\(\int \frac{\sin (x-a)}{\sin (x-b)} d x=A x+B \log |\sin (x-b)|+C \Rightarrow(A, B)=\)

The number of numbers lying between 1000 and 10000 such that every number contains the digits 3 and 7 only once without repetition is

\(\cos 6^{\circ} \sin 24^{\circ} \cos 72^{\circ}=\)

\[
\frac{\cos x}{1+\sin x}+\tan x=
\]

In \(\triangle \mathrm{ABC}\), if \(\mathrm{r}\) is the inradius and \(\mathrm{r}_1, \mathrm{r}_2, \mathrm{r}_3\) are the ex-radii, then \(\frac{1}{4}\left[b^2 \sin 2 C+c^2 \sin 2 B\right]=\)

If the line joining the points \(A(7, p, 2)\) and \(B(q,-2,5)\) is parallel to the line joining the points \(C(2,-3,5)\) and \(D(-6,-15,11)\), then the value of \(p^2+q^2\) is equal to

In a \(\triangle A B C\)
\[
\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4 b^2 c^2}
\]
equals

If the product of the perpendicular distances from any point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) to its asymptotes is 6 and eccentricity of the hyperbola is \(\sqrt{3}\), then the length of the conjugate axis of the hyperbola is

All the words that can be formed using alphabets A, H, L, U, R are written as in dictionary (no alphabet is repeated). Then the rank of the word RAHUL is.......

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

Truth table for the given circuit is

A source of sound \(S\) in the form of a block kept on a smooth horizontal surface is connected to a spring, as shown in the figure. If the spring oscillates with an amplitude of \(50 \mathrm{~cm}\) along horizontal between the wall and the observer \(O\), the maximum frequency heard by the observer is \(12.5 \%\) more than the minimum frequency heard by him. If the mass of the source of sound is \(100 \mathrm{~g}\), the force constant of the spring is (Speed of sound in air is \(340 \mathrm{~ms}^{-1}\) )

The radii and Young's moduli of two uniform wires \(A\) and \(B\) are in the ratio \(2: 1\) and \(1: 2\) respectively. Both wires are subjected to the same longitudinal force. If the increase in length of the wire \(A\) is one percent, the percentage increase in length of the wire \(B\) is