Jee Math Test Series Quiz-2

A value of $\theta$ for which $\frac{2+3 i \sin \theta}{1-2 i \sin \theta}$ is purely imaginary, is:

If $z$ is a non-real complex number, then the minimum value of $\frac{\operatorname{Im} z^5}{(I m z)^5}$ is :

If $z$ is a complex number such that $|z| \geq 2$, then the minimum value of $\left|z+\frac{1}{2}\right|$ :

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

Let $z \neq-i$ be any complex number such that $\frac{z-i}{z+i}$ is a purely imaginary number. Then $z+\frac{1}{z}$ is:

If $z_1, z_2$ and $z_3, z_4$ are 2 pairs of complex conjugate numbers, then
$\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right) \text { equals: }$

Let $w(\operatorname{Im} w \neq 0)$ be a complex number. Then the set of all complex number $\mathrm{z}$ satisfying the equation $\mathrm{w}-\overline{\mathrm{w}} \mathrm{z}=\mathrm{k}(1-\mathrm{z})$, for some real number $\mathrm{k}$, is

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg \left(\frac{1+z}{1+\bar{z}}\right)$ equals:

Let $z$ satisfy $|z|=1$ and $z=1-\bar{z}$.
Statement $1: z$ is a real number.
Statement 2 : Principal argument of $z$ is $\frac{\pi}{3}$

Let $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$, where $z$ is any non-zero complex number.
The set $\mathrm{A}=\{a:|z|=1$ and $z \neq \pm 1\}$ is equal to:

If $Z_1 \neq 0$ and $Z_2$ be two complex numbers such that $\frac{Z_2}{Z_1}$ is a purely imaginary number, then $\left|\frac{2 Z_1+3 Z_2}{2 Z_1-3 Z_2}\right|$ is equal to:

$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$ is equal to

Let $Z$ and $W$ be complex numbers such that $|Z|=|W|$, and arg $Z$ denotes the principal argument of $Z$.
Statement 1:If $\arg Z+\arg W=\pi$, then $Z=-\bar{W}$.
Statement 2: $|Z|=\mid W$, implies $\arg Z-\arg \bar{W}=\pi$.

Let $Z_1$ and $Z_2$ be any two complex number.
Statement 1: $\left|Z_1-Z_2\right| \geq\left|Z_1\right|-\left|Z_2\right|$
Statement 2: $\left|Z_1+Z_2\right| \leq\left|Z_1\right|+\left|Z_2\right|$

The number of complex numbers $\mathrm{z}$ such that $|z-1|=|z+1|=|z-i|$ equals

The conjugate of a complex number is $\frac{1}{i-1}$ then that complex number is

If, for a positive integer $n$, the quadratic equation, $x(x+1)+(x+1)(x+2)+\ldots . .+(x+\overline{n-1})(x+n)=10 n$ has two consecutive integral solutions, then $\mathrm{n}$ is equal to :

The sum of all the real values of $x$ satisfying the equation $2^{(x-1)\left(x^2+5 x-50\right)}=1$ is:

Let $p(x)$ be a quadratic polynomial such that $p(0)=1$. If $p(x)$ leaves remainder 4 when divided by $x-1$ and it leaves remainder 6 when divided by $x+1$; then :

The sum of all real values of $x$ satisfying the equation $\left(x^2-5 x+5\right)^{x^2+4 x-60}=1$ is:

If $x$ is a solution of the equation, $\sqrt{2 x+1}-\sqrt{2 x-1}=1,\left(x \geq \frac{1}{2}\right)$, then $\sqrt{4 x^2-1}$ is equal to :

$\text { Let } \alpha \text { and } \beta \text { be the roots of equation } x^2-6 x-2=0 \text {. If } a_n=\alpha^n$
$-\beta^n \text {, for } \mathrm{n} \geq 1 \text {, then the value of } \frac{\mathrm{a}_{10}-2 \mathrm{a}_8}{2 \mathrm{a}_9} \text { is equal to: }$

If the two roots of the equation, $(a-1)\left(x^4+x^2+1\right)+$ $(a+1)\left(x^2+x+1\right)^2=0$ are real and distinct, then the set of all values of ‘ $a$ ‘ is :

If $2+3 i$ is one of the roots of the equation $2 x^3-9 x^2+k x-13$ $=0, k \in R$, then the real root of this equation:

If $a \in \mathrm{R}$ and the equation
$-3(x-[x])^2+2(x-[x])+a^2=0$
(where $[x]$ denotes the greatest integer $\leq x$ ) has no integral solution, then all possible values of a lie in the interval:

The equation $\sqrt{3 x^2+x+5}=x-3$, where $x$ is real, has;

The sum of the roots of the equation, $\mathrm{x}^2+|2 \mathrm{x}-3|-4=0$, is:

If $\alpha$ and $\beta$ are roots of the equation, $x^2-4 \sqrt{2} k x+2 e^{4 l n k}-1=0$ for some $k$, and $\alpha^2+\beta^2=66$, then $\alpha^3+\beta^3$ is equal to:

If $\frac{1}{\sqrt{\alpha}}$ and $\frac{1}{\sqrt{\beta}}$ are the roots of the equation, $a x^2+b x+1=0(a \neq 0, a, b, \in R)$, then the equation, $x\left(x+b^3\right)+\left(a^3-3 a b x\right)=0$ as roots:

If $f(x)=\left(\frac{3}{5}\right)^x+\left(\frac{4}{5}\right)^x-1, x \in R$, then the equation $f(x)=0$ has :

If $a, b, c$ are distinct +ve real numbers and $a^2+b^2+c^2=1$ then $a b+b c+c a$ is

The region represented by $\{z=x+i y \in \mathrm{C}:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality:

Consider the two sets :
$A=\left\{m \in \mathbf{R}\right.$ : both the roots of $x^2-(m+1) x+m+4=0$ are real $\}$ and $B=[-3,5)$.
Which of the following is not true?

If $\mathrm{A}=\{x \in \mathrm{R}:|x|<2\}$ and $\mathrm{B}=\{x \in \mathrm{R}:|x-2| \geq 3\}$; then :

Let $S$ be the set of all real roots of the equation, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$. Then $S$ :

All the pairs $(x, y)$ that satisfy the inequality $2 \sqrt{\sin ^2 x-2 \sin x+5} \cdot \frac{1}{4 \sin ^2 y} \leq 1$ also satisfy the equation:

The number of integral values of $\mathrm{m}$ for which the quadratic expression, $(1+2 \mathrm{~m}) x^2-2(1+3 \mathrm{~m}) x+4(1+\mathrm{m}), x \in \mathrm{R}$, is always positive, is :

If $\alpha \in\left(0, \frac{\pi}{2}\right)$ then $\sqrt{x^2+x}+\frac{\tan ^2 \alpha}{\sqrt{x^2+x}}$ is always greater than or equal to

The set of all real numbers $x$ for which $x^2-|x+2|+x>0$, is

If $a_1, a_2 \ldots \ldots, a_n$ are positive real numbers whose product is a fixed number $\mathrm{c}$, then the minimum value of $a_1+a_2+\ldots \ldots+a_{n-1}+2 a_n$ is

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^3}$ equals :

$\lim _{x \rightarrow 3} \frac{\sqrt{3 x}-3}{\sqrt{2 x-4}-\sqrt{2}}$ is equal to :

$\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x} \text { is equal to : }$

$\lim _{x \rightarrow 0} \frac{e^{x^2}-\cos x}{\sin ^2 x} \text { is equal to : }$

$\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ is equal to:

If $\lim _{x \rightarrow 2} \frac{\tan (x-2)\left\{x^2+(k-2) x-2 k\right\}}{x^2-4 x+4}=5$
then $\mathrm{k}$ is equal to:

$\lim _{x \rightarrow 0}\left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right)$

Let $f: R \rightarrow[0, \infty)$ be such that $\lim _{x \rightarrow 5} f(x)$ exists and $\lim _{x \rightarrow 5} \frac{(f(x))^2-9}{\sqrt{|x-5|}}=0$
Then $\lim _{x \rightarrow 5} f(x)$ equals :

$\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}\right)$

Let $f: R \rightarrow R$ be a positive increasing function with $\lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1$ then $\lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=$