Mathematics II - Unit Wise Questions

1. For all rational values of n, is equal to

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.

2. If  then f(x) is said to be

a) Removable discontinuity    

b) An ordinary discontinuity    

c) Infinite discontinuity    

d) Finite discontinuity

3. Derivative of tan-1x is equal to

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.

4. The value of is equal to,

a) e^x    b) 1    c) 0    d) -1

15. Evaluate:

5. The differential equation:is known as

a) Second degree second order

b) Second degree first order

c) First degree second order

d) First order second degree

6. One important condition to satisfy Rolle's Theorem by a function f(x) in [a, b] is

a) f(a) > f(b)    b) f(a) < f(b)    c) f(a) = f(b)    d) f(a) = f(b) ≠ 0

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

7. Formula for the composite trapezoidal rule is

8.While applying Simpson's 3/8 rule the number of sub-interval should be

a) Odd    b) 8    c) Even    d) Multiple of 3

9. In Gauss Elimination method the given system of simultaneous equation is transformed into

a) Lower tri-angular equation    b) Unit matrix    c) transpose matrix    d) upper triangular matrix

10. In Newton-Raphson method, if x_n is an approximate solution of f(x) = 0 and f /(xn) ≠ 0 the next approximation is given by

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.