If \(2.4^{2 \mathrm{n}+1}+3^{3 \mathrm{n}+1}\) is divisible by \(k\) for all \(n \in N\), then \(k=\)

\(\int_{\frac{1}{\sqrt[5]{31}}}^{\frac{1}{\sqrt[5]{242}}} \frac{1}{\sqrt[5]{x^{30}+x^{25}}} d x=\)

In \(\triangle \mathrm{ABC}, \frac{\sin 2 \mathrm{~A}+\sin 2 \mathrm{~B}+\sin 2 \mathrm{C}}{\cos \mathrm{A}+\cos \mathrm{B}+\cos \mathrm{C}-1}=\)

Let \(u, v\) and \(w\) be non-coplanar vectors. Then the points corresponding to which of the following vectors are collinear.

\({ }^{2 \mathrm{n}} \mathrm{C}_4:{ }^{\mathrm{n}} \mathrm{C}_3=99: 4 \Rightarrow \mathrm{n}=\)

If \(A, B\) and \(C\) are mutually exclusive and exhaustive events of a random experiment such that \(P(B)=\frac{3}{2} P(A)\) and \(P(C)=\frac{1}{2} P(B)\), then \(P(A \cup C)\) equals to

The area (in square unit) of the triangle formed by \(x+y+1=0\) and the pair of straight lines \(x^2-3 x y+2 y^2=0\) is

A girl of height \(150 \mathrm{~cm}\) with her eye level at \(140 \mathrm{~cm}\) stands in front of plane mirror of height \(75 \mathrm{~cm}\) fixed to a wall. The lower edge of the mirror is at a height of \(85 \mathrm{~cm}\) above her feet level. The height of her image the girl can see in the mirror is

If \(\frac{x^2+x+1}{x^2+2 x+1}=A+\frac{B}{x+1}+\frac{C}{(x+1)^2}\), then \(A-B\) is equal to

Match the following List I with List II.
\(\begin{array}{|c|c|c|}
\hline & \text { List I } & \text { List II } \\
\hline \text { (A) } & \text { Transverse wave (i) } & \begin{array}{l}
\text {Vibrations parallel to the } \\
\text {direction of propagation }
\end{array} \\
\hline \text { (B) } & \begin{array}{l}
\text {Longitudinal wave }
\end{array} & \begin{array}{l}
\text {Vibrations perpendicular to } \\
\text {the direction of propagation }
\end{array} \\
\hline \text { (C) } & \text { Beats } & \begin{array}{l}
\text {Superposition of waves } \\
\text {travelling in the opposite directions }
\end{array} \\
\hline \text { (D) } & \text { Stationary waves (iv) } & \begin{array}{l}
\text {Superposition of waves } \\
\text {travelling in same direction }
\end{array} \\
\hline
\end{array}\)
The correct answer is

The efficiency of a heat engine that works between the temperatures where Celsius-Fahrenheit scales coincides and Kelvin-Fahrenheit scales coincides is (approximately)

Which of the following conditions are not suitable for a spontaneous reaction?

The solution of \(\left(x^2+y^2\right) d x=2 x y d y\) is :