The value of \(\int_{-a}^a f(x) d x\) where \(f(x)=\frac{7^x}{1+7^x}\), is :

If \(\vec{a}\) and \(\vec{b}\) are unit vectors, then the angle between \(\vec{a}\) and \(\vec{b}\) for \(\vec{a}+\sqrt{3 b}\) to be a unit vector is given by :

A integrating factor of the differential equation \(\frac{d y}{d x}+\frac{y}{x}=\frac{1}{x^2}(x>0)\) is equal to

The co-ordinates of the point at which the line \(\frac{x-3}{3}=\frac{y+1}{2}=\frac{z-4}{-2}\) crosses \(x y\) plane, are

A die is tossed once. If the random variable X is defined as \(\mathbf{X}=\left\{\begin{array}{ll}1, & \text { if the die result in an odd number } \\ -1, & \text { if the die result in an even number }\end{array}\right.\) then the variance of X is

The function \(f(x)=-\frac{3}{4} x^4-8 x^3-\frac{45}{2} x^2+163\) has a local maxima at

If A and B are symmetric matrices of the same order, then

The random variable X has the following probability distribution

X	0	1	2	3
P(X)	a	a	b	b

such that E\(\left(x^2\right)\) = 2E(x), then the value of b is:

The constraints of the given shaded feasible region below of an L.P.P., for non-negative variable constraints \(x\) and \(y\) are

Read the passage carefully and answer the questions
Carbohydrates such as cane sugar, glucose, starch etc. belong to a class of naturally occurring organic compounds. Structurally these are polyhydroxy aldehydes or ketones or the compounds which produce such units on hydrolysis. Carbohydrates which are sweet in taste are called sugars. Theses sugars can be reducing or non-reducing. Monosaccharides cannot be hydrolyzed to simpler units whereas Oligosaccharides yield two to ten monosaccharide units on hydrolysis. Polysaccharides yield large number of monosaccharide units on hydrolysis. This class of biomolecules is essential for living systems as they serve as storage molecules in plants and animals. In addition, they provide raw materials for important industries like furniture, textiles, paper, lacquers and breweries. Two aldopentose form important constituents of nucleic acids. Carbohydrates are also called saccharides (Greek: sakcharon which means sugar). Sugar which is used in household and in milk also comes under this class of biomolecules.
Sucrose on hydrolysis yields

Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white balls. A bag is selected at random and then a ball is drawn from it. The probability that the ball drawn is black is :

In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of \( 3 \times 10^{10} \text{ Hz} \) and its amplitude is 30 V/m. The amplitude of the oscillating magnetic field is:

An electron is projected with uniform velocity along the axis of a current carrying long solenoid. Which of the following is true ?