The random variable X has a probability distribution \(P(X=r)=\left\{\begin{array}{ll}r k, & \text { if } r \leq 2, \\ (r-1) k, & \text { if } 2 < r \leq 4, \text { where } r \in N \cup\{0\} \text { and } k \in R . \\ 0, & \text { otherwise },\end{array}\right.\), where \(n\) is the set of natural numbers
(A) \(k=\frac{1}{9}\)
(B) \(P(2 \leq X \leq 3)=\frac{1}{2}\)
(C) \(P(X=4)=\frac{1}{3}\)
(D) \(P(X>1)=\frac{7}{8}\)
Choose the correct answer from the options given below:

Match List-I with List-II

List-I	List-II
(A) \(\int_0^1 \frac{x^2}{1+x^3} d x\)	(I) 0
(B) \(\int_0^\pi 3 \sin x d x\)	(II) \(2 \log _e\left(\frac{3}{2}\right)\)
(C) \(\int_{-1}^1 \sin ^5 x \cos ^6 x d x\)	(III) 6
(D) \(\int_2^3 \frac{4}{x^2-1} d x\)	(IV) \(\frac{1}{3} \log _e 2\)

Choose the correct answer from the options given below:

If \(\left|\begin{array}{cc}2 x & 2 \\ 4 & x\end{array}\right|=10\), then \(x\) is:

Which of the following statements is/are true?
(A) \(\left(\tan ^{-1} y-x\right) d y=\left(1+y^2\right)\) dx is a differential equation where variables are separable.
(B) (1+ \(x^2\))dy + 2xydx = cot xdx(x ≠ 0) is a first order linear differential equation.
(C) (4x + 6y + 5)dy - (3y + 2x + 4)dx = 0 is not a homogeneous differential equation.
(D) (xy)dx - (x + \(\left.y^2\right)\)dy = 0 is a homogeneous differential equation.
Choose the correct answert from the option given below :

Consider \(z=x+3 y\) subject to the constraints \(2 x+3 y \geq 6,2 x+y \leq 8, x \geq 0, y \geq 0\). The minimum value of \(z\) is :

If the length of a simple pendulum is decreased by \(3 \backslash \%\), the percentage change in its period T is
[Note : \(T=2 \pi \sqrt{\frac{L}{g}}\) where \(L\) is the length of pendulum and \(g\) is constant]

Let y be a discrete random variable, taking values \(y_1, y_2, y_3, ..., y_n\) with probabilities \(p_1, p_2, p_3, ..., p_n\) respectively. Then the standard deviation of y is given by:

A convex lens of glass (n = 1.5) has a focal length 18 cm in air. When it is immersed in water (n = 4/3) the focal length increases to:

Let \(\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}\), and \(\vec{c}=2 \hat{i}+\hat{j}+4 \hat{k}\).
A vector \(\vec{d}\) which is perpendicular to both \(\vec{a}\) and \(\vec{b}\) and satisfies \(\vec{c} \cdot \vec{d}=14\) is :

Read the passage carefully and answer the Questions
The term vitamin was coined from the word vital +amine since the earlier identified compounds had amino groups.Later work showed that most of them didn't contain amino groups, so the letter "e" was dropped and ther term vitamin is used these days.
Vitamins are classified into two groups depending upon their solubility in water or fat. Fat soluble vitamins are soluble in fat and oils but insoluble in water. They are stored in liver and adipose (fat storing) tissues. Water-soluble vitamins are soluble in water. They must be supplied regularly in the diet as they are readily excreated in urine.
What is Xerophthalmia disease?

A 100 W bulb working on 200 V and a 200 W bulb on 100 V, have their resistances in the ratio

There is a Building which is tilted and equations of two different walls are \(2 x-y+z=4\) and \(x+y+2 z=3\). Angle between the walls is :

A first order reaction is found to have a rate constant, k = 5.5 × 10^-14 s^-1. The half life of the reaction will be :