From 7 men and 4 women a committee of 6 is to be formed. In how many ways can this be done (i) when the committee contains exactly 2 women, (ii) at least 2 women?
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Find the values of x for which the function
f ( x) = x 2 − 2x + 1
is increasing or decreasing. Sketch the graph of y = f (x) .
Evaluate the following integrals:
i) 
ii) ∫ x ln | x + 1| dx
A training director wishes to find out whether a particular training programme increases the efficiency of the employees. He plans to take a random sample of employees and record their efficiency rating before the training starts. After the completion of the programme, the efficiency of the same sample of employees is recorded as given in the table below. At 0.05 level of significance do you reject or accept the null hypothesis
H 0 : µd = 0 against H1 : µd > 0
where µd = 0 is the mean of the difference?for a population of size N = 120 with a mean of 7.5 and a standard deviation of 1.5, find the estimated standard error of the mean for the following sample sizes:
- ) n = 9 ii) n = 25
Records show that the probability is 0.00006 that a car will have a flat tire while driving through a certain tunnel. Find the probability that at least 2 of 10,000 cars passing through this tunnel will have flat tires.
Let x be the size of a certain population of predators and y the size of the population of prey upon which they feed. As function of time t, x = t 2 + 4 and
y = 2t 2 − 3t . Let u be the number of prey to each predator. Find the rate of change of u .
If p is the length of the perpendicular from the origin to the line x/a + y/b = 1 prove that 1/p2 = 1/a2 + 1/b2.