Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $\mathrm{P}=\left[\mathrm{p}_{\mathrm{ij}}\right]$ be a $\mathrm{n} \times \mathrm{n}$...

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}$. Then the probability that the problem...

If $3^x=4^{x-1}$, then $x=$

Which of the given statement(s) about N, O, P and Q with respect to M is (are) correct ?

With reference to the scheme given, which of the given statments(s) about T, U, V and W is (are) correct?

Let $w=\frac{\sqrt{3}+i}{2}$ and $P=\left\{w^n: n=1,2,3, \ldots.\right\}$. Further $H_1=\left\{z \in C: \operatorname{Re} z>\frac{1}{2}\right\}$ and $H_2=\left\{z \in C: \operatorname{Re} z<-\frac{1}{2}\right\}$, where C is the set of all complex numbers. If $\mathrm{z}_1 \in \mathrm{P} \cap \mathrm{H}_1, \mathrm{z}_2 \in \mathrm{P} \cap \mathrm{H}_2$ and O represents the origin, then $\angle \mathrm{z}_1 O \mathrm{z}_2=$

In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle...

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2=9$, then $|2 \vec{a}+5 \vec{b}+5 \vec{c}|$ is

Two lines $\mathrm{L}_1: \mathrm{x}=5, \frac{\mathrm{y}}{3-\alpha}=\frac{\mathrm{z}}{-2}$ and $\mathrm{L}_2: \mathrm{x}=\alpha, \frac{\mathrm{y}}{-1}=\frac{\mathrm{z}}{2-\alpha}$ are coplanar. Then $\alpha$ can take value(s)

The value of $6+\log _{\frac{3}{2}}\left(\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \ldots .}}}\right)$ is

Let $f: \operatorname{IR} \rightarrow \operatorname{IR}$ be defined as $f(x)=|x|+\left|x^2-1\right|$. The total number of points at which $f$ attains either a...

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum...

For the given aqueous reaction which of the statement(s) is (are) true ?

Let $S$ be the focus of the parabola $y^2=8 x$ and let $P Q$ be the common chord of the...

Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, parallel to the straight line $2 x-y=1$. The points of contacts of the...

If S be the area of the region enclosed by $\mathrm{y}=\mathrm{e}^{-\mathrm{x}^2}, \mathrm{y}=0, \mathrm{x}=0$, and $\mathrm{x}=1$. Then

The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the...

Let $\theta, \phi \in[0,2 \pi]$ be such that $...