The number of positive odd divisors of 216 is

If \(a x^2+2 h x y-2 a y^2+3 x+15 y-9=0\) represents a pair of lines intersecting at \((1,1)\), then ah \(=\)

The center of ellipse \(x^2+2 y^2-4 x+12 y+14=0\) is

\(\lim _{x \rightarrow \infty} \frac{3 x+4 \cos ^2 x}{\sqrt{x^2-5 \sin ^2 x}}=\)

The number of solutions of the trigonometric equation \(1+\cos x \cdot \cos 5 x=\sin ^2 x\) in \([0,2 \pi]\) is

\(P\) and \(Q\) are points on the straight line passing through the point \(A(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})\) and parallel to the vector \(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\), If \(A P=A Q=3\), then the vector equation of the plane \(O P Q\) is

Match the following
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=p x+q \\
(p \neq 0), \forall x \in R
\end{array} & \begin{array}{l}
\text {I. } f \text { is neither one-one nor onto }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
f: R \rightarrow R^{+} \cup\{0\} \text { is such that } f(x)=x^2, \forall x \in R
\end{array} & \begin{array}{l}
\text {II. } f \text { is both one-one and onto }
\end{array} \\
\hline \text { (C) } & \begin{array}{l}
f: N \rightarrow N \text { is such that } f(n)=n^2+2 n+3, \forall n \in N
\end{array} & \begin{array}{l}
\text {III. } f \text { is one-one but not onto }
\end{array} \\
\hline \text { (D) } & \begin{array}{l}
f: R \rightarrow R \text { is such that } f(x)=2\left(\cos ^2 5 x+\sin ^2 5 x\right) \\
\forall x \in R
\end{array} & \begin{array}{l}
\text {IV. } f \text { is onto but not one-one }
\end{array} \\
\hline && V. f \text{ is a constant function and also a bijection} \\
\hline
\end{array}\)
The correct answer is

Assertion (A) : For an endothermic dissolution process, increase in temperature increases the solubility in a nearly saturated solution.
Reason (R) : In saturated solution dynamic equilibrium exists between dissolved solute and undissolved solute.

Consider the following statements A and B and identify the correct answers given below.
A. Peltier coefficient is numerically equal to the potential difference across the junctions of the thermocouple through which current is flowing.

B. According to Thomson, energy is neither absorbed nor evolved at the junction of a thermocouple but is observed or evolved only along the lengths of both the conductors.

Solve the following equation \(\sin x+\sqrt{3} \cos x=\sqrt{2}\)

What is ' \(Z\) ' in the following reaction sequence ?
\(\mathrm{C}_3 \mathrm{H}_6 \xrightarrow[\mathrm{CCl}_4]{\mathrm{Br}_2} \underset{\text { (colourless) }}{\mathrm{X}} \xrightarrow[\text { (ii) } \mathrm{NaNH}_2, \Delta]{\text { (i) } \mathrm{KOH} / \text { alcohol }} \mathrm{Y} \xrightarrow[\substack{\mathrm{Hg}^{2+}, \mathrm{H}^{+} \\ 333 \mathrm{~K}}]{\mathrm{H}_2 \mathrm{O}} \mathrm{Z}\)

If \(\tan A+\tan B=x\) and \(\cot A+\cot B=y\), then \(\tan (\mathrm{A}+\mathrm{B})=\)

\(\int \frac{x-1}{(x-2)(x-3)} d x\) is equal to