In a plane there are 37 straight lines of which 13 pass through point \(A\) and 11 pass through the point \(B\). Moreover, no three lines (apart from the lines passing through \(A\) and \(B\) ) pass through same point and no two are parallel. What is the number of points of intersection of the straight lines?

Match the following
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=p x+q \\
(p \neq 0), \forall x \in R
\end{array} & \begin{array}{l}
\text {I. } f \text { is neither one-one nor onto }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
f: R \rightarrow R^{+} \cup\{0\} \text { is such that } f(x)=x^2, \forall x \in R
\end{array} & \begin{array}{l}
\text {II. } f \text { is both one-one and onto }
\end{array} \\
\hline \text { (C) } & \begin{array}{l}
f: N \rightarrow N \text { is such that } f(n)=n^2+2 n+3, \forall n \in N
\end{array} & \begin{array}{l}
\text {III. } f \text { is one-one but not onto }
\end{array} \\
\hline \text { (D) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=2\left(\cos ^2 5 x+\sin ^2 5 x\right) \\
\forall x \in R
\end{array} & \begin{array}{l}
\text {IV. } f \text { is onto but not one-one }
\end{array} \\
\hline && V. f \text{ is a constant function and also a bijection} \\
\hline
\end{array}\)
The correct answer is

\(\int \frac{\sin ^6 x}{\cos ^8 x} d x=\)

If the slope of one of the pair of lines represented by \(2 x^2+3 x y+K y^2=0\) is 2 , then the angle between the pair of lines is

The values of \(x\) for which the inequality \(\frac{8 x^2+16 x-51}{(2 x-3)(x+4)}>3\) holds, are

\(\mathrm{S}=\{\mathrm{z} \in \mathrm{C} /|\mathrm{z}-1+\mathrm{i}|=1\}\) represents

\(\int \frac{2 \tan (x)}{1+2 \tan ^2(x)} d x=\)

If \(Q\) denotes the set of all rational numbers and \(f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}\) for any \(\frac{p}{q} \in Q\), then observe the following statements.
I. \(f\left(\frac{p}{q}\right)\) is real for each \(\frac{p}{q} \in Q\)
II. \(f\left(\frac{p}{q}\right)\) is a complex number for each \(\frac{p}{q} \in Q\).
Which of the following is correct?

If \(f: R \rightarrow R\) is an even function having derivatives of all orders, then an odd function among the following is

When the temperature of a gas is raised from \(27^{\circ} \mathrm{C}\) to \(90^{\circ} \mathrm{C}\). The increase in the rms velocity of the gas molecules is

If \(x=\log \left[\cot \left(\frac{\pi}{4}+\theta\right)\right]\), then the value of \(\sin h x\) is

Observe the following solutions
I. Black coffee
II. \(0.2 \mathrm{M} \mathrm{NaOH}\)
III. Lemon Juice
IV. Lime water V. Human Saliva VI. Tomato Juice
The number of solutions having \(\mathrm{pH}\) range of \(1-7\) and \(7-14\), in the above list, is respectively.

In Young's double slit experiment the slits are \(3 \mathrm{~mm}\) apart and are illuminated by light of two wavelengths \(3750 Å\) and \(7500 Å\). The screen is placed at \(4 \mathrm{~m}\) from the slits. The minimum distance from the common central bright fringe on the screen at which the bright fringe of one interference pattern due to one wavelength coincide with the bright fringe of the other is