(Students are expected to solve either part B or part C )

Part B

Q22) Find the projection of = 2 - + on = - 2 + .

Q23) Find the centre and radius of the following sphere:
2x2 + 2y2 + 2z2 - 4x - 6y + 2z + 3 = 0.

Q24) Find the magnitude and direction of resultant of two forces of magnitude 10N and 15N, which are inclined to each other at an angle of 60^o.

Q25) Find the two like parallel forces, acting at a distance of 3m apart which is equivalent to a force 12N acting at a distance of 0.5m from one of the forces.

Q26) A particle moving with uniform acceleration passes over 720cm in the 11th second and 960cm in the 15th second of its motion .Find the initial velocity of the particle.

Q27) Find so that the four points with positive vector -6 + 3 + 2 ,  3 + + 4, 5 + 7 + 3 and -13 + 17 - are coplanar.

Q28) Solve the differential equation:
dx/dy = (x + 2y - 1)/(x + 2y + 1)

Q29) Find the vector equation of plane passing through the intersection of planes
. (2 - 7 + 4) = 3 and  . (3 - 5 + 4) + 11 = 0
and passing through the point (-2, 1, 3)

Q30) A particle is projected with a velocity of 100m/sec, at an angle of elevation of 30^o find:
(i) the equation of its path.
(ii) the length of the latus rectum of its path.
(iii) the greatest height attained.
(iv) the time of flight and
(v) the horizontal range.
[take g = 10m/s²]

Part C

Q22) For a certain distribution the arithmetic mean is 45, median is 48 and Karl Pearson's coefficient of Skewness is -0.4. Calculate the S.D. of the distribution.

Q23) An unbiased coin is tossed 6 times. Find using Binomial distribution, the probability of getting at least 5 heads.

Q24) Find the present value of an annuunity of Rs. 1,200 payable at the end of each of 6 month for 3 years when the interest is earned at 8% per year compounded semi-annually. [Take (1.04)⁶ = 1.2653]

Q25) The cost function of a firm is given by
C = 4x² - x + 70
find (i) the average cost (ii) the marginal cost , when x = 3.

Q26) X draws a bill on Y for Rs. 3,000 on 1st March, 1999 payable after 3 months. The bill is discounted by X, as he is in need of money, Compute the discount in each of the following cases:
(i) The bill is discounted for Rs 2,840
(ii) The bill is discounted at 8% per annum.

Q27) Ten percent of bulbs produced in a certain factory turn out to be defective. Find the probability using poisson distribution, that in a sample of 10 bulbs chosen at random, (i) Exactly 2 (ii) more than 2 bulbs, will be defective (Take e^-1 = 0.368).

Q28) A began a business unit with Rs. 85,000. After 2 month B joined with a certain capital. At the end of the year, the profit were divided in the ratio of 5 : 2. Find how much money was invested by B.

Q29) A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X , 2 units of labour and 3 unit of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Y are priced as Rs 100 and Rs 120 per unit respectively, how should the producer use his resources to maximise the total revenue? Solve the problem, graphically.

Q30) A company has two plants to manufacture scooters. Plant - I manufactures 70% of the scooters and Plant - II manufactures 30%. At Plant - I, 80% of the scooters are rated of standard quality and at Plant - II 90% of the scooters are found to be of standard quality. A scooter is chosen at random and is found to be of standard quality. Find the probability that it has come from Plant - II.