 + y2 = 2  cos   i sin 

(a) 4x – 3

(b) 3x – 4

4. If   1  i 3 and n is a positive integer which is not a multiple of 3, then  2 n  2n  n  22 n  (a) 1

(b) –1

(c) 0

(d)  2

5. Let w  1 be a cube root of unity and E = 2(1 + w)(1 + w2) + 3(2w + 1)(2w2 + 1) + 4(3w + 1)(3w2 + 1) + .... + (n + 1) (nw + 1)(nw2 + 1) Then E is equal to

(a)

n2  n  1

2

4

(b)

n2  n  1

2

4

n

(c)

n2  n  1 4

2

n

(d)

n2  n  1 4

2

  n  1

 x  y 6. If x  cos   i sin  and y  cos   i sin , then x  y is equal to       (a) i tan    2 

   (b) i tan    2  10

 2i 5   7. Let z1 and z2 be given by z1    2i 5 

 1 2  (a) 2cos  20cos  3  www.vidyamandir.com

 (c) i tan    2 

 (d) i tan    2 

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 2i 5   . Then |z1 + z2| is equal to and z2   2i 5 

 1 2  (b) 2sin 10cos  3 

 1 2  (c) 2cos 10cos  3 

 1 2  (d) 2sin  20cos  3 

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CLASS WORK SHEET No-1 IIT JEE2013-VCC Topic: Complex Number

MATHEMATICS

n

8. If 1  z   a0  a1 z  a2 z 2  ...  an z n , where a0, a1, a2, ...., an, are real, then 2

 a0  a2  a4  a6  ...   a1  a3  a5  a7  ... (a) 2n

(b) a02  a12  a22  ....  an2

(c) 2 n

2

 (d) 2n 2

2

MULTIPLE CORRECT ANSWER TYPE QUESTIONS 9. If x and y are real numbers and (a) x = 3

1  i  x  2i   2  3i  y  i  i 3 i

3i

(b) y = 1

(c) y = –1

10. The complex number(s) satisfying the equations (a) 6 – 8i

(d) x = –3

z  12 5 z4  and  1 is (are) z  8i 3 z 8

(b) 6 + 17i

11. If x is a real number such that 0 < x < 2 and

then

(c) 6 + 8i

(d) 6 – 17i

sin  x / 2   cos  x / 2    i tan x is real, then the possible value(s) of x is 1  2i sin  x / 2 

(are) (a) 0

(c) /4

(b) 2

(d) /4

MATRIX-MATCH TYPE QUESTIONS 12. Match the items in Column I with those in Column II Column I (A) If z = x + iy, z1/3 = a – ib and

Column II

x y     a 2  b2  , then  is a b

(p) 10

(B) If |z – i| < 1, then the value of |z + 12 – 6i| is less than

(q) 14

(C) If |z1| = 1 and |z2| = 2, then |z1 + z2|2 + |z1 – z2|2 is equal to

(r) 1

(D) If z = 1 + i, then 4(z4 – 4z3 + 7z2 – 6z + 3) is equal to

(s) 4 (t) 5

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CLASS WORK SHEET No-1 IIT JEE2013-VCC Topic: Complex Number

MATHEMATICS

13. Match the items in Column I with those in Column II. w  1 is a cube root of unity.. Column I (A) The value of

Column II

1 1  w 1  w2 1  w4 1  w8  is 3

(p) –128

(B) w(1 + w – w2)7 is equal to

(q) 6

(C) The least positive integer n such that (1 + w2)n = (1 + w4)n is

(r) 0

(D)

1 1 1   is equal to 1  2w 2  w 1  w

(s) 128 (t) 3

COMPREHENSION-TYPE QUESTIONS Passage: (14 to 16) A complex number z is pure real if an only if z  z and is pure imaginary if and only if z   z. Answer the following questions: 14. If x and y are real numbers and the complex number

 2  i  x  i 1  i  y  2i 4i



4i

is pure real, the relation between x and

y is (a) 8x – 17y = 16 15. If z 

(b) 8x + 17y = 16

(c) 17x – 8y = 16

(d) 17x – 8y = –16

3  2i sin     0     is pure imaginary, then  is equal to 1  2i sin   2

(a) /4

(b) /6

(c) /3

(d) /12

z1  z2 16. If z1 and z2 are complex numbers such that z  z  1 then 1 2 (a) z1/z2 is pure real

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(b) z1/z2 is pure imaginary (c) z1 is pure real

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(d) z1 and z2 are pure imaginary

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