When the coordinate axes are rotated by an angle ${\tan }^{-1}\left(\frac{3}{4}\right)$ about the origin, then the equation $x^2+y^2=9$ is transformed to the equation

If \(f(x)=x e^{x(1-x)}, x \in \mathbb{R}\), then \(f(x)\) is

If \(\Delta=a^2-(b-c)^2\), is the area of the \(\triangle A B C\), then \(\tan A\) is equal to

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

Let \(Q\) be the set of all rational numbers in \([0,1]\) and \(f:[0,1] \rightarrow[0,1]\) be defined by
\[
f(x)=\left\{\begin{array}{r}
x \text { for } x \in Q \\
1-x \text { for } x \notin Q
\end{array}\right.
\]
Then, the set \(S=\{x \in[0,21]:(f o f)(x)\}\) is equal to

\(\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3}\) is equal to

If \(y=\frac{\sin \mathrm{h}^{-1} x}{\sqrt{1+x^2}}\), then \(\left(1+x^2\right) \mathrm{y}_2+3 \mathrm{xy}_1+\mathrm{y}=\)

If \(t_n\) denotes the number of triangles formed with \(n\) points in a plane, no three of which are collinear and if \(t_{n+1}-t_n=36\), then \(n\) is equal to

In which of the following reactions, the concentration of product is higher than the concentration of reactant at equilibrium? ( \(K=\) equilibrium constant \()\)

In the nuclear fusion reaction \({ }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+\mathrm{n}\), if the repulsive potential energy between the two nuclei is \(2.07 \times 10^{-14} \mathrm{~J}\), then the temperature at which the gases must be heated to initiate the reaction is \(\left(\right.\) Boltzmann constant \(\left.=1.38 \times 10^{-23} \mathrm{JK}^{-1}\right)\)

If the increase in the side of square is \(6 \%\), then the approximate percentage increase in its area ......

Identify the alkyne in the following sequence of reactions,

The X formed in the following reaction sequence and its structural type are respectively