Find the number of marked points on the plane, if when connected pairwise by line segments, the total number of line segments formed is $15.$

If the sum of the roots of the quadratic equations is 1 and sum of the squares of the roots is 13 , then find that equation.

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

\(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-10 x-24=\) 0 . If \(\alpha(\beta+\gamma), \beta(\gamma+\alpha)\) and \(\gamma(\alpha+\beta)\) are the roots of the equation \(x^3+p x^2+q x+r=0\), then \(q=\)

If the values of \(k\) for which the equation \(x^2+2(k+2) x+\) \(6 \mathrm{k}+7=0\) has equal roots are \(\mathrm{k}_1\) and \(\mathrm{k}_2\), then \(\mathrm{k}_1^2+\mathrm{k}_2^2=\)

If \(\int \frac{\mathrm{dx}}{\sin ^3 \mathrm{x}+\cos ^3 \mathrm{x}}=\mathrm{A} \log \left|\frac{\sqrt{2}+\mathrm{t}}{\sqrt{2}-\mathrm{t}}\right|+\mathrm{B} \operatorname{Tan}^{-1}(\mathrm{t})+\mathrm{c}\), then \(\left(\frac{\mathrm{B}}{\mathrm{A}}, \mathrm{t}\right)=\)

\(\int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x=\)

If \(\mathrm{S}_1, \mathrm{~S}_2\) and \(\mathrm{S}_3\) are the tensions at liquid-air, solid-air and solid-liquid interfaces respectively, and \(\theta\) is the angle of contact at the solid-liquid interface, then

If \(\mathrm{c}\) and \(\mathrm{d}\) are arbitrary constants, then \(y=e^{2 x}(\cosh \sqrt{2} x+d \sinh \sqrt{2} x)\) is the general solution of the differential equation

If \(\mathbf{a}=\hat{i}+\hat{j}+\hat{k}, \mathbf{c}=\hat{j}-\hat{k}, \mathbf{a} \times \mathbf{b}=\mathbf{c}, \mathbf{a} \cdot \mathbf{b}=3\), then \(\mathbf{b}=\)

If \(\mathbf{a}\) makes an acute angle with \(\mathbf{b}, \mathbf{r} \cdot \mathbf{a}=0\) and \(\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\), then \(\mathbf{r}=\)

The mean deviation from the mean of the series \((a),(a+d),(a+2 d), \ldots \ldots \ldots . .,(a+2 n d)\) is

A slab consists of two identical plates of copper and brass. The free face of the brass is at \(0^{\circ} \mathrm{C}\) and that of copper at \(100^{\circ} \mathrm{C}\). If the thermal conductivities of brass and copper are in the ratio \(1: 4\), then the temperature of interface is