Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.9

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.9

Choose the correct or the most suitable answer from the given four alternatives:

Question 1.
in + in+1 + in+2 + in+3 is ______
(a) 0
(b) 1
(c) -1
(d) i
(a) 0

Question 2.
The value of $$\sum_{i=1}^{13}\left(i^{n}+i^{n-1}\right)$$ is ______
(a) 1 + i
(b) i
(c) 1
(d) 0
(a) 1 + i

Question 3.
The area of the triangle formed by the complex numbers z, iz, and z + iz in the Argand’s diagrams is ______
(a) $$\frac { 1 }{ 2 }$$ |z|
(b) |z|2
(c) $$\frac { 3 }{ 2 }$$|z|2
(d) 2|z|2
(a) $$\frac { 1 }{ 2 }$$ |z|
Hint: Area of triangle = $$\frac { 1 }{ 2 }$$ bh
= $$\frac { 1 }{ 2 }$$ |z| |iz|
= $$\frac { 1 }{ 2 }$$ |z|2

Question 4.
The conjugate of a complex number is $$\frac{1}{i-2}$$. Then, the complex number is _____
(a) $$\frac{1}{i+2}$$
(b) $$\frac{-1}{i+2}$$
(c) $$\frac{-1}{i-2}$$
(d) $$\frac{1}{i-2}$$
(b) $$\frac{-1}{i+2}$$
Hint:

Question 5.
If $$z=\frac{(\sqrt{3}+i)^{3}(3 i+4)^{2}}{(8+6 i)^{2}}$$ then |z| is equal to ______
(a) 0
(b) 1
(c) 2
(d) 3
(c) 2
Hint:

Question 6.
If z is a non zero complex number, such that 2iz2 = $$\bar{z}$$ then |z| is ______
(a) $$\frac { 1 }{ 2 }$$
(b) 1
(c) 2
(d) 3
(a) $$\frac { 1 }{ 2 }$$
Hint:

Question 7.
If |z – 2 + i| ≤ 2, then the greatest value of |z| is _______
(a) √3 – 2
(b) √3 + 2
(c) √5 – 2
(d) √5 + 2
(d) √5 + 2
Hint:

Question 8.
If $$\left|z-\frac{3}{z}\right|$$, then the least value of |z| is ______
(a) 1
(b) 2
(c) 3
(d) 5
(a) 1
Hint:

Question 9.
If |z| = 1, then the value of $$\frac{1+z}{1+\bar{z}}$$ is _______
(a) z
(b) $$\bar{z}$$
(c) $$\frac { 1 }{ z }$$
(d) 1
(a) z
Hint:

Question 10.
The solution of the equation |z| – z = 1 + 2i is _______
(a) $$\frac { 3 }{ 2 }$$ – 2i
(b) –$$\frac { 3 }{ 2 }$$ + 2i
(c) 2 – $$\frac { 3 }{ 2 }$$i
(d) 2 + $$\frac { 3 }{ 2 }$$i
(a) $$\frac { 3 }{ 2 }$$ – 2i

Question 11.
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1 z2 + 4z1 z3 + z2 z3| = 12, then the value of |z1 + z2 + z3| is ______
(a) 1
(b) 2
(c) 3
(d) 4
(b) 2
Hint: |z1 + z2 + z3| = 2

Question 12.
If z is a complex number such that z ∈ C/R and z + $$\frac { 1 }{ z }$$ ∈ R, then |z| is ______
(a) 0
(b) 1
(c) 2
(d) 3
(b) 1
Hint: We have

Question 13.
z1, z2 and z3 are complex numbers such that z1 + z2 + z3 = 0 and |z1| = |z2| = |z3| = 1 then $$z_{1}^{2}+z_{2}^{2}+z_{3}^{2}$$ is ______
(a) 3
(b) 2
(c) 1
(d) 0
(d) 0
Hint:

Question 14.
If $$\frac{z-1}{z+1}$$ is purely imaginary, then |z| is _______
(a) $$\frac { 1 }{ 2 }$$
(b) 1
(c) 2
(d) 3
(b) 1

Question 15.
If z = x + iy is a complex number such that |z + 2| = |z – 2|, then the locus of z is _____
(a) real axis
(b) imaginary axis
(c) ellipse
(d) circle
(b) imaginary axis
Hint:
|z + 2| = | z – 2|
⇒ |x + iy + 2| = |x + iy – 2|
⇒ |x + 2 + iy|2 = |x – 2 + iy|2
⇒ (x + 2)2 + y2 = (x – 2)2 + y2
⇒ x2 + 4 + 4x = x2 + 4 – 4x
⇒ 8x = 0
⇒ x = 0

Question 16.
The principal argument of $$\frac{3}{-1+i}$$ is ______
(a) $$-\frac{5 \pi}{6}$$
(b) $$-\frac{2 \pi}{3}$$
(c) $$-\frac{3 \pi}{4}$$
(d) $$\frac{-\pi}{2}$$
(c) $$-\frac{3 \pi}{4}$$
Hint:

The complex number lies in III quadrant. $$\theta=-(\pi-\alpha)=-\left(\pi-\frac{\pi}{4}\right)=\frac{-3 \pi}{4}$$

Question 17.
The principal argument of (sin 40° + i cos 40°)5 is ______
(a) -110°
(b) -70°
(c) 70°
(d) 110°
(a) -110°
Hint:
(sin 40° + i cos 40°)5
= (cos 50° + i sin 50°)5
= (cos 250° + i sin 250°)
To find the principal argument the rotation must be in a clockwise direction which coincides with 250°
θ = -110°

Question 18.
If (1 + i) (1 + 2i) (1 + 3i)…(1 + ni) = x + iy, then 2.5.10……(1 + n2 ) is _____
(a) 1
(b) i
(c) x2 + y2
(d) 1 + n2
(c) x2 + y2

Question 19.
If ω ≠ 1 is a cubic root of unity and (1 + ω)7 = A + B ω, then (A, B) equal to ______
(a) (1, 0)
(b) (-1, 1)
(c) (0, 1)
(d) (1, 1)
(d) (1, 1)
Hint:

Question 20.
The principal argument of the complex number $$\frac{(1+i \sqrt{3})^{2}}{4 i(1-i \sqrt{3})}$$ is _____
(a) $$\frac{2 \pi}{3}$$
(b) $$\frac{\pi}{6}$$
(c) $$\frac{5 \pi}{6}$$
(d) $$\frac{\pi}{2}$$
(d) $$\frac{\pi}{2}$$
Hint:

Question 21.
If α and β are the roots of x2 + x + 1 = 0, then α2020 + β2020 is ______
(a) -2
(b) -1
(c) 1
(d) 2
(b) -1
Hint:
x2 + x + 1 = 0
α and β are the roots of the equation.
There are the two roots of cube roots of unity except 1.

Question 22.
The product of all four values of $$\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)^{\frac{3}{4}}$$ is ____
(a) -2
(b) -1
(c) 1
(d) 2
(c) 1

Question 23.
If ω ≠ 1 is a cubic root of unity and $$\left|\begin{array}{cccc}{1} & {1} & {1} \\ {1} & {-\omega^{2}-1} & {\omega^{2}} \\ {1} & {\omega^{2}} & {\omega^{7}}\end{array}\right|$$ = 3k, then k is equal to ______
(a) 1
(b) -1
(c) i√3
(d) -i√3
(d) -i√3
Hint:

Question 24.
The value of $$\left(\frac{1+\sqrt{3} i}{1-\sqrt{3} i}\right)^{10}$$ is ______
(a) cis $$\frac{2 \pi}{3}$$
(b) cis $$\frac{4 \pi}{3}$$
(c) -cis $$\frac{2 \pi}{3}$$
(d) -cis $$\frac{4 \pi}{3}$$
(a) cis $$\frac{2 \pi}{3}$$
Hint:

Question 25.
If ω = cis $$\frac{2 \pi}{3}$$, then the number of distinct roots of $$\left|\begin{array}{ccc}{z+1} & {\omega} & {\omega^{2}} \\ {\omega} & {z+\omega^{2}} & {1} \\ {\omega^{2}} & {1} & {z+\omega}\end{array}\right|=0$$ are _____
(a) 1
(b) 2
(c) 3
(d) 4