If the angles of a triangle $ABC$ are in the ratio $1: 2: 3 ,$ then the corresponding sides are in the ratio

\(\int_{-5 \pi}^{5 \pi}(1-\cos 2 x)^{\frac{5}{2}} d x=\)

In a triangle ABC . if \(r_1=2 r_2=3 r_3\), then \(\sin \mathrm{A}: \sin \mathrm{B}: \sin \mathrm{C}=\)

The number of ways of awarding 9 scholarships among three students so that each may have 3 scholarships is

Choose the correct option about the matrices given below
\(\begin{aligned} & A=\left[\begin{array}{ccc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1\end{array}\right] \\ & B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{array}\right] \\ & C=\left[\begin{array}{ccc}\cos \frac{\pi}{6} & 0 & \sin \frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\end{array}\right]\end{aligned}\)
\(D=\left[\begin{array}{ccc}
\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\
-\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\)

In \(\triangle A B C\), if \(r_1=2 r_2=3 r_3\), then

The length of the common chord of the circles of radii 15 and 20, whose centres are 25 unit of distance apart, is

Eight spherical rain drops of the same mass and radius are falling down with a terminal speed of \(6 \mathrm{~cm}-\mathrm{s}^{-1}\). If they coalesce to form one big drop, what will be the terminal speed of bigger drop? (Neglect the buoyancy of the air)

Activation energy for the hydrolysis of sucrose by acid is \(\mathrm{X} \mathrm{kJ} \mathrm{mol}^{-1}\) whereas activation energy for the hydrolysis of sucrose by sucrase is \(\mathrm{YkJ} \mathrm{mol}^{-1}\). X and Y respectively are

The length of a rod under longitudinal tension \(T_1\) is \(L_1\) and that under longitudinal tension \(T_2\) is \(L_2\). What is the actual length of the rod, in the absence of tensions?

Which anong the following is a bactericidal antibiotic?

The minimum horizontal speed with which a body must be projected, so that it goes around a smooth vertical circular track of radius \(4 \mathrm{~m}\) is \(\left(g=9.8 \mathrm{~ms}^{-2}\right)\)

Identify the correct statements from the following
i. Reaction of hydrogen with fluorine occurs even in dark.
ii. Manufacture of ammonia by Haber process is an endothermic reaction
iii. HF is electron rich hydride