Mathematics II 2018

Group B

Attempt any SIX questions. [6x5 = 30]

11. If a function f(x) is defined as:

f(x) = 3x2 + 2            if x < 1

2x + 3                        if x > 1

4                                if x = 1

Discuss the continuity of function at x = 1.

12.  Find the derivative of sin3x by using definition.

13. Using L-Hospital's rule evaluate:

14. If demand function and cost function are given by

P(Q) = 1-3Q and

C(Q) = Q² – 2Q respectively, Where Q is the quality (number) of the product then

find output of the factor for the maximum profit.

15. Evaluate:

16. Solve:

17. Examine the consistency of the system of equation and solve if possible.

x₁ + x₂ - x₃ = 1

2x₁ + 3x₂ + 3x₃ = 3

x₁ - 3x₂ + 3x₃ = 2

Group-C

Attempt any two questions [2x10=20]

18. Define Homogeneous equation and solve the following system of equations using Inverse Matrix Method.

-2x + 2y + z = -4

-8x + 7y – 4x = -47

9x – 8y + 5z = 55

19. State Rolle's Theorem and interpret it geometrically. Verify Rolle's theorem for

f(x) = x² – 4 in  - 3 ≤ x ≤ 3

20. Using Composite Trapezoidal Rule, compute with four intervals. Find the absolute error of approximation from its actual value.