If Rolle's Theorem is applicable for the function:
\[ f(x) = \begin{cases} x^p \log x, & x \neq 0 \\ 0, & x = 0 \end{cases} \]
on the interval \([0,1]\), then a possible value of \( p \) is:
A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is: \
Suppose \( \theta_1 \) and \( \theta_2 \) are such that \( (\theta_1 - \theta_2) \) lies in the 3rd or 4th quadrant. If \[ \sin\theta_1 + \sin\theta_2 = \frac{21}{65} \quad \text{and} \quad \cos\theta_1 + \cos\theta_2 = \frac{27}{65} \] then \[ \cos\left(\frac{\theta_1 - \theta_2}{2}\right) = \]
Given below are two statements, one is labelled as Assertion (A) and the other one labelled as Reason (R).Assertion (A): \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty = \sqrt{4} \] Reason (R): \[ |x| <1, \quad (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \]
With respect to the roots of the equation \( 3x^3 + bx^2 + bx + 3 = 0 \), match the items of List-I with those of List-II.
The number of ways of arranging all the letters of the word "COMBINATIONS" around a circle so that no two vowels come together is
A man has 7 relatives, 4 of them are ladies and 3 gents; his wife has 7 other relatives, 3 of them are ladies and 4 gents. The number of ways they can invite them to a party of 3 ladies and 3 gents so that there are 3 of man's relatives and 3 of wife's relatives, is
If three numbers are randomly selected from the set \( \{1,2,3,\dots,50\} \), then the probability that they are in arithmetic progression is:
