NDA MATHEMATICS QUIZ - 3

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Question 1:

If $z = (t - 4) + i \sqrt{25 - t^{2}}$ then locus of $z$ will be:

यदि $z = (t - 4) + i \sqrt{25 - t^{2}}$ है तो $z$ का बिंदु पथ क्या होगा ?

circle

parabola

straight line

an ellipse

Question 2:

If $z_{1} = \cos \frac{π}{2} + i \sin \frac{π}{2}, z_{2} = \cos \frac{π}{2^{2}} + i \sin \frac{π}{2^{2}}$ ,

$z_{3} = \cos \frac{π}{2^{3}} + i \sin \frac{π}{2^{3}} \dots .$ up to $\infty$ then $z_{1} z_{2} z_{3}$

$\dots \dots$ up to $\infty$ . . . . . . . . .

अगर $z_{1} = \cos \frac{π}{2} + i \sin \frac{π}{2}, z_{2} = \cos \frac{π}{2^{2}} + i \sin \frac{π}{2^{2}}$ ,

$z_{3} = \cos \frac{π}{2^{3}} + i \sin \frac{π}{2^{3}} \dots .$ तक $\infty$ फिर $z_{1} z_{2} z_{3}$

$\dots \dots$ $\infty$ तक का मान क्या होगा ?

- 1

- 1

2.0

+ 1

+ 1

3.0

Question 3:

Which of the following is not correct?

Arg (\frac{z_{1}}{z_{2}}) = Arg z_{2} - Arg z_{1}

Arg (\frac{z_{1}}{z_{2}}) = Arg z_{2} - Arg z_{1}

| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |

| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |

Arg | z_{1} \cdot z_{2} | = Arg | z_{1} | \cdot Arg | z_{2} |

Arg | z_{1} \cdot z_{2} | = Arg | z_{1} | \cdot Arg | z_{2} |

z^{2} = z \bar{z}

z^{2} = z \bar{z}

Question 4:

The locus of $z$ satisfying the inequality

$\log_{\frac{1}{5}} | (z - 1) | > \log_{\frac{1}{5}} | z + 1 |$

$z$ का बिन्दुपथ जो कि असमानता $\log_{\frac{1}{5}} | (z - 1) | > \log_{\frac{1}{5}} | z + 1 |$ को संतुष्ट करता है निम्न में से किस प्रकार का होगा ?

Re (z) > 0

Re (z) > 0

Re (z) < 0

Re (z) < 0

Re (z) = 0

Re (z) = 0

Im (z) = 0

Im (z) = 0

Question 5:

If $ω$ be the cube root of unity then

${(1 + ω - ω^{2})}^{7} + {(1 - ω + ω^{2})}^{7}$ is

अगर $ω$ इकाई का घनमूल हो तो

${(1 + ω - ω^{2})}^{7} + {(1 - ω + ω^{2})}^{7}$ का मान क्या होगा ?

128

Question 6:

If $z = \frac{(1 + i)^{2}}{2 - i}$ then $Im (z) =$ . . . . . . . .

अगर $z = \frac{(1 + i)^{2}}{2 - i}$ तो $Im (z) =$ . . . . . . . . . .

\frac{1}{5}

\frac{1}{5}

\frac{3}{5}

\frac{3}{5}

\frac{4}{5}

\frac{4}{5}

- \frac{2}{5}

- \frac{2}{5}

Question 7:

Value of $\sum_{r = 1}^{6} (\sin \frac{2 π r}{7} - i \cos \frac{2 π r}{7})$ is:

$\sum_{r = 1}^{6} (\sin \frac{2 π r}{7} - i \cos \frac{2 π r}{7})$ का मान क्या होगा ?

i

i

- 1

- 1

Question 8:

If $a + i b = \sqrt{\frac{x + i y}{p + i q}}$ then ${(a^{2} + b^{2})}^{2}$ equals:

अगर $a + i b = \sqrt{\frac{x + i y}{p + i q}}$ तो ${(a^{2} + b^{2})}^{2}$ किसके बराबर होगा ?

{(\frac{x^{2} + y^{2}}{p^{2} + q^{2}})}^{2}

{(\frac{x^{2} + y^{2}}{p^{2} + q^{2}})}^{2}

\frac{x^{2} + y^{2}}{p^{2} + q^{2}}

\frac{x^{2} + y^{2}}{p^{2} + q^{2}}

\sqrt{\frac{x^{2} + y^{2}}{p^{2} + q^{2}}}

\sqrt{\frac{x^{2} + y^{2}}{p^{2} + q^{2}}}

\sqrt{\frac{p^{2} + q^{2}}{x^{2} + y^{2}}}

\sqrt{\frac{p^{2} + q^{2}}{x^{2} + y^{2}}}

Question 9:

If $| z_{1} | = | z_{2} | = | z_{3} | = 1$ then $| z_{1} + z_{2} + z_{3} | =$

अगर $| z_{1} | = | z_{2} | = | z_{3} | = 1$ तब $| z_{1} + z_{2} + z_{3} | =$

| z_{1} | + | z_{2} | + | z_{3} |

| z_{1} | + | z_{2} | + | z_{3} |

| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |

| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |

| {\bar{z}}_{1} | + | {\bar{z}}_{2} | + | {\bar{z}}_{3} |

| {\bar{z}}_{1} | + | {\bar{z}}_{2} | + | {\bar{z}}_{3} |

| z_{1} | + 2 | z_{2} | + 3 | z_{3} |

| z_{1} | + 2 | z_{2} | + 3 | z_{3} |

Question 10:

Square root of $4 - 6 \sqrt{- 5}$ is:

$4 - 6 \sqrt{- 5}$ का वर्गमूल क्या होगा ?

3 - \sqrt{5} i

3 - \sqrt{5} i

1 + 3 \sqrt{2} i

1 + 3 \sqrt{2} i

3 + \sqrt{5} i

3 + \sqrt{5} i

2 - \sqrt{5} i

2 - \sqrt{5} i