# Mathematics - II

### Examination 2018

#### Group B

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:

$$ \lim_{x\to\infty} {{ 2x^2 + 3x} \over {1 + 5x^2}} $$

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: \[ \int {dx\over{1-sinx}} \] \[ \int_{1}^{0} (x^2 + 5)dx \]

16. Solve:

${dy \over dx } = {{xy + y} \over {xy + x}}$

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

x_{1} + x_{2} - x_{3} = 1

2x_{1} + 3x_{2} + 3x_{3} = 3

x_{1} - 3x_{2} + 3x_{3} = 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) = x^{2} – 4 in - 3 ≤ x ≤ 3

20. Using Composite Trapezoidal Rule $\int_{2}^{0}(2x^3 - 1)dx$, compute with four intervals. Find the absolute error of approximation from its actual value.

### Examination 2019

#### Group B

Attempt any Six question

2. Write expansions for log(1+x) and e^{x} and use the expansion e^{x} to prove $\underset{x\to \mathrm{\infty}}{lim}\frac{{e}^{x}-1}{x}=1$

3. Find Derivative of $\sqrt{2x-3}$ using defination.

4. Show that the rectangle of largest possible area for a given perimeter 'P' is a square.

5. Evaluate the integral, $\int (3\mathrm{sinx}-4{)}^{2}\mathrm{cosxdx}$

6. Solve the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{x}^{2}+{y}^{2}}{2{x}^{2}}$

7. Evaluate the limit $\underset{x\to \mathrm{\infty}}{lim}\frac{\mathrm{xcos}\theta -\theta \mathrm{cosx}}{x-\theta}$

8. Using Newton Raphson's Method to find the square root of 153 correct to three places of decimals.

#### Group C

Attempt any Two question

9. Using simplex method to find the optimal solution of the following linear programming problem.

maximize, z = 2x_{1} + 12x_{2} + 8x_{3}

subject to

2x_{1} - 2x_{2} + x_{3} ≤ 100

x_{1} - 2x_{2} + 5x_{3} ≤ 80

10x_{1} + 5x_{2} + 4x_{3} ≤ 80,

x_{1}, x_{2}, x_{3} ≥ 0

10. a) State Mean value Theorem and interpret it geometrically

b) Find the intervals in which the function f(x) = x^{3} - 3x^{2} + 5 concave upwards and concave downward.

11. a) Find the area bounded by the parabola y^{2} = 4x and y axis between the point y = 0 to y = 2.

b) Solve the differential equation $\frac{dx}{dy}+2tanx.y=sinx$