There are four machines and it is known that exactly two of them are faulty. They are tested one by one, in a random order till both the faulty machines are identified. Then, the probability that only two tests are need is

An integrating factor of the differential equation \(\left(x^2+1\right) \frac{d y}{d x}+x y=x^3\) is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is

The value of \(\left\{i^{22}-\left(\frac{1}{i}\right)^{35}\right\}^2\) is

The distance between the pair of lines \(x^2+2 \sqrt{2} x y+2 y^2+4 x+4 \sqrt{2} y+1=0\), is

If one ticket is selected at random from 30 tickets each with a distinct number from 1 to 30 , then the probability that the number on the selected ticket is a multiple of 3 or 5 is

If \(\frac{1}{6} \sin \theta, \cos \theta\) and \(\tan \theta\) are in geometric progression, then the solution set of \(\theta\) is

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

If \(b+c=3 a\), then \(\cot \frac{B}{2} \cot \frac{C}{2}\) is equal to :

If the radical axis of the circles \(x^2+y^2+2 g x+2 f y+c=0\) and \(2 x^2+2 y^2+3 x+8 y+2 c=0\) touches the circle \(x^2+y^2+2 x+2 y+1=0\), then

The oxidation state and covalency of \(\mathrm{Al}\) in \(\left[\mathrm{AlCl}\left(\mathrm{H}_2 \mathrm{O}\right)_5\right]^{2+}\) are and respectively

The distance between the lines represented by \(x^2+2 x y+y^2-8 m x-8 m y-9 m^2=0\) is

The minimum wavelength of X-rays produced by 20 kV electrons is nearly