BCA 2nd Semester
Math II 2022 Board Question Paper

"A function \(f(x)\) is defined as \[ f(x) = \begin{cases} 2x + 3 & \text{for } -\frac{3}{2} \leq x < 0 \\ 3 - 2x & \text{for } 0 \leq x \leq \frac{3}{2} \\ -3 - 2x & \text{for } x > \frac{3}{2} \end{cases} \] show that \(f(x)\) is continuous at \(x = 0\) and discontinuous at \(x = \frac{3}{2}\)."

Find \(\frac{dy}{dx}\) when \(x = a(t + \sin t)\), \(y = a(1 - \cos t)\).

State L'Hospital's Rule. Use it to evaluate: $$\lim_{x \to 0} \frac{xe^x - In(1+x)}{x^2}$$

State Rolle's Theorem. Verify Rolle's Theorem for the function \(f(x) = x^2 - 9\) in the interval \(-3 \leq x \leq 3\).

Evaluate:
a) \(\int_0^2 \frac{dx}{x^2-25}\)
b) \(\int \frac{2x+5}{\sqrt{x^2}\ + 5x} dx\)

Solve: \(\sin x \frac{dy}{dx} + (\cos x) y = -\sin x \cos x\).

Using simplex method, find optimal solution of \(Z = 7x1 + 5x2\) Subject to   \(x1 + 2x2 \leq 6\)       \(4x1 + 3x2 \leq 6\)       \(x1 \geq 0, x2 \geq 0\).

What do you mean by stationary points and inflection points? Using derivatives, find two numbers whose sum is 20 and sum of whose squares is minimum.

Using Simpson's \(\frac{1}{3}\) rule evaluate \(\int_0^1 \frac{1}{1+x} {dx}\) with 3 points of intervals. Find error of approximation. How many points are to be considered to make the approximation value within 10-5?