Let \(X\) denote the number of hours a student studies on a selected day. The probability distribution of \(X\) is given by (where \(k\) is some unknown constant)
\(P\left(X=x_i\right)=\left\{\begin{array}{ll}0.5, & \text { if } x_i=0 \\k x_i, & \text { if } x_i=1 \\k\left(4-x_i\right), & \text { if } x_i=2 \text { or } 3 \\0, & \text { otherwise }\end{array}\right.\)
Then the value of \(k\) is

If \(x, y\) and \(z\) are non-zero distinct numbers, then \(\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1\end{array}\right|\) is equal to

The area of the triangle with vertices \((1,4),(2,7)\) and \((4,13)\) is:

Anush takes a loan of ₹ \( 150000\) @ \(16 \%\) annual interest for 5 years. His EMI (Equally Monthly Installmant) on monthly basis under flat rate system is :

Assume P, Q, R and W are matrices of order \(3 \times 3, a \times 4, b \times c\) and \(d \times a\) respectively. If PQ + WR is well is well defined, find the value of ab + cd

(A) \(\frac{d}{d x} a^x=\frac{a^x}{\log _e a}\)
(B)\(\frac{d}{d x} e^{a x}=a e^{a x}\)
(C) \(\frac{d}{d x} x^x=x \cdot x^{x-1}\)
(D) \(\frac{d}{d x} x^a=a x^{a-1}\)
(E) \(\frac{d}{d x} a^a=0\)
Choose the correct answer from the options given below:

The integral \(\int \frac{d x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}\) equals __________ .

Arrange the following alcohols in increasing order of their boiling points.
(A) Propan-1-ol
(B) Butan-2-ol
(C) Butan-1-ol
(D) Pentan-1-ol
Choose the correct answer from the options given below:

A problem in mathematics is given to three students whose chances of solving it are \(\frac{1}{2}, \frac{1}{3}\) and \(\frac{1}{4}\) respectively, the probability that the problem is solved is ______________.

A non-polar dielectric material is placed in an external electric field. Choose the correct statement(s) from the following:
(A) There will be no change in the positions of positive and negative charges inside the dielectric.
(B) The positive charges of molecules will shift opposite to the direction of the electric field.
(C) The negative charges of molecules will shift in the direction of the electric field.
(D) The positive charges of the molecules will shift in the direction of the field and negative charges of the molecules will shift opposite to the field.
Choose the correct answer from the options given below:

A bag contains 3 white, 4 black and 2 red balls. If 2 balls are drawn at random (without replacement), then the probability that both the balls are white is:

The solution set of \(6 \leq-3(2 x-4)<12, x \in R\) is :

Which of the following reactions is the most suitable for the preparation of n-propylbenzene?