The value of
\(\left\{x \in \mathbb{R} \mid\left[\log (1.6)^{1-x^2}-(0.625)^{6(1+x)}\right] \in \mathbb{R}\right\}\) is

\(\lim _{x \rightarrow 0} \frac{x^2 \sin ^2(3 x)+\sin ^4(6 x)}{(1-\cos 3 x)^2}=\)

\(\tan ^{-1}\left[\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+e^x} d x\right]=\)

The distance of the point \((6,4 \sqrt{3})\) from the focus of \(y^2=8 x\) 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) \(-1\)

If the circle \(x^2+y^2+2 g x+2 f y+c=0(c>0)\) touches both the coordinate axes and lies in the third quadrant, then the length of the chord intercepted by the circle on the line \(x+y+\sqrt{c}=0\) is

In \(\triangle A B C\), if \(\mathrm{A}=60^{\circ}\) and \(\mathrm{B}=105^{\circ}\) then
\[
\begin{aligned}
& \frac{2 \mathrm{R}^2(b-c) \sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}}{(b+c)(s-a \cos \mathrm{C}-c \cos \mathrm{A})(\mathrm{s}-a \cos \mathrm{B}-b \cos \mathrm{A})}= \\
\end{aligned}
\]

Three closed vessels \(A, B\) and \(C\) are at the same temperature \(T\) and contain gases. Vessel \(A\) contains only \(\mathrm{O}_2, B\) contains only \(\mathrm{N}_2\) and \(\mathrm{C}\) contains a mixture of equal quantities of \(\mathrm{O}_2\) and \(\mathrm{N}_2\). If the rms speed of \(\mathrm{O}_2\) molecules in vessel \(A\) is \(v_1\) and that of \(\mathrm{N}_2\) molecules in vessel \(B\) is \(v_2\) then the rms speed of \(\mathrm{O}_2\) molecules in vessel \(\mathrm{C}\) is

Two stretched strings A and B when vibrated together produce 4 beats per second. If the tension applied to the string A increased, the number of beats produced per second is increased to 7. If the frequency of string B is 480 Hz initially. The frequency of string A is

Which one of the following is false?

The temperature of the sink of a Carnot's engine is \(300 \mathrm{~K}\) and the efficiency of the engine is 0.25 . If the temperature of the source of the engine is increased by \(100 \mathrm{~K}\), the efficiency of the engine increases by

A long current carrying wire produces a magnetic field of 1 \(\mathrm{T}\) at a distance of \(r\). The magnetic field (a) \(\frac{\mathrm{r}}{2}\) (b) \(2 \mathrm{r}\) and (c) \(3 \mathrm{r}\) is

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