General Instructions: The question paper consists of three Sections A, B and C. Section. In addition to Section as, every student has to attempt either Section B or Section C. 1. For Section A Question numbers 1 to 8 are of 3 marks each. Question numbers 9 to 15 are of 4 marks each. Question numbers 16 to 18 are of 6 marks each. 2. For Section B/Section C Question numbers 19 to 22 are of 3 marks each. Question numbers 23 to 25 are of 4 marks each. Question numbers 26 is of 6 marks. 3. All questions are compulsory. 4. Internal choices have been provided in some questions. You have to attempt only one of the choices in such questions. 5. Use of calculator is not permitted. However, you may ask for logarithmic and statistical tables, if required. SECTION - A

Q1. If Q2. Using properties of determinates prove that

Q3. A machine operates of all of its three components function. The probability that the first component fails during the year is 0.14, the second component fails is 0.10 and the third component fails is 0.05. What is the probability that the machine will fail during the year? Q4. A coin is biased so that the head is 3 times as likely to occur as a tail. If the coin is tossed twice, find the probability for the number of tails. Or

The probability that a person will get an electric contract is not get plumbing contract is

and the probability that he well is what is the

if the probability of getting at least one contract

probability that he will get both?

Q5. Evaluate: Q6. Evaluate: Q7. Solve the following initial value problem:

Q8. Solve the following differential equation: Q9. Examine the validity of the following argument:

Or Construct a combinatorial circuit for the following Boolean expression:

• •

Q10. Evaluate:

Q11. If

Q12. If Q13. Find the intervals in which the function increasing, (b) strictly decreasing. Q14. Evaluate: given by

. is (a) strictly

Q15. Prove that if Or

is an odd function, them

Use it to evaluate

Q16. If Use it solve the following system of equations:

Q17. Find the equations of the tangent and normal to the curve where the x-axis. Or and

at

Also, find the points of intersection where both tangent and normal cut

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is the volume of the sphere.

of

Q18. Using integration, find the area of the smaller region bounded by the ellipse and the straight line SECTION - B Q19. If Or Let and find a vector which is perpendicular to both and and and then show that

Q20. Using vectors, prove that the diagonals of a rhombus are perpendicular bisectors of reach other. Q21. A ball projected vertically upwards takes second to reach a height metres. If seconds and that

is the time taken by the ball to reach from this point to the ground, prove that the maximum height reached is

Q22. The velocity v of a particle moving along a straight line when at a distance of x from the origin, is given by show that the acceleration of the particle is

Q23. Find the equation of the sphere passing through the origin and making intercepts on coordinate axes, respectively. Or Find the vector and Cartesian forms of the equation of the plane containing two lines and Q24. Two forces of magnitudes P+ Q and P-Q make an angle resultant makes an angle with one another and their

with the bisector of the angle between them show that

Q25. ABCD is a square. Along the sides AB, CB, DA forces act equal to 6, 5, 8 and 12 N respectively. Find the algebraic sum of their moments about O, the centre of the square, if the side of the square is 4 m. Q26. Find the Cartesian and vector equations of the planes passing through the intersection of the panes which are at unit distance from the origin. SECTION - C Q19. Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Find the mean and variance for the...