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 \(\lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=\)