The radius of the circle given by
\(\begin{aligned}
x^2+y^2 & +z^2+2 x-2 y-4 z-19=0 \\
& =x+2 y+2 z+7, \text { is }
\end{aligned}\)

If the latusrectum of an ellipse subtends a right angle at the centre of that ellipse, then the eccentricity of that ellipse is

If \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is a real number and \(0 < \theta < 2 \pi\), then \(\theta\) is equal to

\(\frac{\cos 10^{\circ}+\cos 80^{\circ}}{\sin 80^{\circ}-\sin 10^{\circ}}=\)

The system \(x+2 y+3 z=4,4 x+5 y+3 z=5,3 x+4 y+3 z=\lambda\) is consistent and \(3 \lambda=n+100\), then \(n=\)

The area of the triangle formed by the line \(x+y=4\) and angular bisectors of the pair of lines \(x^2-y^2+2 y-1=0\) is......... sq. units.

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

A circular wheel with 10 spokes with its plane vertical along East-West is rotating about its natural axis with a uniform speed 100 revolutions per minute in the earth's magnetic field. Radius of the wheel is \(0.3 \mathrm{~m}\). If the emf induced between the centre of the wheel and the rim is \(3 \pi \mu \mathrm{V}\), angle of dip at the place is
(Vertical component of the earth's magnetic field \(\mathrm{B}_{\mathrm{V}}=15 \mu \mathrm{T}\) )

If in a \(\triangle A B C, \frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}\), then \(\angle C\) is equal to

Which halogen can be oxidised by concentrated nitric acid?

If the range of a random variable \(X\) is \(\{0,1,2,3,4, \ldots \ldots\}\) with \(P(X=k)=\frac{(k+1) a}{3^k}\) for \(k \geq 0\), then \(a\) is equal to

\(4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}}=\)

A car covers a distance at speed of $60 \mathrm{km} {\mathrm{h}}^{-1}$. It returns and comes back to the original point moving at a speed of $V$. If the average speed for the round trip is $48 \mathrm{km} {\mathrm{h}}^{-1}$, then the magnitude of $v$ is