Complex Numbers - Maharashtra Board STD 11 Maths Questions

Q 1MCQ1 Mark

If z = x + iy and |z – zi| = 1, then

A
z lies on X-axis
B
z lies on Y-axis
C
z lies on a rectangle
✓
z lies on a circle

Answer: D.

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Q 2MCQ1 Mark

If $-1+\sqrt{ } 3 i = re ^{ i \theta}$, then $\theta=$

A
$-\frac{2 \pi}{3}$
B
$\frac{\pi}{3}$
C
$-\frac{\pi}{3}$
✓
$\frac{2 \pi}{3}$

Answer: D.

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Q 3MCQ1 Mark

If $\arg ( z )=\theta$, then $\arg \overline{( z )}=$

✓
-θ
B
θ
C
π – θ
D
π + θ

Answer: A.

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Q 4MCQ1 Mark

The modulus and argument of $(1+ i \sqrt{ } 3)^8$ are respectively

A
2 and $\frac{2 \pi}{3}$
B
256 and $\frac{8 \pi}{3}$
✓
256 and $\frac{2 \pi}{3}$
D
64 and $\frac{4 \pi}{3}$

Answer: C.

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Q 5MCQ1 Mark

If $\omega(\neq 1)$ is a cube root of unity and $(1+\omega)^7=A+B \omega$, then $A$ and $B$ are respectively the numbers

A
0, 1
✓
1, 1
C
1, 0
D
-1, 1

Answer: B.

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Q 6Solve the following Question.(1 Marks)1 Mark

Simplify
$\frac{ i ^{238}+ i ^{236}+ i ^{234}+ i ^{232}+ i ^{2290}}{ i ^{2228}+ i ^{226}+ i ^{224}+ i ^{222}+ i ^{2200}}$

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Q 7Solve the following Question.(1 Marks)1 Mark

Simplify
$\left(i^{65}+\frac{1}{i^{145}}\right)$

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Q 8Solve the following Question.(1 Marks)1 Mark

Simplify
$\frac{ i ^{29}+ i ^{39}+ i ^{49}}{ i ^{30}+ i ^{40}+ i ^{50}}$

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Q 9Solve the following Question.(1 Marks)1 Mark

Evaluate:
$i^{131}+i^{49}$

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Q 10Solve the following Question.(1 Marks)1 Mark

Evaluate:
$\left(1-i+i^2\right)^{-15}$

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Q 11Solve the Following Question.(2 Marks)2 Marks

If $\omega$ is the cube root of unity, then find the value of $\left(\frac{-1+i \sqrt{3}}{2}\right)^{18}+\left(\frac{-1-i \sqrt{3}}{2}\right)^{18}$

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Q 12Solve the Following Question.(2 Marks)2 Marks

Show that $\frac{1-2 i}{3-4 i}+\frac{1+2 i}{3+4 i}$ is real.

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Q 13Solve the Following Question.(2 Marks)2 Marks

Show that $z=\left(\frac{-1+\sqrt{-3}}{2}\right)^3$ is a rational number.

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Q 14Solve the Following Question.(2 Marks)2 Marks

Convert the complex numbers in polar form and also in exponential form.
z = -6 + √2 i

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Q 15Solve the Following Question.(2 Marks)2 Marks

Show that $\left(\frac{1+i}{\sqrt{2}}\right)^8+\left(\frac{1-i}{\sqrt{2}}\right)^8=2$

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Q 16Solve the Following Question.(3 Marks)3 Marks

If $\omega$ is a complex cube root of unity, prove that $\left(1-\omega+\omega^2\right)^6+\left(1+\omega-\omega^2\right)^6=128$.

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Q 17Solve the Following Question.(3 Marks)3 Marks

Simplify $\left[\frac{1}{1-2 i}+\frac{3}{1+i}\right]\left[\frac{3+4 i}{2-4 i}\right]$

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Q 18Solve the Following Question.(3 Marks)3 Marks

If $x+$ iy $=\frac{a+i b}{a-i b}$, prove that $x^2+y^2=1$.

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Q 19Solve the Following Question.(3 Marks)3 Marks

Show that $\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{10}+\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)^{10}=0$

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Q 20Solve the Following Question.(3 Marks)3 Marks

Find the real numbers $x$ and $y$ such that $\frac{x}{1+2 i}+\frac{y}{3+2 i}=\frac{5+6 i}{-1+8 i}$

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Q 21Solve the Following Question.(4 Marks)4 Marks

If $\alpha$ and $\beta$ are complex cube roots of unity, prove that $(1-\alpha)(1-\beta)\left(1-\alpha^2\right)\left(1-\beta^2\right)=9$.

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Q 22Solve the Following Question.(4 Marks)4 Marks

Convert the complex numbers in polar form and also in exponential form.

$\frac{-3}{2}+\frac{3 \sqrt{3} i}{2}$

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Q 23Solve the Following Question.(4 Marks)4 Marks

Convert the complex numbers in polar form and also in exponential form.

$z=\frac{2+6 \sqrt{3} i}{5+\sqrt{3} i}$

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Q 24Solve the Following Question.(4 Marks)4 Marks

Find the square roots of
6 + 8i

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Q 25Solve the Following Question.(4 Marks)4 Marks

Find the square roots of
3 – 4i

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Q 26Solve the Following Question.(5 Marks)5 Marks

If $x=a+b, y=\alpha a+\beta b$ and $z=a \beta+b \alpha$, where $\alpha$ and $\beta$ are complex cube roots of unity, show that $x y z=a^3+b^3$.

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Q 27Solve the Following Question.(5 Marks)5 Marks

Convert the complex number $z =\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ in the polar form.

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Q 28Solve the Following Question.(5 Marks)5 Marks

Find the modulus and argument of the complex number $\frac{1+2 i}{1-3 i}$.

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Q 29Solve the Following Question.(5 Marks)5 Marks

Solve the following quadratic equations:

$(2+i) x^2-(5-i) x+2(1-i)=0$

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Q 30Solve the Following Question.(5 Marks)5 Marks

Solve the following quadratic equations:

$x^2-(5-i) x+(18+i)=0$

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Complex Numbers question types

Complex Numbers questions

Generate a Complex Numbers paper free

Complex Numbers Maths - Complex Numbers

Complex Numbers question types

Complex Numbers questions

Generate a Complex Numbers paper free