\((\vec{a} \cdot \hat{i})^2+(\vec{a} \cdot \hat{j})^2+(\vec{a} \cdot \hat{k})^2\) is equal to:

The probability distribution of a random variable \(x\) is, \(P(x)=\frac{k}{2^x}, x=0,1,2,3\). Then
Match List-I with List-II

List-I	List-II
(A) k	(I) \(\frac{2}{15}\)
(B) \(P ( x =1)\)	(II) \(\frac{1}{5}\)
(C) \(P (1<x<3)\)	(III) \(\frac{8}{15}\)
(D) \(P(x \geq 2)\)	(IV) \(\frac{4}{15}\)

Choose the correct answer from the options given below :

If E and F are independent events associated with an experiment, then which one of the following statements is correct?

A machine costing Rs.50,000 has a useful life of 4 years .The estimated scrap value is Rs 10,000. The rate of depreciation per annum is :

If the area of the triangle with vertices \(A(1,3), B(2,0), C(k, 2 k)\) is \(\frac{k^2}{2}\) sq. units, then the value(s) of \(k\) is/are:

The region represented by the system of inequalities \(x \geq 0 ; y \geq 0\),
\(2 x+y \geq 8\) and \(x+2 y \geq 10\) is :

The number of vectors of unit length perpendicular to both the vectors \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + 3\hat{k}\) is:

If \(x, y\) and \(z\) are non-zero distinct numbers, then \(\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1\end{array}\right|\) is equal to

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in 'Swiss cheese' are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
Baculoviruses are -

The function \(f:[-1,1] \rightarrow \mathbb{R}\) (set of real numbers) given by \(f(x)=\frac{x}{x+3}\) is

The standard deviation of a sampling distribution of a statistic is also known as:

If a fair coin is tossed 10 times, then the probability of obtaining at least one head is:

The present value of a sequence of payments of Rs. \(2000\) made at the end of every \(6\) months and continuing forever, if money is worth \(8 \%\) per annum compounded semi-annually, is: