B.C.A. (Sem - II)
END SEMESTER EXAMINATION - 2020
Mathematics - 1
2BCA-1
TIME : 3 HOURS
MAXIMUM MARKS : 70
NOTE: Candidates are required to give answer in their own words as far as practicable.
The questions are of equal value
Answer any five questions.
Group-A
Q-1 (a) Evaluate Lim
$$ \lim_{x \to 0} \frac{x e^x - \log(1+x)}{x^2} $$
(b) Find the relationship between a and b so that function \( f(x) \) define by
$$ f(x) = \begin{cases} ax + 1, & \text{if } x \leq 3 \\ bx + 3, & \text{if } x > 3 \end{cases} $$
is continuous at \( x = 3 \).
Q-2 (a) Find the maximum and minimum values of $$ 3x^4 - 2x^3 - 6x^2 + 6x + 1 $$ in the intervals (0,2).
(b) Find the equation of tangent at any point \((x,y)\) to the curve
$$ x^{2/3} + y^{2/3} = a^{2/3} $$
Q-3 Find \( \frac{dy}{dx} \) in the following cases :
(a) \( y = x^2 \log x \)
(b) \( x^y = y^x \)
Q-4 Integrate the following :
(a) $$ \int \frac{e^{\tan^{-1}x}}{1+x^2} dx $$
(b) $$ \int_{1}^{2} \frac{x \, dx}{(x+1)(x+2)} $$
Q-5 (a) Solve
\[
\begin{cases}
x + y + z = 6 \\
x + 2y + 3z = 10 \\
x + 2y + 5z = 12
\end{cases}
\]
in matrix method
(b) If
\[ A = \begin{bmatrix} 1 & 3 & 0 \\ -1 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 2 & 3 \\ -1 & 1 & 2 \end{bmatrix} \]
compute \( AB \) and \( BA \). Show that \( AB \neq BA \).
Group-B
Q-6 (a) Find the root of the equation \[ x^3 - 4x - 9 = 0 \] using the bisection method in four stages.
(b) Find by Newton-Raphson method, the real root of the equation \[ 3x = \cos x + 1 \]
Q-7 (a) Apply Gauss elimination method to solve
\[ \begin{cases} x + y + z = 9 \\ 2x - 3y + 4z = 13 \\ 3x + 4y + 5z = 40 \end{cases} \]
(b) Solve \[ 10x + y + z = 12, \quad x + 10y + z = 12, \quad x + y + 10z = 12 \] by iterative method.
Q-8 Evaluate \[ \int_{0}^{1} \frac{dx}{1+x^2} \] using
(i) Trapezoidal rule taking \(h = \tfrac{1}{4}\)
(ii) Simpson's 1/3 rule taking \(h = \tfrac{1}{4}\)
Q-9 (a) Derive Newton's forward difference interpolation formula.
(b) From the following table, estimate the number of students who obtained marks between 40 and 45 :
| Marks | 30 - 40 | 40 - 50 | 50 - 60 | 60 - 70 | 70 - 80 |
|---|---|---|---|---|---|
| Number of Students | 31 | 42 | 51 | 35 | 31 |
Q-10 Apply Runge-Kutta fourth order method, to find an approximate value of \(y\) when \(x=0.2\), given that
\[ \frac{dy}{dx} = x + y, \quad y = 1 \text{ when } x=0 \]
--x--