The area (in sq. units) of the triangle formed by the straight line \(x+y=3\) and the angular bisectors of the pair of straight lines \(x^2-y^2+2 y=1\), is

\(\cot \left[\sum_{n=3}^{32} \cot ^{-1}\left(1+\sum_{k=1}^n 2 k\right)\right]=\)

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

The coefficient of \(x^3\) in the expansion of \((1-x)^{3 / 2}\), \((|x| < 1)\) is

The value of $\frac{\sin \theta +\sin 3\theta }{\cos \theta +\cos 3\theta }$ is

The sum of all 4-digit numbers that can be formed using the digits 2, 3, 4, 5, 6 without repetition, is

The equation \(x^2-5 x y+p y^2+3 x-8 y+2=0\) represents a pair of straight lines. If \(\theta\) is the angle between them, then \(\sin \theta\) is equal to

Let \(f: R \rightarrow R\) be defined by
\(f(x)=\left\{\begin{array}{ccc}\alpha+\frac{\sin [x]}{x}, & \text { if } & x>0 \\ 2, & \text { if } & x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & \text { if } & x < 0\end{array}\right.\)
where, \([x]\) denotes the integral part of \(x\). If \(f\) continuous at \(x=0\), then \(\beta-\alpha\) is equal to

In \(\triangle A B C\), if \(a=13, b=14\) and \(\cos \frac{C}{2}=\frac{3}{\sqrt{13}}\), then \(2 r_1=\)

Which of the following solutions has the lowest freezing point?

Pressure of \(1 \mathrm{~g}\) ideal gas \(X\) at \(300 \mathrm{~K}\) is \(2 \mathrm{~atm}\). When \(2 \mathrm{~g}\) of another gas \(Y\) is introduced in the same vessel at same temperature, the pressure become \(\mathrm{l}\) atm. The correct relationship between molar masses of \(X\) and \(Y\) is

When the load applied to a wire is increased from 5 kg wt to 8 kg wt , the elongation of the wire increases from 1 mm to 1.8 mm . The work done during the elongation of the wire is (Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )

Match the following.
\begin{array}{|c|c|c|c|}
\hline & List I (Substance) & & List II (Use) \\
\hline (A) & Methanoic acid & (i) & Textile \\
\hline (B) & Ethanoic acid & (ii) & Polymer \\
\hline (C) & Hexanedioic acid & (iii) & Food preservative \\
\hline (D) & Sodium benzoate & (iv) & Vinegar \\
\hline
\end{array}