Let \(M\) be the set of all \(2 \times 2\) matrices with entries from the set R of real numbers. Then the function \(f: M \rightarrow R\) defined by \(f(A)=|A|\) for every \(A \in M\) is

If the circle \(x^{2}+y^{2}=9\) and \(x^{2}+y^{2}+20 x+2 y+1=0\) touch each other internally, then \(\alpha=\)

For a given Linear Programming problem, the objective function is
\(Z=3 x+2 y\)
Subject to constraints are
\(\begin{aligned}
& 4 x+3 y \leq 60 \\
& x \geq 3 \\
& y \leq 2 x \\
& y \geq 0
\end{aligned}\)
P is one of the corner points of the feasible region for the given Linear Programming problem. Then the coordinate of P is

\(\int e^{2x}\cos(5x + 3)dx =\)

\(\text { If } I_n=\int_\limits0^{\dfrac{\pi}{4}} \tan ^n x d x \text {, for } n \geq 2 \text {, then } I_n+I_{n-2}=\)

\(\text { If } f(x)=\left\{\begin{array}{cc}
\dfrac{1-\sin x}{(\pi-2 x)^2} & , \quad \text { if } x \neq \dfrac{\pi}{2} \\
\lambda, & \text { if } x=\dfrac{\pi}{2}
\end{array}\right.
\)
Then \(f(x)\) will be continues function at \(x=\dfrac{\pi}{2}\), then \(\lambda=\)

The value of \(\tan 67 \frac{1^{\circ}}{2}+\cot 67 \frac{1^{\circ}}{2}\) is

If \(x\) occurs is the expansion of \(\left(x+\frac{1}{x}\right)^{n}\), then its coefficient is

Two masses of \(1 g\) and \(9 g\) are moving with equal kinetic energy. The ratio of magnitude of their momentum is

In Young's double slit experiment, the intensity of light at a point on the screen where the path difference is \(\lambda\) is \(\mathrm{K}\) units (\(\lambda\) is the wavelength of light used). The percentage change in intensity at a point where the path difference is \(\dfrac{\lambda}{6}\) and the above point is

A slit of width \(10 \times 10^{-7} \mathrm{~m}\) is illuminated by light of wavelength \(500 \mathrm{~nm}\). Angular position of the first minimum is

Match the metals in Column II with their characteristic properties given in Column I.
\(\begin{array}{|c|l|c|c|} \hline & \text { Properties } & & \text { Metals } \\ \hline \text { A } & \text { Lanthanoid with low value of } 3^{\text {rd }} \text { ionisation enthalpy. } & P & V \\ \hline \text { B } & \text { f-Block element with electronic configuration }[\mathrm{Rn}] 6 \mathrm{~d}^2 7 \mathrm{~s}^2 & Q & G d \\ \hline \text { C } & 3 \mathrm{~d} \text {-series metal whose } \mathrm{M}^{3+}{ }_{(\mathrm{aq})} \text { is colourless. } & R & T h \\ \hline \text { D } & 3 \mathrm{~d} \text {-series metal with highest } \Delta \mathrm{H}_{(\text {atomisation })} & S & S c \\ \hline \end{array}\)

A bar magnet is oscillating in the earth's magnetic field with a period \(T\). What happens to its period and motion, if its mass is quadrupled?