\(\lim _{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}=\)

If $3\sin xy+4\cos xy=5$, then $\frac{dy}{dx}$ is equal to______

If \(\int x^3 e^{5 x} d x=\frac{e^{5 x}}{5^4}[f(x)]+C\), then \(f(x)\) is equal to

If \(f(t)=\int_0^t \tan ^{(2 n-1)} x d x, n \in N\), then \(f(t+\pi)=\)

\[
\int_{-4 \pi}^{4 \pi} \tan ^9 x \sin ^6 x \cos ^3 x d x=
\]

If the equation of the circle lying in the first quadrant, touching both the coordinate axes and the line \(\frac{x}{3}+\frac{y}{4}=1\) is \((x-c)^2+(y-c)^2=c^2\), then \(c=\)

If the distance between the foci of an ellipse is 6 and the length of the minor axis is 8 , then the eccentricity is

In a steady state, the temperature at the end $A$ and end $B$ of a $20 \mathrm{cm}$ long rod $AB$ are $100°\mathrm{C}$ and $0°\mathrm{C}$. The temperature of a point, $9 \mathrm{cm}$ from $A$ is

If the maximum and minimum voltages of an A.M wave are \(\mathrm{V}_{\max }\) and \(\mathrm{V}_{\text {min }}\) respectively, Then the modulation factor ' \(m\) ' is

If \(\frac{x-4}{x^2-5 x+6}\) can be expanded in the ascending powers of \(x\), then the coefficient of \(x^3\) is

The length of the subnormal at any point on the curve \(y=\left(\frac{x}{2024}\right)^k\) is constant if the value of \(k\) is

A swimming pool has depth of 3 meters. The pressure at the bottom of the pool due to water alone is (Density of water is \(1000 \mathrm{~kg} \mathrm{~m}^{-3}\), Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )

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