## Monday, June 19, 2017

2015
MATHEMATICS
Full Marks: 100
Pass Marks: 30

Time: 3 hours
The figures in the margin indicate full marks for the questions

1. Answer the following questions: 1x10=10
1. If , what is the number of relations on ?
2. Find the principal value of .
3. If , what is the order of the matrix ?
4. If is a nonsingular matrix such that , what is ?
5. What is the co-factor of 7 in the determinant 6. Is the derivative of an even function even?
7. Is the function increasing?
8. What are the direction cosines of the vector ?
9. If the distance of a place from the origin be ‘d’ and direction cosines of the normal to the plane through origin be , what are the co-ordinates of the foot of the normal?
10. What are the equations of the planes parallel to and at a distance ‘a’ from it?
2. A function is defined by . Is the function one-one, and onto? Justify your answer. 2+2=4
Or
Let be the set of all lines in the and be the relation in defined by . Show that is an equivalence relation. Find the set of all lines related to the line . 3+1=4
3. If prove that 4. If , find a matrix such that , where is the unit matrix.   4
Or
Using elementary row operation, find the inverse of the matrix 4
5. If then find the values of for which is valid. 4
6. If a function is differentiable at a point, prove that it is continuous at that point. 4
Or
Using Rolle’s Theorem, find at what points on the curve in the tangent is parallel to .
7. Evaluate any one of the integrals: 4
1. 2. 8. Prove that , when is an odd function. Hence evaluate 4
9. Solve (any one): 4
1. 2. 10. Find the equation of a curve passing through the origin, given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the co-ordinates of the point. 4
11. Using vectors prove that angle in a semicircle is a right angle. 4
Or
Using vectors prove that 4 12. Find the vector equation of a plane in normal form. 4
Or
Find the equation of a plane passing through a given point and perpendicular to a given vector in vector form.
13. Assume that each child born is equally like to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls, given that 4
1. The youngest is a girl,
2. At least one is a girl?
Or
The probability of a shooter hitting a target is 3/4. How many minimum numbers of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?
14. If x, y, z are all different and 6 Prove that Or
If 6 Then find the value of 15. Find the maximum and minimum value of the following functions; if exist. 3+3=6
1. 2. Or
Find the maximum area of an isosceles triangle inscribed in the ellipse with its vertex at one end of the major axis. 6
16. Evaluate: 6 Or
Evaluate as the limit of a sum.
17. Find the area bounded by And 6
Or
Find the ratio in which the area bounded by the curves and is divided by the line .

18. Show that the lines And are coplanar
And, find the equation of the plane containing both these lines. 6

19. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making, while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s time. If the profit on racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity. 6