If the circle \(x^2+y^2+2 g x+2 f y+c=0(c>0)\) touches both the coordinate axes and lies in the third quadrant, then the length of the chord intercepted by the circle on the line \(x+y+\sqrt{c}=0\) is

If two unbiased six-faced dice are thrown simultaneously until a sum of either $7$ or $11$ occurs, then the probability that $7$ comes before $11$ is

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}$ and if $6\hat{i}+2\hat{j}+3\hat{k}={\lambda }_1(\overrightarrow{a}\times \overrightarrow{b})+{\lambda }_2(\overrightarrow{b}\times \overrightarrow{c})+{\lambda }_3(\overrightarrow{c}\times \overrightarrow{a}),$ then $\left({\lambda }_1, {\lambda }_2, {\lambda }_3\right)=$

The center of ellipse \(x^2+2 y^2-4 x+12 y+14=0\) is

Suppose \(X\) follows a binomial distribution with parameters \(n\) and \(p\), where \(0 < p < 1\). If \(\frac{P(X=r)}{P(X=n-r)}\) is independent of \(n\) for every \(r\), then \(p\) is equal to

\(\begin{aligned} & \lim _{n \rightarrow \infty} n\left[\frac{1}{\left(3 n^2+8 n+4\right)}+\frac{1}{3 n^2+16 n+16}+\ldots\right. \\ & \left.+\frac{1}{15 n^2}\right]=\end{aligned}\)

Let \(\mathbf{u}\) and \(\mathbf{v}\) be two vectors in a plane. Then any vector \(\mathbf{w}\) in the plane can be written as \(w=a \mathbf{u}+b \mathbf{v}\) for some scalars ' \(a\) ' and ' \(b\) ' if and only if

The elastic energy stored per unit volume in terms of longitudinal strain ' \(\epsilon\) ' and Young's modulus ' \(Y\) ' is

Match the following List I with List II.
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \text { Transverse wave (i) } & \begin{array}{l}
\text {Vibrations parallel to the } \\
\text {direction of propagation }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
\text {Longitudinal wave }
\end{array} & \begin{array}{l}
\text {Vibrations perpendicular to } \\
\text {the direction of propagation }
\end{array} \\
\hline \text { (C) } & \text { Beats } & \begin{array}{l}
\text {Superposition of waves } \\
\text {travelling in the opposite directions }
\end{array} \\
\hline \text { (D) } & \text { Stationary waves (iv) } & \begin{array}{l}
\text {Superposition of waves } \\
\text {travelling in same direction }
\end{array} \\
\hline
\end{array}\)
The correct answer is

\(\lim _{x \rightarrow-\infty} \log _e(\cosh x)+x=\)

If \(\int \frac{1}{\cot \frac{x}{2} \cot \frac{x}{3} \cot \frac{x}{6}} d x=A \log \left|\cos \frac{x}{2}\right|+B \log \left|\cos \frac{x}{3}\right|+C \log \left|\cos \frac{x}{6}\right|+k\) then \(\mathrm{A}+\mathrm{B}+\mathrm{C}=\)

The terminal velocity of a liquid drop of radius ' \(r\) ' falling through air is \(v\). If two such drops are combined to form a bigger drop, the terminal velocity with which the bigger drop falls through air is (ignore any buoyant force due to air)

Two blocks of ice when pressed together join to form one block because