For a polynomial $\mathrm{g}(\mathrm{x})$ with real coefficient, let $\mathrm{m}_{\mathrm{g}}$ denote the number of distinct real roots of $\mathrm{g}(\mathrm{x})$. Suppose S...

Let $\mathrm{x} \in \mathbb{R}$ and let $...

A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer...

In a triangle PQR , let $\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{QR}}, \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{RP}}$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{PQ}}$. If $|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{~b}}|=4$ and $\frac{\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}})}{\overrightarrow{\mathrm{c}} \cdot(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})}=\frac{|\overrightarrow{\mathrm{a}}|}{|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}|}$, then the value of...

Let f : [0, 2] -> R be the function defined by
$...

The inductors of two $L R$ circuits are placed next to each other, as shown in the figure. The values...

A cubical solid aluminium (bulk modulus $=-V \frac{\mathrm{dP}}{\mathrm{dV}}=70 \mathrm{GPa}$ ) block has an edge length of 1 m on the...

Let $a_1, a_2, a_3, \ldots$. be a sequence of positive integers in arithmetic progression with common difference 2 . Also,...

Two capacitors with capacitance values $\mathrm{C}_1=2000 \pm 10 \mathrm{pF}$ and $C_2=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage...

Let $m$ be the minimum possible value of $\log _3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$, where $y_1, y_2, y_3$ are real numbers for which $y_1+y_2+y_3=9$....

In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in...

A spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be...

Starting at time $t=0$ from the origin with speed $1 \mathrm{~ms}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$...

Let $\mathrm{L}_1$ and $\mathrm{L}_2$ be the following straight line.
$\mathbf{L}_1: \frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $\mathrm{L}_2: \frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$
Suppose the straight line...

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two...

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are lengths of the sides of...

Let S be the set of all complex numbers z satisfying $\left|\mathrm{z}^2+\mathrm{z}+1\right|=1$. Then which of the following statements is/are TRUE...

Let M be a 3 X 3 invertible matrix with real entries and let I denote the $3 \times 3$...