If \(p\) and \(q\) are distinct prime numbers and if the equation \(x^2-p x+q=0\) has positive integers as its roots, then the roots of the equation are

If \(\int x^3 e^{5 x} d x=\frac{e^{5 x}}{5^4}[f(x)]+C\), then \(f(x)\) is equal to

In \(\triangle \mathrm{ABC},\left(r_2+r_3\right) \operatorname{cosec}^2 \frac{A}{2}=\)

If a random variable X satisfies poisson distribution with a mean value of 5 , then probability that \(\mathrm{X} \lt 3\) is

The range of the real valued function $f\left(x\right)=\sqrt{\frac{x^2+2x+8}{x^2+2x+4}}$ is

If two cards are drawn randomly from a pack of 52 playing cards, then the mean of the probability distribution of number of kings is

Two bodies are projected from points \((0,0)\) and \((\sqrt{3}-1,0)\) with velocities \(10 \mathrm{~ms}^{-1}\) and \(\mathrm{v} \mathrm{ms}^{-1}\) respectively, as shown in the figure. The time after which they collide in space is ________.

A circular coil connected to a battery of emf $E$ produced a certain magnetic induction field at its centre. The coil is unwound, stretched to double its length rewound into a coil of $\frac{1^{\mathrm{rd}}}{3}$ of the original radius and connected to a battery of emf $E^{\text{'}}$ to produce same field at the centre. Then $E^{\text{'}}$ is

Using mathematical induction, the numbers $a_n$'s are defined by $a_0=1, a_{n+1}= 3n^2+n+a_n, \left(n\ge 0\right)$, then $a_n=$

Identify the initiation step \(/ \mathrm{s}(\mathrm{X})\) and termination \(\mathrm{step} / \mathrm{s}(\mathrm{Y})\) of free radical chlorination of \(\mathrm{CH}_4\) from the following.
a) \(\dot{\mathrm{C}}_3+\dot{\mathrm{C}} \mathrm{l} \rightarrow \mathrm{CH}_3-\mathrm{Cl}\)
b) \(\dot{\mathrm{Cl}}+\dot{\mathrm{C}} l \rightarrow \mathrm{Cl}-\mathrm{Cl}\)
c) \(\dot{\mathrm{C}}_3+\mathrm{Cl}_2 \rightarrow \mathrm{CH}_3-\mathrm{Cl}+\dot{\mathrm{Cl}}\)
d) \(\mathrm{Cl}_2 \rightarrow 2 \dot{\mathrm{Cl}}\)
e) \(\mathrm{CH}_4+\dot{\mathrm{Cl}} \rightarrow \dot{\mathrm{C}} \mathrm{H}_3+\mathrm{HCl}\)
f) \(\mathrm{CH}_3+\dot{\mathrm{C}}_3 \rightarrow \mathrm{CH}_3-\mathrm{CH}_3\)

The magnitude of maximum acceleration is \(\pi\) times that of maximum velocity of a simple harmonic oscillator. The time period of the oscillator in seconds is

Two bolts, two nuts and three needles are in a box. Two parts are choosen at random from the box. What is the probability that one is a
bolt and one is a needle?