In a triangle ABC, if \(\sin \frac{\mathrm{A}}{2}=\frac{1}{4} \sqrt{\frac{3}{5}}, \mathrm{a}=2, \mathrm{c}=5\) and b is an integer, then the area (in sq. units) of triangle ABC is

Express \(\frac{d t}{d x}=\frac{t}{\left(x+t e^{-2 x / 4}\right)}\) in the form of \(\frac{d x}{d t}=\phi\left(\frac{x}{t}\right)\)

If \(\int \frac{3 x+1}{(x-1)^3(x+1)} d x=\mathrm{A} \cdot \log \left|\frac{x+1}{x-1}\right|+\frac{\mathrm{B}}{x-1}+\frac{\mathrm{C}}{(x-1)^2}+\mathrm{D}\) then \(\mathrm{A}+\mathrm{B}+\mathrm{C}=\)

If \(\mathrm{a}>0, \mathrm{~b}>0\) then \(\lim _{n \rightarrow \infty}\left(\frac{a+b^{1 / n}-1}{a}\right)^n=\)

Find the area of the circle
\[
(x+1)(x+2)+(y-1)(y+3)=0
\]

In how many ways can $5$ balls be placed in $4$ tins if any number of balls can be placed in any tin?

The mean and standard deviation of a binomial variate \(X\) are 4 and \(\sqrt{3}\) respectively. Then \(P(X \geq 1)\) is equal to

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

If $l{x}^2+3xy-2y^2-5x+5y+k=0$ represents a pair of perpendicular lines, then

If \(\frac{y}{x} \cos ^4 \alpha+\frac{x}{y} \sin ^4 \alpha=2 \sin ^2 \alpha \cdot \cos ^2 \alpha\), then \(\frac{d y}{d x}=\)

Identify the correct statements:
i. Oxidation of \(\mathrm{NaBH}_4\) with \(\mathrm{I}_2\) gives \(\mathrm{B}_2 \mathrm{H}_6\)
ii. \(\mathrm{B}_2 \mathrm{H}_6\) burns in oxygen and releases an enormous amount of energy
iii. \(\mathrm{B}_2 \mathrm{H}_6\) on hydrolysis gives a tribasic acid

An electron having charge $1.6\times {10}^{-19} \mathrm{C}$ and mass $9\times {10}^{-31} \mathrm{kg}$ is moving with $4\times {10}^6 \mathrm{m} {\mathrm{s}}^{-1}$ speed in a magnetic field $2\times {10}^{-1} \mathrm{T}$ in a circular orbit. The force acting on electron and the radius of the circular orbit is _____

Which of the following do not exist?