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Questions >> TN XII Math >> Complex Numbers

Find all the values of $\bigg(\large\frac{1}{2}$ $-i\large\frac{\sqrt 3}{2}\bigg)^{\Large\frac{3}{4}}$ and hence prove that the product of the values is 1.

answered Jun 13, 2013 by sreemathi.v

1 answer

Solve : $x^4-x^3+x^2-x+1=0$

answered Jun 13, 2013 by sreemathi.v

1 answer

Solve : $x^4+4=0$

answered Jun 13, 2013 by sreemathi.v

1 answer

Prove that if $\omega^3=1$ ,then (iii) $\large\frac{1}{1+2\omega}-\frac{1}{1+\omega}+\frac{1}{2+\omega}$ $=0$

answered Jun 12, 2013 by sreemathi.v

1 answer

Prove that if $\omega^3=1$ ,then (ii) $\bigg(\large\frac{-1+i\sqrt 3}{2}\bigg)^5+\bigg(\large\frac{-1-i\sqrt 3}{2}\bigg)^5$ $=-1$

answered Jun 12, 2013 by sreemathi.v

1 answer

Prove that if $\omega^3=1$ ,then (i) $(a+b+c)(a+b\omega+c\omega^2)(a+b\omega^2+c\omega)=a^3+b^3+c^3-3abc$

answered Jun 12, 2013 by sreemathi.v

1 answer

If $x=a+b$ , $y=a\omega+b\omega^2,z=a\omega^2+b\omega$ show that (ii) $x^3+y^3+z^3=3(a^3+b^3)$ where $\omega$ is the complex cube root of unity.

answered Jun 12, 2013 by sreemathi.v

1 answer

If $x=a+b$ , $y=a\omega+b\omega^2,z=a\omega^2+b\omega$ show that (i) $xyz=a^3+b^3$ where $\omega$ is the complex cube root of unity.

answered Jun 12, 2013 by sreemathi.v

1 answer

Find all the value of the following : $(-\sqrt 3-i)^{\large\frac{2}{3}}$

answered Jun 12, 2013 by sreemathi.v

1 answer

Find all the value of the following : $(8i)^{\large\frac{1}{3}}$

answered Jun 12, 2013 by sreemathi.v

1 answer

Find all the value of the following : $(i)^{\large\frac{1}{3}}$

answered Jun 12, 2013 by sreemathi.v

1 answer

If $x\;=\cos\;\alpha +i\sin\;\alpha\;;\; y\;= \cos\;\beta + i\sin \;\beta$ prove that $x^{m}y^{n} \;+\large \frac{1}{x^{m}y^{n}}$ $= \;2\cos \;\left ( m\alpha + n\beta \right )$

answered Jun 12, 2013 by sreemathi.v

1 answer

If $x+\large\frac{1}{x}$ $=2\cos\;\theta$ and $y+\large\frac{1}{y}$ $=2 \cos \;\phi$ show that $\large\frac{x^{m}}{y^{n}} - \frac{y^{n}}{x^{m}}$ $= 2i \;sin\left ( m\theta -n\phi \right )$

answered Jun 12, 2013 by sreemathi.v

1 answer

If $x+\large\frac{1}{x}$ $=2\cos\;\theta$ and $y+\large\frac{1}{y}=$ $2 \cos \;\phi$ show that $\large\frac{x^{m}}{y^{n}}+ \frac{y^{n}}{x^{m}}$ $= 2 \cos \left ( m\theta -n\phi \right )$

answered Jun 12, 2013 by sreemathi.v

1 answer

If $x +\large\frac{1}{x} =$ $2\; cos\;\theta$ prove that $x^{n} -\large \frac{1}{x^{n}}=$ $2i\; sin\;n\theta$

answered Jun 12, 2013 by sreemathi.v

1 answer

If $\; x+\large\frac{1}{x}=$ $\;2\;\cos\;\theta \;$ prove that $\;x^{n}+\large\frac{1}{x^{n}}$ $= \;2\;\cos\;n\theta\;$

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\alpha$ and $\beta$ are the roots of $x^{2}-2x+4=0$ . Prove that $\alpha ^{n}-\beta ^{n}=i2^{n+1}sin\large\frac{n\pi }{3}$ and calculate $\alpha ^{9}-\beta ^{9}$ , where n $\in$ N?

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-2px+\left ( p^{2} + q^{2}\right )=0$ and $tan \; \theta =\large \frac{q}{y+p}$ show that $\large\frac{\left (y+\alpha \right )^{n}-\left ( y+\beta \right )^{n}}{\alpha -\beta }$ = $q^{n-1}\large\frac{sin \;n\theta }{sin\;^{n}\theta }$

answered Jun 11, 2013 by sreemathi.v

1 answer

Prove that $\left ( 1+\mathit{i} \right )^{4n}$ and $\left ( 1+\mathit{i} \right )^{4n+2}$ are real and purely imaginary respectively.

answered Jun 11, 2013 by sreemathi.v

1 answer

Prove that $\left ( 1+ \cos \; \theta + \mathit{i}\sin \; \theta \right )^{\mathit{n}} + \left ( 1+ \cos \; \theta - \mathit{i}sin \; \theta \right )^{\mathit{n}} = 2^{\mathit{n+1}} \cos ^{\mathit{n}}\left ( \large\frac{\theta}{2} \right )$ $\cos \;\large\frac{n\theta }{2}$

answered Jun 11, 2013 by sreemathi.v

1 answer

Prove that $\left ( 1+ \mathit{i\sqrt{3}} \right )^{\mathit{n}} + \left ( 1- \mathit{i\sqrt{3}} \right )^{\mathit{n}} = 2^{n+1} \cos \; \large\frac{n\pi}{3}$

answered Jun 11, 2013 by sreemathi.v

1 answer

Prove that $\left ( 1+\mathit{i} \right )^{\mathit{n}} + \left ( 1-\mathit{i} \right )^{\mathit{n}} = 2^{\large\frac{n+2}{2}} \cos \; \large\frac{n\pi}{4}$

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\cos\;\alpha + \cos\;\beta + \cos\;\gamma = 0 = \sin \;\alpha + \sin \;\beta + \sin \;\gamma$ , prove that $\cos^{2}\;\alpha + \cos^{2}\;\beta + \cos^{2}\;\gamma = \sin^{2} \alpha + \sin^{2}\beta + \sin^{2}\gamma = \large\frac{3}{2}$ .

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\cos\;\alpha + \cos\;\beta + \cos\;\gamma = 0 = \sin \;\alpha + \sin \;\beta + \sin \;\gamma$ , prove that $\sin 2\alpha + \sin 2\beta + \sin 2\gamma = 0$

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\cos \alpha +\cos \beta + \cos \gamma = 0 = \sin \alpha + \sin \beta + \sin \gamma$ , prove that $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 0$

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\cos \alpha +\cos \beta + \cos \gamma = 0 = \sin \alpha + \sin \beta + \sin \gamma$ , prove that $\sin 3\alpha + \sin 3\beta + \sin 3\gamma = 3 \sin\left ( \alpha +\beta + \gamma \right )$

answered Jun 11, 2013 by sreemathi.v

1 answer

If $\cos \alpha + cos \beta + cos \gamma = 0 = sin \alpha + sin \beta + sin \gamma$ , prove that $cos 3\alpha + cos 3\beta + cos 3\gamma = 3 cos\left ( \alpha +\beta + \gamma \right )$

answered Jun 11, 2013 by sreemathi.v

1 answer

Simplify: $\Large\frac{\left (cos \;\alpha + \mathit{i} sin \; \alpha \right )^{3}}{\left (sin \;\beta + \mathit{i} cos \;\beta \right)^{4} } \Large$

answered Jun 11, 2013 by sreemathi.v

1 answer

Simplify: $\Large\frac{\left ( cos2 \theta -\mathit{i}sin2 \theta \right )^{7} \left ( cos 3 \theta +\mathit{i} sin3 \theta \right )^{-5}}{\left (cos 4 \theta + \mathit{i} sin4 \theta \right )^{12} \left ( cos 5 \theta -\mathit{i}sin5 \theta \right )^{-6}}\Large$

answered Jun 11, 2013 by sreemathi.v

1 answer

Solve : $6x^{4}-25x^{3}+32x^{2}+3x-10=0$ given that one of the roots is $2-i$ .

answered Jun 11, 2013 by sreemathi.v

1 answer

Solve the equation $x^{4}-4x^{3}+11x^{2}-14x+10=0$ if one root is $1+2i$ .

answered Jun 10, 2013 by sreemathi.v

1 answer

Solve the equation $x^{4}-8x^{3}+24x^{2}-32x+20$ = 0 if $3+\mathit{i}$ is a root.

answered Jun 10, 2013 by sreemathi.v

1 answer

$P$ represents the variable complex number $z$ .Find the locus of $P$ ,if $arg\bigg(\large\frac{z-1}{z+3}\bigg)=\large\frac{\pi}{2}$

answered Jun 10, 2013 by sreemathi.v

1 answer

$P$ represents the variable complex number $z$ .Find the locus of $P$ ,if $\mid 2z-3\mid=2$

answered Jun 10, 2013 by sreemathi.v

1 answer

$P$ represents the variable complex number $z$ .Find the locus of $P$ ,if $Re\bigg(\large\frac{z-1}{z+i}\bigg)$ $=1$

answered Jun 10, 2013 by sreemathi.v

1 answer

$P$ represents the variable complex number $z$ .Find the locus of $P$ ,if $\mid z-5i\mid=\mid z+5i\mid$

answered Jun 10, 2013 by sreemathi.v

1 answer

$P$ represents the variable complex number $z$ .Find the locus of $P$ ,if $Im\bigg[\large\frac{2z+1}{iz+1}\bigg]$ $=-2$

answered Jun 10, 2013 by sreemathi.v

1 answer

If arg $\left ( z-1 \right )$ = $\large\frac{\pi}{6}$ and arg $\left ( z+1 \right )$ = 2 $\large\frac{\pi}{3}$ then prove that $\left | z \right |=1$

answered Jun 10, 2013 by sreemathi.v

1 answer

Express the following complex numbers in polar form. $1 - \mathit{i}$

answered Jun 10, 2013 by sreemathi.v

1 answer

Express the following complex numbers in polar form. $-1 - \mathit{i}$

answered Jun 10, 2013 by sreemathi.v

1 answer

Express the following complex number in polar form. $-1 + \mathit{i}\sqrt{3}$

answered Jun 10, 2013 by sreemathi.v

1 answer

Express the following complex number in polar form. $2 + 2\sqrt{3}\mathit{i}$

answered Jun 10, 2013 by sreemathi.v

1 answer

Prove that the points representing the complex numbers $\left ( 7+5\mathit{i} \right )$ , $\left ( 5+2\mathit{i} \right )$ , $\left ( 4+7\mathit{i} \right )$ and $\left ( 2+4\mathit{i} \right )$ form a parallelogram. (Plot the points and use midpoint formula).

answered Jun 10, 2013 by sreemathi.v

1 answer

Prove that the triangle formed by the points representing the complex numbers $(10+8i),(-2+4i)$ and $(-11+31i)$ on the Argand plane is right angled.

answered Jun 10, 2013 by sreemathi.v

1 answer

If $z^2=(0,1)$ find $z$ .

answered Jun 7, 2013 by sreemathi.v

1 answer

Find the square root of $(-8-6i)$

answered Jun 7, 2013 by sreemathi.v

1 answer

If $\left ( 1+\mathit{i} \right )\left ( 1+2\mathit{i} \right )\left ( 1+3\mathit{i} \right )... \left ( 1+\mathit{ni} \right )=\mathit{x+iy}$ , show that 2.5.10 ... $\left ( 1+n^{2} \right )$ = $x^{2}+y^{2}$

answered Jun 7, 2013 by sreemathi.v

1 answer

For what values of $\mathit{x}$ and $\mathit{y}$ , the numbers $-3+\mathit{ix^{2}y}$ and $\mathit{x^{2}}+y+4\mathit{i}$ are complex conjugate of each other $?$

answered Jun 7, 2013 by sreemathi.v

1 answer

Find the real values of $\mathit{x}$ and $\mathit{y}$ for which the following equation: $\sqrt{x^{2}+3x+8} + \left ( x+4 \right )\mathit{i} = \mathit{y}\left (2 + \mathit{i} \right )$

answered Jun 7, 2013 by sreemathi.v

1 answer

Find the real values of $\mathit{x}$ and $\mathit{y}$ for which the following equation: $\large\frac{\left ( 1+\mathit{i} \right )\mathit{x}-2\mathit{i}}{3+\mathit{i}} + \frac{\left ( 2-3\mathit{i} \right )\mathit{y}+\mathit{i}}{3-\mathit{i}} =$ $\mathit{i}$

answered Jun 7, 2013 by sreemathi.v

1 answer

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