BT-401 Mathematics - III - Jun 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aFind the real root of the equation $x^3-3x-5=0$ correct to four places of decimals by Newton Raphson method.
   How to answer

   • Define Newton-Raphson iteration $x_{n+1}=x_n-f/f'$
   • Locate initial root interval for $f(x)=x^3-3x-5$
   • Iterate to 4 decimal accuracy
   • State final real root
2. 7 Marks
   bEvaluate $$\Delta^6(ax-1)(bx^2-1)(cx^3-1); h=1$$
   How to answer

   • Expand product $(ax-1)(bx^2-1)(cx^3-1)$ as degree 6 polynomial
   • Use $\Delta^n$ with $h=1$ annihilates degree < n
   • Compute 6th finite difference = $6! \times$ leading coefficient
   • Simplify to $720abc$
3. 7 Marks
   aUsing Newton's divided difference formula, find $f(8)$ and $f(15)$ from the following table.

   

   How to answer

   • Construct divided-difference table for given x, F(x)
   • Write Newton divided-difference interpolating polynomial
   • Substitute $x=8$ to get $f(8)$
   • Substitute $x=15$ to get $f(15)$
4. 7 Marks
   bFind the real root of the equation $xe^x-3=0$ correct to three decimal places by Regula Falsi method.
   How to answer

   • Define $f(x)=xe^x-3$ and bracket root
   • Write Regula-Falsi formula
   • Iterate until stable to 3 decimals
   • State root
5. 7 Marks
   aUse Simpson's 1/3 and 3/8 rule to evaluate the following. $$\int_0^1 \frac{dx}{1+x^2}$$ Hence obtain the approximate value of $\pi$ in each case.
   How to answer

   • Take suitable n for 1/3 and 3/8 rules for $\int_0^1 dx/(1+x^2)$
   • Apply Simpson 1/3 formula
   • Apply Simpson 3/8 formula
   • Use $\int_0^1 dx/(1+x^2)=\pi/4$ to get $\pi$
6. 7 Marks
   bCompute the value of the following integral by trapezoidal Rule. $$\int_{0.2}^{1.4}(Sinx-\log_e x+e^x)dx$$
   How to answer

   • Identify integrand $\sin x - \log_e x + e^x$ limits 0.2 to 1.4
   • Choose subintervals and compute ordinates
   • Apply trapezoidal formula
   • Compute final value
7. 7 Marks
   aSolve the following system by Gauss Siedel iteration method. $$27x+6y-z=85$$ $$6x+15y+2z=72$$ $$x+y+54z=110$$
   How to answer

   • Check diagonal dominance and rewrite equations for iteration
   • Write Gauss-Seidel iteration formulas
   • Start with initial guess and iterate
   • State converged x, y, z
8. 7 Marks
   bUse Simpson's rule to find approximate value of the following integral. $$\int_1^2 \frac{dx}{x}$$
   How to answer

   • Identify $f(x)=1/x$, limits 1 to 2
   • Divide interval into even number of strips
   • Apply Simpson's rule formula
   • Compute approximate value
9. 7 Marks
   aUse Euler's method compute the solution of $\frac{dy}{dx}=y^2-x^2$ with $y(0)=1$ at $x=0.1$.
   How to answer

   • Identify $f(x,y)=y^2-x^2$, $x_0=0,y_0=1$
   • Use Euler formula $y_1=y_0+h f(x_0,y_0)$
   • Compute with $h=0.1$
   • State y(0.1)
10. 7 Marks
    bUse Euler's Modified method to solve the equation $\frac{dy}{dx}=x+y$, $y(0)=1$, $h=0.2$ compute $y(1)$
    How to answer

    • Write Euler modified predictor-corrector formulas
    • Take $h=0.2$ from $x=0$ to $x=1$
    • Iterate corrector to convergence at each step
    • State $y(1)$
11. 7 Marks
    aUse Laplace transform to solve the Initial Value Problem. $$y'+2y=26\sin 3t,\: y(0)=3$$
    How to answer

    • Take Laplace transform of $y'+2y=26\sin 3t$
    • Use $y(0)=3$ to get $Y(s)$
    • Take inverse Laplace via partial fractions
    • State $y(t)$
12. 7 Marks
    bEvaluate $$L(t \cos at)$$
    How to answer

    • Recall $L\{\cos at\}=s/(s^2+a^2)$
    • Use multiplication by t property $L\{t f(t)\}=-d/ds F(s)$
    • Differentiate and simplify
    • State result
13. 7 Marks
    aEvaluate $$L^{-1}\left[\frac{1}{(1+s)^3}\right]$$
    How to answer

    • Write $1/(1+s)^3$ in standard form
    • Use shift / known $L^{-1}\{1/s^n\}$
    • Apply first shifting theorem
    • State $e^{-t}t^2/2$
14. 7 Marks
    bEvaluate $$L\{t^2e^{-3t}\}$$
    How to answer

    • Recall $L\{t^n\}=n!/s^{n+1}$
    • Use first shifting $L\{e^{at}f(t)\}=F(s-a)$
    • Substitute $n=2, a=-3$
    • State $2/(s+3)^3$
15. 7 Marks
    aFind mean of Poisson distribution.
    How to answer

    • Write Poisson pmf $P(X=r)=e^{-\lambda}\lambda^r/r!$
    • Define mean $E[X]=\sum r P(r)$
    • Simplify series to $\lambda$
    • State mean = variance = $\lambda$
16. 7 Marks
    bWrite Short note on Exponential Distribution.
    How to answer

    • Define exponential pdf $f(x)=\lambda e^{-\lambda x}$
    • Give mean, variance and memoryless property
    • Mention applications in waiting time
    • Conclude short note