The negation of the statement " \( 72 \) is divisible by \( 2 \) and \( 3 \) " is

The least and the greatest distances of the point \((10,7)\) from the circle
\[
x^{2}+y^{2}-4 x-2 y-20=0 \text { are }
\]

If \( f(x)=\left\{\begin{array}{cc}\frac{\sin 3 x}{e^{2 x}-1} & x \neq 0 \\ k-2 & x=0\end{array}\right. \) is continuous at \( x=0 \), then \( k= \)

If the vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) is

Let R be the relation in the set \(\mathbb{N}\) given by \(R = \{(a, b) : a = b - 2, b > 6\}\). Which of the following is the correct answer?

The value of \( \lim _{x \rightarrow 0} \frac{[x]}{x} \) is

The projection of \(\overrightarrow{\mathbf{a}}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) on \(\overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) is

The function \(x^x ; x\gt0\) is strictly increasing at

The ionisation energy of \(\mathrm{Li}^{2+}\) is equal to

Match List-I with List-II
\begin{array}{|l|l|l|l|}\hline & \text{List-I (Types of redox reactions)} & & \text{List-II (Examples)} \\\hline \text{a}. & \text{Combination reaction} & \text{(i)} & \mathrm{Cl}_{2(\mathrm{~g})}+2 \mathrm{Br}_{(\mathrm{aq})}^{-} \rightarrow 2 \mathrm{Cl}_{(\mathrm{aq})}^{-}+\mathrm{Br}_{2(\mathrm{l})} \\\hline \text{b.} & \text{Decomposition reaction} & \text{(ii)} & 2 \mathrm{H}_2 \mathrm{O}_{2(\mathrm{aq})} \rightarrow 2 \mathrm{H}_2 \mathrm{O}_{(1)}+\mathrm{O}_{2(\mathrm{~g})} \\\hline \text{c.} & \text{Displacement reaction} & \text{(iii)} & \mathrm{CH}_{4(\mathrm{~g})}+2 \mathrm{O}_{2(\mathrm{~g})} \xrightarrow{\Delta} \mathrm{CO}_{2(\mathrm{~g})}+2 \mathrm{H}_2 \mathrm{O}_{(1)} \\\hline \text{d.} & \text{Disproportionation reaction} & \text{(iv)} & \mathrm{H}_2 \mathrm{O}_{(1)} \xrightarrow{\Delta} 2 \mathrm{H}_{2(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})} \\\hline\end{array}

Choose the correct answer from the options given below.

\( \sum_{r=1}^{n}(2 r-1)=x \) then
\( \lim _{n \rightarrow \infty}\left[\frac{1^{3}}{x^{2}}+\frac{2^{3}}{x^{2}}+\frac{3^{3}}{\chi^{2}}+\ldots+\frac{n^{3}}{\chi^{2}}\right]= \)

For the parabola \(\mathrm{y}^{2}=4 \mathrm{x}\), the point \(P\) whose focal distance is 17 , is

A wire of resistance \(R\) is connected across a cell of emf \((\varepsilon)\) and internal resistance \((r)\). The current through the circuit is \(I\). In time \(t\), the work done by the battery to establish the current \(I\) is