Assertion (A): \(\lim _{x \rightarrow 0} \frac{1}{x}=\infty\)
Reason (R): As the value of \(x\) decreases, the value of \(\frac{1}{x}\) increases

The function \(f(x)=\sqrt{\frac{3 x^2-5 x-2}{2 x^2-7 x+5}}\) has discontinuous points at \(\mathrm{x}=\)

If one side of an isosceles triangle is given by \(y=2\) and the base is provided by the points \((2,0)\) and \((0,2)\), then its area (in sq. units) is

If a card is drawn at random from a well shuffled pack of 52 playing cards, then the probability that it is either an ace or a spade card is

The shortest distance between the line passing through the point \(\bar{i}+2 \bar{j}+3 \bar{k}\) and parallel to the vector \(2 \bar{i}+3 \bar{j}+4 \bar{k}\) and the line passing through the point \(2 \bar{i}+4 \bar{j}+5 \bar{k}\) and parallel to the vector \(3 \bar{i}+4 \bar{j}+5 \bar{k}\), is

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

Define \(f: R \rightarrow R\) by
\[
\left.f(x)=\cos \left(\tan ^{-1} \sin \left(\tan ^{-1} x\right)\right)\right) \text {, then } \lim _{x \rightarrow \infty}(f o f) x
\]
is equal to

If the system of equations \(a_1 x+b_1 y+c_1 z=0\), \(a_2 x+b_2 y+c_2 z=0, \quad a_3 x+b_3 y+c_3 z=0\) has only trivial solution, then the rank of \(\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]\) is

What is the value of \(x\) if the mean of \(8,6,7\), \(5, x\) and 4 is 7 ?

The standard deviation of first 10 multiples of 4 is

If \((l, m)\) is the circumcentre of an equilateral triangle inscribed in the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) having vertices at points with eccentric angles \(\theta_1, \theta_2\) and \(\theta_3\), then \(\frac{2}{3}\left[\cos \left(\theta_1-\theta_2\right)+\cos \left(\theta_2-\theta_3\right)+\cos \left(\theta_3-\theta_1\right)\right]=\)

Identify the reaction in which diborane is produced on industrial scale?

When \(10^{-3} \mathrm{M}\) solution of glucose in water, freezes at \(-0.0186^{\circ} \mathrm{C}\), then at what temperature \(10^{-3} \mathrm{M}\) solution of \(\mathrm{NaCl}\) will freeze?