If the interval in which the function \(f(x)=4 x^3-6 x^2-72 x+30\) is strictly decreasing, is \(( a , b )\) then \(a + b\) is equal to

If \(A\) is an invertible matrix of order 2 , then \(\operatorname{det}\left((\operatorname{adj} A)^{-1}\right)\) is equal to

A linear programming problem is as follows:
Maximize/minimize objective function \(z=2 x-y+5\) subject to constraints:
\(3 x+4 y \leq 60, \quad x+3 y \leq 30, \quad x \geq 0, y \geq 0 .\)
If the corner points of feasible region are \(A(0,10), B(12,6), C(20,0), O(0,0)\), then which of the following is true:

If \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=2 \hat{i}+2 \hat{j}+\hat{k}\) then
Match List-I with List-II

List-I	List-II
(A) Projection of \(\vec{a}\) on \(\vec{b}\) is	(I) \(-7 \hat{i}+4 \hat{j}+6 \hat{k}\)
(B) \(\vec{a} \times \vec{b}\) is	(II) \(\frac{1}{\sqrt{101}}(-7 \hat{i}+4 \hat{j}+6 \hat{k})\)
(C) Unit vector along \(\vec{a}+\vec{b}\) is	(III) \(\frac{5}{3}\)
(D) Unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) is	(IV) \(\frac{1}{\sqrt{33}}(4 \hat{i}+\hat{j}+4 \hat{k})\)

Choose the correct answer from the options given below :

The following data are from a simple random sample: 6, 8, 11, 9, 15, 17,
then the point estimate of the population mean is:

If \(f(x) = \begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix}\), then \(f(0)\) is:

The variance of the number of heads in two tosses of a coin is :

The vapour pressure of pure benzene is } \(0.900\) bar. If a non-volatile electrolytic solid is added, the vapour pressure of the solution becomes \(0.200\) bar. If the number of moles of solute and solvent in the solution are \(1\) and \(2\) respectively, what is the value of van't Hoff factor (i)?

A circular loop is placed in the plane of the paper in a magnetic field B. A current is induced in the loop in an anticlockwise direction. This is because the magnetic field B is

Match List-I and List-II

List-I	List-II
A. Phenol + CH3Cl is presence of anhy AlCl 3	I. Salicylaldehyde
B. Phenol + Br2 water	II. 2,4,6 -Tribromophenol
C. Phenol + CH3Cl/OH	III. Salicylic acid
D. Phenol + CO2 \(/ NaOH\)	IV. p-cresol

Choose the correct answer from the options given below :

The value of \(\lambda\) so that the lines \(\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}\) are at right angle, is :

Read the following passage carefully and answer the given questions.
Immune System in the Body
The human immune system consists of lymphoid organs, tissues, cells and soluble molecules like antibodies. The primary lymphoid organs are bone marrow and thymus where immature lymphocytes differentiate into antigen-sensitive lymphocytes. After maturation the lymphocytes migrate to secondary lymphoid organs like spleen, lymph nodes, tonsils, Peyer's patches of small intestine and appendix. The secondary lymphoid organs provide the sites for interaction of lymphocytes with the antigen, which then proliferate to become effector cells. The bone marrow is the main lymphoid organ where all blood cells including lymphocytes are produced. Each antibody molecule has four peptide chains, two small called light chains and two longer called heavy chains. Hence, an antibody is represented as H2L2. Different types of antibodies are produced in our body. IgA, IgM, IgE, IgG are some of them. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs.
The site of interaction of lymphocytes with antigen is

A point charge – 5 µC is at a distance 5 cm directly above the centre of a square of side 10 cm.
The magnitude of electric flux through this square is: