In which year did Japan undergo the Meiji Restoration?

Important Questions on Complex Numbers and Quadratic Equations

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

A mass $M$ attached to a horizontal spring executes $\mathrm{S}.\mathrm{H}.\mathrm{M}.$ of amplitude $A_1$. When the mass $M$ passes through its mean position, then a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\left(\frac{A_1}{A_2}\right)$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The value of sum $\sum _{n=1}^{13}\left(i^n+i^{n+1}\right), \text{where} i=\sqrt{-1},$equals

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

A body of mass $M$ moving with velocity $v$ explodes into two equal parts. If one part comes to rest and the other part moves with velocity $v_0$; what would be the value of $v_0$?

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

For all complex numbers $z$ of the form $1+i\alpha , \alpha \in R,$ if $z^2=x+iy,$ then 

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Amongst the following statements, that which was not proposed by Dalton was :

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

For positive integers $n_1, n_2$ the value of the expression ${\left(1+i\right)}^{n_1}+{\left(1+i^3\right)}^{n_1}+{\left(1+i^5\right)}^{n_2}+{\left(1+i^7\right)}^{n_2}, i=\sqrt{-1}$ is a real number if and only if

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If $\left|\frac{1-iz}{z-i}\right|=1,$ then $z$ lies on

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The value of sum $\sum _{n=1}^{13}(i^n+i^{n+1})$, where $i=\sqrt{-1}$, equals

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If $z$ is a complex number such that $\mathrm{Re}\left(z\right)=\mathrm{I}\mathrm{m}\left(z\right)$, then

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If $\mathrm{\Delta }\mathrm{ }=\left|\begin{array}{lll}\sqrt{6}& 2i& 3+\sqrt{6}\\ \sqrt{12}& \sqrt{3}+\sqrt{8}i& 3\sqrt{2}+\sqrt{6}i\\ \sqrt{18}& \sqrt{2}+\sqrt{12}i& \sqrt{27}+2i\end{array}\right|$ , then $\mathrm{\Delta }$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The complex number which satisfy the equation $\mathrm{z}+\sqrt{2}\left|\mathrm{z}+1\right|+\mathrm{i}=0\mathrm{ }$ is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Dividing $f\left(z\right)$ by $z-i$, we obtain the remainder $i$ and dividing it by $z+i$, we get the remainder $1+i$. The remainder upon the division of $f\left(z\right)$ by $z^2+1$ is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The maximum value of $\left|z\right|$where $z$satisfies the condition $\left|z+\frac{2}{z}\right|=2$, is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If $z=x-iy$ and $z^{\frac{1}{3}}=p+iq$, then $\frac{\frac{x}{p}+\frac{y}{q}}{\left(p^2+q^2\right)}$ is equal to: (where $i=\sqrt{-1}$)

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If the complex number $z$ satisfy the equation $\left(i-z\right)\left(1+2i\right)+\left(1-iz\right)\left(3-4i\right)=1+7i,$ then $z+\overline{z}+z\overline{z}$ is equal to -

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The relation between the real numbers $\mathrm{a}$ and $\mathrm{b}$ , which satisfy the equation $\frac{1-\mathrm{i}\mathrm{x}}{1+\mathrm{i}\mathrm{x}}=\mathrm{a}\mathrm{ }–\mathrm{i}\mathrm{b}$ , for some real value of $\mathrm{x}$ , is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Evaluate ${\left(-\sqrt{-1}\right)}^{4n+3},n\in N$

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The equation, whose roots are $i,\frac{1+i}{\sqrt{2}},\frac{i-1}{\sqrt{2}},$ their conjugates, $-1 \mathrm{and}\mathrm{ }1$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Cotyledons are also called-