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# Mathematics - II

### Examination 2019

#### Group B

Attempt any Six question

2. Write expansions for log(1+x) and ex and use the expansion ex 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, $\int \left(3\mathrm{sinx}-4{\right)}^{2}\mathrm{cosxdx}$

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 fo 153 correc 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 = 2x1 + 12x2 + 8x3
subject to
2x1 - 2x2 + x3 ≤ 100
x1 - 2x2 + 5x3 ≤ 80
10x1 + 5x2 + 4x3 ≤ 80,
x1, x2, x3 ≥ 0

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

b) Find the intervals in which the function f(x) = x3 - 3x2 + 5 concave upwards and concave downward.

11. a) Find the area bounded by the parabola y2 = 4x and y axis between the point y = 0 to y = 2.

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

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