Mathematics Multiple Choice Questions on “Complex Numbers”.
1. The value of x and y if (3y – 2) + i(7 – 2x) = 0
(a) x = 7/2, y = 2/3
(b) x = 2/7, y = 2/3
(c) x = 7/2, y = 3/2
(d) x = 2/7, y = 3/2
Clarification: x = 7/2, y = 2/3
Hint:
Given, (3y – 2) + i(7 – 2x) = 0
Compare real and imaginary part, we get
3y – 2 = 0
⇒ y = 2/3
and 7 – 2x = 0
⇒ x = 7/2
So, the value of x = 7/2 and y = 2/3
2. Is i(iota) a root of 1+x2=0?
a) True
b) False
Answer: a
Clarification: 1+x2 = 0
1 + i2 = 1 – 1 = 0.
So, it is a root of 1 + x2 = 0.
3. In z=4+i, what is the real part?
a) 4
b) i
c) 1
d) 4+i
Answer: a
Clarification: In z=a+bi, a is real part and b is imaginary part.
So, in 4+i, real part is 4.
4. In z=4+i, what is imaginary part?
a) 4
b) i
c) 1
d) 4+i
Answer: c
Clarification: In z=a+bi, a is real part and b is imaginary part.
So, in 4+i, imaginary part is 1.
5. (x+3) + i(y-2) = 5+i2, find the values of x and y.
a) x=8 and y=4
b) x=2 and y=4
c) x=2 and y=0
d) x=8 and y=0
Answer: b
Clarification: If two complex numbers are equal, then corresponding parts are equal i.e. real parts of both are equal and imaginary parts of both are equal.
x+3 = 5 and y-2 = 2
x = 5-3 and y = 2+2
x=2 and y=4.
6. If z1 = 2+3i and z2 = 5+2i, then find sum of two complex numbers.
a) 4+8i
b) 3-i
c) 7+5i
d) 7-5i
Answer: c
Clarification: In addition of two complex numbers, corresponding parts of two complex numbers are added i.e. real parts of both are added and imaginary parts of both are added.
So, sum = (2+5) + (3+2) i = 7+5i.
7. 0+0i is ______________________for complex number z.
a) additive inverse
b) additive identity element
c) multiplicative identity element
d) multiplicative inverse
Answer: b
Clarification: On adding zero (0+0i) to a complex number, we get same complex number so 0+0i is additive identity element for complex number z i.e. z+0 = z.
8. 1+0i is _________________ for complex number z.
a) additive inverse
b) additive identity element
c) multiplicative identity element
d) multiplicative inverse
Answer: c
Clarification: On multiplying one (1+0i) to a complex number, we get same complex number so 1+0i is multiplicative identity element for complex number z i.e. z*1=z.
9. -z is _________________ for complex number z.
a) additive inverse
b) additive identity element
c) multiplicative identity element
d) multiplicative inverse
Answer: a
Clarification: On adding negative of complex number (-z) to complex number z, we get additive identity element zero i.e. z+(-z)=0.
10. 1/z is _________________ for complex number z.
a) additive inverse
b) additive identity element
c) multiplicative identity element
d) multiplicative inverse
Answer: d
Clarification: On multiplying reciprocal of complex number (1/z) to complex number z, we get multiplying inverse one i.e. z*1=z.
11. If z1 = 2+3i and z2 = 5+2i, then find z1-z2.
a) -3+1i
b) 3-i
c) 7+5i
d) 7-5i
Answer: a
Clarification: In subtracting one complex number from other, difference of corresponding parts of two complex numbers is calculated. So, z1-z2 = (2-5) + (3-2) i = -3+1i.
12. Value of i(iota) is ____________
a) -1
b) 1
c) (-1)1/2
d) (-1)1/4
Answer: c
Clarification: Iota is used to denote complex number.
The value of i (iota) is (sqrt{-1}) i.e. (-1)1/2.
13. Find real θ such that (3 + 2i × sin θ)/(1 – 2i × sin θ) is imaginary
(a) θ = nπ ± π/2 where n is an integer
(b) θ = nπ ± π/3 where n is an integer
(c) θ = nπ ± π/4 where n is an integer
(d) None of these
14. If {(1 + i)/(1 – i)}n = 1 then the least value of n is
(a) 1
(b) 2
(c) 3
(d) 4
15. If arg (z) < 0, then arg (-z) – arg (z) =
(a) π
(b) -π
(c) -π/2
(d) π/2
16. if x + 1/x = 1 find the value of x2000 + 1/x2000 is
(a) 0
(b) 1
(c) -1
(d) None of these
17. The value of √(-144) is
(a) 12i
(b) -12i
(c) ±12i
(d) None of these
18. If the cube roots of unity are 1, ω, ω², then the roots of the equation (x – 1)³ + 8 = 0 are
(a) -1, -1 + 2ω, – 1 – 2ω²
(b) – 1, -1, – 1
(c) – 1, 1 – 2ω, 1 – 2ω²
(d) – 1, 1 + 2ω, 1 + 2ω²
19. (1 – w + w²)×(1 – w² + w4)×(1 – w4 + w8) × …………… to 2n factors is equal to
(a) 2n
(b) 22n
(c) 23n
(d) 24n
20. The modulus of 5 + 4i is
(a) 41
(b) -41
(c) √41
(d) -√41