If \(\frac{1}{2} \leq x \leq 1\), then \(\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)\) is equal to

The angles \(A, B, C\) of a triangle \(A B C\) are in AP. If \(A B=6, B C=7\), then \(A C=\)

The number of pairs of consecutive positive even integers such that the sum of their squares is 290 is

The centre of a circle is \((2,-3)\) and the circumference is \(10 \pi\). Then its equation is

\(A B C D\) is a parallelogram such that \(L\) is mid-point of \(B C\), then \(\mathbf{A L}\) is equal to

Let \(x^2+y^2=16\) be the equation of the auxiliary circle of a hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) and let \((4 \sqrt{2}, 3)\) be a point on the hyperbola. Then, the eccentricity of the hyperbola is

The value of the sum \(1 \times 2 \times 3+2 \times 3 \times 4+3 \times 4 \times 5+\ldots\) upto \(n\) terms is equal to

The domain of the real valued function \(f(x)=\frac{\sqrt{2-x}+\sqrt{1+x}}{\sqrt{x+3}}\) is

The perpendicular distance from the point \(3 \bar{i}-2 \bar{j}+\bar{k}\) to the line joining the points \(\bar{i}-3 \bar{j}+5 \bar{k}\) and \(2 \bar{i}+\bar{j}-4 \bar{k}\) is

The mean deviation about the mean of the set of first ' \(n\) ' natural numbers, when ' \(n\) ' is an even number is equal to

Light of wavelength $300 \mathrm{nm}$ in medium $A$ enters into medium $B$ through a plane surface. If the frequency of light is $5\times {10}^{14} \mathrm{Hz}$ and the ratio of speed in medium $A$ to that is medium $B$ is $\frac{4}{5}$, the absolute refractive index of medium $B$ is,

In the given circuit, if the potential difference between $A$ and $B$ is $80 \mathrm{V},$ then the equivalent capacitance between $A$ and $B$, and the charge on $10 \mu \mathrm{F}$ capacitor respectively, are

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)