The derivative of \(f(\cot x)\) with respect to \(g(cosec x)\) at \(x=\frac{\pi}{4}\) (where \(\left.f^{\prime}(1)=2, g^{\prime}(\sqrt{2})=4\right)\) is:

Out of the given statement, choose the correct statement.
(A) The direction ratios of the vector \(\vec{a}=3 \hat{i}-\hat{j}+4 \hat{k}\) is \(3,-1,4\).
(B) If \(\theta\) is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \cos \theta\).
(C) The unit vector in the direction of vector \(\vec{a}=\hat{i}+2 \hat{j}-2 \hat{k}\) is \(\hat{a}=\frac{1}{3}(\hat{i}+2 \hat{j}-2 \hat{k})\).
(D) If \(\vec{a}=3 \hat{i}\) and \(\vec{b}=4 \hat{j}\) then \(\vec{a} \cdot \vec{b}=12\).
(E) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a triangle then its area is given of \(\frac{1}{2}|\vec{a} \times \vec{b}|\).
Choose the correct answer from the options given below :

An unbiased die is rolled. If the random variable \(X\) is defined as,
\(X=\left\{\begin{array}{ll}1 & \text { if the outcome is a number less than or equal to } 3 \\0 & \text { if the outcome is a number greater than } 3\end{array}\right.\)
Then the probability distribution of \(X\) is :

The solution of the differential equation \(\frac{d r}{d t}=-r t, r(0)=r_0\) is

The area included between \(y^2=4 x, y=x+1\) and x-axis is :

If \(x=2 t^3\) and \(y=3 t^4\), then \(\frac{d^2 y}{d x^2}\) is equal to :

Match List - I with List - II.

LIST-I (Equation of curves)	LIST-II (Slope of tangent at \(x=2\))
(A) \(y=x^3-x\)	(I) 8
(B) \(y=(x-2)^2\)	(II) \(frac{2}{3}\)
(C) \(y=2 x^2+3\)	(III) \(11\)
(D) \(y=\sqrt{4 x+1}-7\)	(IV) \(0\)

Choose the correct answer from the options given below :

Consider the LPP : Maximize \(z=5 x+3 y\) subject to \(3 x+5 y \leq 15,5 x+2 y \leq 10, x, y \geq 0\).
The optimal feasible solution occurs at

The area bounded by the curve \(x = |y|\) and the line \(x = 2\) is:

Match List-l with List-ll:

List-1 (Molecule)	List-II (Van't Hoff Factor)
(A) Glucose	(I) 3
(B) \(K _2 SO _4\)	(II) 5
(C) \(Ca _3\left( PO _4\right)_2\)	(III) 1
(D) \(K _3\left[ Fe ( CN )_6\right]\)	(IV) 4

Choose the correct answer from the options given below:

Which of the following is the correct equation to find the area of the shaded region, with reference to the figure given below ?

Which of the following is NOT a basic step in the genetic modification of an organism?

If \(A\) is a square matrix of order \(3 \times 3\) and \(|A|=4\). The value of \(\mid(\operatorname{adj} A)\). \(A \mid\) is: