If \(0 < \theta < \frac{\pi}{2}\), then solution of the equation \(\sin \theta-3 \sin 2 \theta+\sin 3 \theta=\cos \theta-3 \cos 2 \theta+\cos 3 \theta\) is

If \(\alpha\) is a root of the equation \(25 \cos ^2 \theta+5 \cos \theta-12=0\), for \(\frac{\pi}{2} < \alpha < \pi\), then \(\sin 2 \alpha=\)

The smallest positive value of \(x\) (in degrees) for which \(\tan \left(x+100^{\circ}\right)=\tan \left(x+50^{\circ}\right) \tan (x) \tan \left(x-50^{\circ}\right)\) is

The angle between the tangents drawn from the point \((1,4)\) to the parabola \(y^2=4 x\) is

If in the angles of a triangle are in the ratio \(1: 1: 4\), then the ratio of the perimeter of the triangle to its largest side is

If \(a\) and \(b\) are any two real numbers, then
\(\left|\begin{array}{ccc}
2 a-2 b-4 & 4 a & 4 a \\
4 & 2-b-a & 4 \\
2 b & 2 b & b-a-2
\end{array}\right|=\)

The angles of a triangle are in the ratio \(3: 5: 10\). Then the ratio of the smallest side to the greatest side is :

Balance the following equation by choosing the correct option. \(x \mathrm{KNO}_3+y \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} \longrightarrow p \mathrm{~N}_2+q \mathrm{CO}_2+r \mathrm{H}_2 \mathrm{O}+s \mathrm{~K}_2 \mathrm{CO}_3\)

Area of the triangle formed by the lines \(3 x^2-4 x y+y^2=0,2 x-y=6\) is

\(L_1\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}-3 \hat{\mathbf{k}}\). \(L_2\) is a line passing through the points with position vectors \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\). Then the distance between \(L_1\) and \(L_2\) is

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

At room temperature, gaseous hydrogen is bombarded with a beam of electrons of 13.6 eV energy. The series to which the emitted spectral line belongs to

How many of the following metals give oxides and nitrides when burnt in air?
\(\mathrm{Be}, \mathrm{Na}, \mathrm{Mg}, \mathrm{Ba}, \mathrm{Sr}, \mathrm{Li}, \mathrm{K}\)