The direction cosines of the line of intersection of the planes \(x+2 y+z-4=0\) and \(2 x-y+z-3=0\) are

If the origin is shifted to a point P by the translation of axes to remove the \(y\)-term from the equation \(x^2-y^2+2 y-1\) \(=0\). then the transformed equation of it is

\(\int_{-\pi}^\pi \frac{x \sin x}{1+\cos ^2 x} d x=\)

If \(\alpha, \beta, \gamma\) are the roots of \(f(x)=x^3-9 x^2+26 x\) -24, then \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\) are the roots of

\(f(x)\) is a quadratic expression such that \(f(x)\) is negative when \(x \in\left(-\infty,-\frac{5}{3}\right) \cup(3, \infty)\) and positive when \(x \in\left(-\frac{5}{3}, 3\right) \cdot g(x)\) is another quadratic expression such that \(g(x)\) is negative when \(x \in\left(3, \frac{9}{2}\right)\) and positive when \(x \in R-\left[3, \frac{9}{2}\right]\). Then, the sign of \(f(x) g(x)\) in \([0,5]\) is

Assertion (A): The difference of the slopes of the lines represented by
\(\mathrm{y}^2-2 \mathrm{xysec}^2 \alpha+\left(3+\tan ^2 \alpha\right)\left(-1+\tan ^2 \alpha\right) \mathrm{x}^2=0 \text { is } 4\)
Reason (R): The difference of the slopes represented by
\(a x^2+2 h x y+b y^2=0 \text { is } \frac{2 \sqrt{h^2-a b}}{|b|}\)

The sum of the squares of the distances of a moving point from 2 fixed points \(A(a, 0)\) and \(B(-a, 0)\) is equal to a constant \(2 c^2\), then the equation of its locus is

If $4^{\mathrm{x}}-3^{\mathrm{x}-\frac{1}{2}}=3^{\mathrm{x}+\frac{1}{2}}\mathrm{ }-2^{2\mathrm{x}-1}$ then value of $\mathrm{x}$ is equal to

If the lines $2x+y+12=0, kx-3y-10=0$ are conjugate with respect to the circle $x^2+y^2-4x+3y-1=0,$ then $k=$

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\left\{\begin{array}{cl}a-\frac{\sin [x-1]}{x-1} &, \text { if } x>1 \\ 1 &, \text { if } x=1 \\ b-\left[\frac{\sin [x-1]-[x-1]}{\left.([x-1])^3\right]}\right] &, \text { if } x < 1\end{array}\right.\)
where \([t]\) denotes the greatest integer less than or equal to \(t\). If \(f\) is continuous at \(\mathrm{x}=1\), then \(\mathrm{a}+\mathrm{b}=\)

The initial rates of decrease of \(\mathrm{I}_2\) in acetone - iodine reaction catalysed by \(\mathrm{H}^{+}\) are given in the table.
\(\begin{array}{|c|c|c|c|c|}
\hline \begin{array}{c}
\text { Experiment } \\
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{I}_2\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{H}^{+}\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial } \\
{\left[\mathrm{CH}_3 \mathrm{COCH}_3\right]} \\
\left(\mathrm{mol} \mathrm{L}^{-1}\right)
\end{array} & \begin{array}{c}
\text { Initial rate of decrease of } \mathrm{I}_2 \\
\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}\right)
\end{array} \\
\hline 1 & 0.01 & 0.1 & 0.1 & 0.096 \\
2 & 0.01 & 0.2 & 0.1 & 0.192 \\
3 & 0.02 & 0.2 & 0.1 & 0.192 \\
4 & 0.01 & 0.2 & 0.2 & 0.384 \\
\hline
\end{array}\)
The order with respect to \(\mathrm{I}_2, \mathrm{H}^{+}\), acetone and total order of the reaction respectively are:

The sum of the solutions in \((0,2 \pi)\) the equation \(\cos x \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right)=\frac{1}{4}\) is

If, the binding energy of electrons in a metal is \(250 \mathrm{~kJ} / \mathrm{mol}\), what should be the threshold frequency of the striking photons in order to free an electron from the metal surface?