If a sample has \(n\) observations \(x_1, x_2, \ldots, x_n\) with \(m\) constraints on these values, then the degree of freedom of the sample statistic is:

The area of a triangle having the points \(A (1,1,1), B (1,2,3)\) and \(C (2,3,1)\) as its vertices is :

If \(y=\log x^5\), then \(\frac{d^2 y}{d x^2}\) is given by :

Which one of the following set of constraints represents the shaded region given below?

The probability distribution of a random variable X is given below:

x	-5	-4	-3	-2	-1	0	1	2	3	4	5
P(X)	p	2p	3p	4p	5p	7p	8p	9p	10p	11p	12p

Sudhir had Rs 45,000. He lent a part of it at 4% p.a. and the remaining at 10% p.a. simple interest. He earned a total profit at 8% p.a. on the money lent, what sum did he lend at 10% p.a ?

The function \(f(x)=\left\{\begin{array}{ll}\frac{\sin 2 x}{x}+\cos x & , \text { if } x \neq 0 \\ K & , \text { if } x=0\end{array}\right.\) is continuous at \(x = 0\) then the value of K is:

The charge flowing in a conductor varies with time as \(q(t) = (8t^2 - 4t + 5)C\). Then the current changes at the rate of :

The person with karyotype 44+XXY is suffering from:

In the month of January, the number of cases diagnosed of influenza epidemic are given in the following table:

Date	1	2	3	4	5	6	7
Number of cases	2	0	5	12	20	27	46

The 3-day moving averages are:

Read the passage carefully and give the answers to the question :
Does the number of species in a community really matter to the functioning of the ecosystem? This is a question for which ecologists have not been able to give a definitive answer. For many decades, ecologists believed that communities with more species, generally, tend to be more stable than those with less species. According to the International Union for Conservation of Nature and Natural Resources (IUCN) (2004), the total number of plant and animal species described so far is more than 1.5 million.
The scientist who proved that species richness directly correlates with the stability of a community, was

1It is given that 3% of items manufactured by an industry are defective. The probability that a packet of 250 items contains one defective item is: [Given: \(e^{-7.5}\) 0.000553]

If the objective function for an L.P.P. is \(z=3 x-4 y\) and the corner points for the bounded feasible region are \((0,0),(5,0),(6,5),(6,8),(4,10)\) and \((0,8)\), then the minimum value of \(z\) occurs at :