In \(\triangle A B C\), if \(a=8, b=10, c=12\), then \(\frac{r}{R}=\)

If \(\sec ^{-1} \frac{x}{a}-\sec ^{-1} \frac{x}{b}=\sec ^{-1} b \sec ^{-1} a\), then \(x\) is equal to

The number of normals drawn to the parabola \(y^2=4 x\) from the point \((1,0)\) is

\(\int\left[(\log 2 x)^2+2 \log 2 x\right] d x=\)

If \(a^2 x^4+b^2 y^4=c^6\) then maximum value of \(x y=\)

For what values of \(m \in N\), the following divisibility \(x+y / x^m+y^m\) holds?

Minimum value of \(5 \tan ^2 \alpha+\frac{9}{\tan ^2 \alpha}+4 \sec ^2 \alpha\) is

The circumference of a circle is measured as \(56 \mathrm{~cm}\) with an error \(0.02 \mathrm{~cm}\). The percentage error in its area is

In the figure shown, \(\mathrm{AB}\) is a rod of length \(30 \mathrm{~cm}\), area of cross-section \(1 \mathrm{~cm}^2\) and thermal conductivity \(336 \mathrm{SI}\) units. The ends \(\mathrm{A}\) and \(\mathrm{B}\) are at constant temperatures \(20^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\) respectively. A point \(\mathrm{C}\) of the rod is connected to ice at \(0^{\circ} \mathrm{C}\) in a thermally insulated box \(\mathrm{D}\) through a highly conducting wire of negligible heat capacity. The rate at which ice melts in the box is \(\left(\mathrm{L}_{\mathrm{ice}}=80 \mathrm{cal} \mathrm{g}^{-1}\right)\)

A bus is moving with a velocity of \(5 \mathrm{~ms}^{-1}\) towards a huge wall. The driver sounds horn of frequency \(165 \mathrm{~Hz}\). If the speed of sound in air is \(335 \mathrm{~ms}^{-1}\), the number of beats heard per second by the passengers in the bus will be

\(\int_{-1}^{3 / 2}|x \sin \pi x| d x=\)

A message signal of frequency \(8 \mathrm{kHz}\) and peak voltage \(12 \mathrm{~V}\) is used to modulate a carrier of frequency 1.2 MHz and peak voltage \(20 \mathrm{~V}\). The modulation index is

The diagram shows different transitions across the energy levels for an electron in a certain atom. Among these, which transition represents the emission of a photon with the most energy?