\(\int \sin x \sin 2 x \sin 3 x d x\) is equal to:

If the function
\(f(x)=\left\{\begin{array}{ll}a x+2, & x \leq 1 \\x^2+3 x+b, & x>1\end{array}\right.\)
is differentiable at \(x=1\), then the value of \((2 a+b)\) is

(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:

If z = 5x + 8y is the objective function of a LPP and (0,0), (3, 1), (2, 4), (0, 3), (5,0) are corner points of the bounded feasible region, then the maximum value of the objective function is

\(\sin ^{-1}\left(\cos \frac{3 \pi}{5}\right)\) equals

If \(f(x)=\frac{1}{2 x+1}, x \neq-\frac{1}{2}\), then \(f[f(x)]\) is :

A coin is tossed twice and outcomes are recorded. If the random variable \(X\) represents the number of heads in the experiment, then the expectation of X will be :

Which of the following statement is false?

Three resistances of 1 \(\Omega\), 2 \(\Omega\) and 3 \(\Omega\) are given. In order to obtain an equivalent resistance of 11/3 \(\Omega\) they are to be connected as

Year(x)	2011	2012	2013	2014	2015	2016	2017
Sales (in lakh Rs. ) (y)	65	68	70	72	75	67	73

If \(y=70+0.964(x-2014)\) be a straight line trend fitted by the method of the least squares to the above data, then the trend value for the year 2015 is :

If \(A^{-1}\) exists for the matrix \(A=\left[\begin{array}{ccc}1 & \lambda & -1 \\ -1 & 1 & 0 \\ \lambda & 1 & 1\end{array}\right]\) then

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.
An antibody is represented as -

Magnetic moment of \(\text{Co}^{2+}\) will be