The length of \(x\)-intercept made by pair of lines \(2 x^2+x y-6 y^2-2 x+17 y-12=0\) 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

For \(|x| \lt \frac{1}{\sqrt{2}}\), the coefficient of \(x\) in the expansion of \(\frac{(1-4 x)^2\left(1-2 x^2\right)^{1 / 2}}{(4-x)^{3 / 2}}\) is

Tom and Jerry play a game of alternately throwing an unfair coin. First one to get head wins. If Tom starts the game, he has $62.5\%$ chance of winning the game. Suppose this coin is tossed $5$ times, then the probability of getting exactly $3$ heads is

The curve $f\left(x\right)=e^x\sin x$ is defined in the interval $[0,2\pi ].$ The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum, is

If the \(\operatorname{Arg} z_1\) and \(\operatorname{Arg} \overline{z_2}\) are \(\frac{\pi}{3}\) and \(\frac{\pi}{5}\) respectively then the value of \(\operatorname{Arg} z_1+\operatorname{Arg} z_2\) is

Let $a,b$ and $c$ be the lengths of the sides of a triangle with its opposite angles $A,B$ and $C$ respectively. If $a=3, b=4$ and $A={\sin }^{-1}\left(\frac{3}{4}\right),$ then the angle $B$ is

A free body of mass \(8 \mathrm{~kg}\) is moving at \(2 \mathrm{~m} \cdot \mathrm{s}^{-1}\) along a straight line. It splits into two equal parts due to an internal explosion releasing \(16 \mathrm{~J}\) of energy, and neither part deviates from the original line of motion. Finally,

A homogeneous equation of second degree in \(x\) and \(y\) represents which of the following?

Identify the products \((P),(Q)\) and \((R)\) formed from the following reactions.

Let \(A, B\) and \(C\) be three angles of a \(\triangle A B C\) such that \(\cos A+\cos B+\cos C=\frac{3}{2}\), then the \(\triangle A B C\) is

If the circles \(x^2+y^2-2 \lambda x-2 y-7=0\) and \(3\left(x^2+y^2\right)-8 x+29 y=0\) are orthogonal, then \(\lambda=\)

A block \((P)\) is rotating in contact with the vertical wall of a rotor as shown in figures \(\mathrm{A}, \mathrm{B}, \mathrm{C}\). The relation between angular velocities \(\omega_A, \omega_B\) and \(\omega_C\) so that block does not slide down.
\(\left(R_A \lt R_B \lt R_C \text { radii }\right)\)