\(\lim _{x \rightarrow \pi / 6}\left[\frac{3 \sin x-\sqrt{3} \cos x}{6 x-\pi}\right]\) is equal to :

If ${\tan }^{-1}\left[\frac{1}{1+1.2}\right]+{\tan }^{-1}\left[\frac{1}{1+2.3}\right]+\cdots +{\tan }^{-1}\left[\frac{1}{1+n(n+1)}\right]={\tan }^{-1}\left[x\right]$, then $x=$

The mean of four observations is 3 . If the sum of the squares of these observations is 48 , then their standard deviation is

A pair of perpendicular lines passes through the origin and also through the points of intersection of the curve \(x^2+y^2=4\) with \(x+y=a\), where \(a>0\). Then \(a\) is equal to

Let \(a, b \in \mathbf{R}-\{0\}\), and \(I_2\) be the identity matrix of order 2. Then the rank of the (block) matrix \(\left[\begin{array}{ll}a I_2 & b_2 \\ a I_2 & b I_2\end{array}\right]\) is

If the lines \(x+y-2=0,3 x-4 y+1=0\) and \(5 x+k y-7=0\) are concurrent at \((\alpha, \beta)\), then equation of the line concurrent with the given lines and perpendicular to \(\mathrm{kx}+\mathrm{y}-\mathrm{k}=0\) is

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) \(-1\)

Identify the correct statements from the following.
(i) Oxygen shows \(-2,-1,+1\) and +2 oxidation states.
(ii) The thermal stability of \(\mathrm{H}_2 \mathrm{O}, \mathrm{H}_2 \mathrm{Se}\) and \(\mathrm{H}_2 \mathrm{~S}\) follows the order \(\mathrm{H}_2 \mathrm{O} < \mathrm{H}_2 \mathrm{~S} < \mathrm{H}_2\) Se.
(iii) The reducing nature of \(\mathrm{H}_2 \mathrm{Se}, \mathrm{H}_2 \mathrm{~S}\) and \(\mathrm{H}_2 \mathrm{Te}\) follows the order \(\mathrm{H}_2 \mathrm{~S} < \mathrm{H}_2 \mathrm{Se} < \mathrm{H}_2 \mathrm{Te}\)

The radius of the circle $2x^2+2y^2-3x+2y-1=0$ is _________ units.

The domain of the function \(f(x)=\sqrt{\log _e\left(\frac{1}{x^2-4 x+4}\right)}+\sin ^{-1}\left(x^2-2\right)\) is

Complete hydrolysis of Xenon hexafluoride gives HF along with compound X. The hybridisation in X is

In the following unbalanced reaction, the product formed is
\(\mathrm{Al}_2 \mathrm{O}_3+\mathrm{NaOH}(a q)+\mathrm{H}_2 \mathrm{O} \longrightarrow\)

A magnetic field given by \(B(t)=\left(0.2 t-0.05 t^2\right) T\) is directed perpendicular to the plane of a circular coil containing 25 turns of radius \(1.8 \mathrm{~cm}\) and whose total resistance is \(5 \Omega\). The power dissipation at \(3 \mathrm{~s}\) is nearly