Ques 1.

State whether the following statements are true or false giving reasons in support of your answer.

i) A binomial distribution has mean 3 and variance 4.

ii) The function  has no maxima or minima.

iii) The line of regression of x on y is the same as the line of regression of y on x .

iv) The plane  is parallel to the line 

v) A continuous random variable can have probability density function

Ques 2.

a) Define  Is f injective, surjective, monotone? 

Ques 3.

b) A bag contains 3 white balls and 2 red balls. Another bag contains 5 white and 3 red balls. A bag is chosen at random and a ball is drawn from it. Find the probability that it is white.

Ques 4.

c) Compute the correlation co-efficient between X and Y for the following data: 

Ques 5.

a) Find asymptotes of the graph of the function 

Ques 6.

b) Find the term independent of a in the binomial expansion of 

Ques 7.

c) In a certain Poisson distribution the probability of 3 successes is exactly equal to the probability of 4 successes. Find its mean and standard deviation. Also find the probability of more than 1 success for the given distribution. 

Ques 8.

a) Integrate 

Ques 9.

b) Find the mean and the standard deviation of a random variable with the following probability density function: 

Ques 10.

c) A club has 9 member having ages 21, 28, 23, 29, 52, 43, 32, 37 and 30 years. One has to be at least 30 years of age to be eligible for the presidentship of the club. A simple random sample of size 5 is selected to provide an estimate of the population proportion eligible for  presidentship. Find the mean and the standard error of this estimate

Ques 11.

a) Find the radius of the sphere which passes through the points ),1,0,0( ),,0,0,1( )0,1,0( and ).1,0

Ques 12.

b) Verify Euler’s theorem for the function 

Ques 13.

c) Two groups of 10 plants each were grown on two different fertilizers. The average height of first group of plants was 92.44 cm, with a standard deviation of 4 cm. For the second group, the average height was 90 cm with a standard deviation of 2 cm. At 5% level of significance test the hypothesis that first fertilizer is better than the second in terms of the plant growth. 

Ques 14.

a) Find (a × b)⋅ c where

Ques 15.

b) Solve the differential equation cos 

Ques 16.

c) A garden pea plant is genetically mixed for the gene pair Tt, where the gene T (for tall) is dominant over the gene t (for short). The plant produced 40 tall and 20 short offspring. Using  test find out whether the plant was self fertilized or fertilized by a short plant at 5% level of significance. 

Ques 17.

a) It is known that 10 men out of every 100 men and 30 women out of every 1000 women are color blind. In a community, half the population is male. Using Baye’s theorem find the probability that a color blind person chosen at random from among all color blind persons in the community is male.

Ques 18.

b) Show that sin x 1( + cos x) has a maximum at 

Ques 19.

c) Find the sum of the first n terms of the series log 2 + log6 + log18 + log54 + ..... 

Ques 20.

a) If  is defined by  then show that f is a bijection. Find the formula that defines

Ques 21.

b) Obtain 

Ques 22.

c) The difference of mean and variance of a binomial distribution of 9 trails is 4. Find the probability p of success of the binomial distribution. Also find the probability of (i) exactly two successes and (ii) less than 2 successes.

Ques 23.

a) A and B are two events which are independent. The probability that both A and B occur is   and the probability that neither of them occurs is  Find the probability of the occurrences of A and B. 

Ques 24.

b) If  prove tha 

Ques 25.

c) Fit a straight line for regression of Y on X from the following table.

Find the value of y when 

Ques 26.

a) Calculate: (i) Quartile deviation and (ii) Mean Deviation from mean for the following data:

Ques 27.

b) Find two positive numbers x and y such tha  60and  is maximum. 

Ques 28.

c) Find the angle between the planes  and  Alsospecify the type of the angle obtained.