BCA Family An online BCA community

# 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.
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 $\int_{2}^{0}(2x^3 - 1)dx$, 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 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 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 = 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 Theorem 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 $\frac{dx}{dy}+2tanx.y=sinx$

I there is