BT-401 Mathematics III - Jun 2024 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
   aIf the equation $f(x)$ is given as $x^3 - x^2 + 4x - 4 = 0$. Considering the initial approximation at $x = 2$ then find the value of next approximation correct upto 2 decimal places.
   How to answer

   • Define $f(x)=x^3-x^2+4x-4$ and $f'(x)=3x^2-2x+4$.
   • Use Newton-Raphson formula $x_{n+1}=x_n-f(x_n)/f'(x_n)$ with $x_0=2$.
   • Compute $f(2), f'(2)$ and $x_1$.
   • Round to 2 decimal places.
2. 7 Marks
   bWhat is the rate of convergence of Newton Raphson method?
   How to answer

   • State Newton-Raphson iteration formula.
   • Define error $e_n = x_n - \alpha$.
   • Show $e_{n+1} \approx e_n^2 f''(\alpha)/(2f'(\alpha))$ giving second order.
   • Conclude rate/order of convergence is 2 for simple root.
3. 7 Marks
   aFind i) $\Delta e^{ax}$ ii) $\Delta^2 e^x$ iii) $\Delta \log x$
   How to answer

   • Recall forward difference $\Delta f(x)=f(x+h)-f(x)$.
   • Apply to $e^{ax}$, $e^{x}$ to get $(e^{ah}-1)$ factors.
   • Apply second difference for $\Delta^2 e^x$.
   • Use $\Delta \log x = \log(x+h)-\log x = \log(1+h/x)$.
4. 7 Marks
   bProve that $f(4) = f(3) + \Delta f(2) + \Delta^2 f(1) + \Delta^3 f(1)$ taking '1' as the interval of differencing.
   How to answer

   • Write $f(4)-f(3)=\Delta f(3)$ with interval 1.
   • Express $\Delta f(3)=\Delta f(2)+\Delta^2 f(2)$ etc.
   • Expand successively down to argument 1.
   • Collect to get required identity.
5. 7 Marks
   aUsing Simpson's 3/8 rule to solve the integral $$\int_{4}^{5.2} \log_e x \, dx$$
   How to answer

   • State Simpson's 3/8 rule with $n$ multiple of 3.
   • Divide $[4,5.2]$ into suitable subintervals and evaluate $\log_e x$.
   • Apply weights 1,3,3,2,...,1 with factor $3h/8$.
   • Compute numerical value of integral.
6. 7 Marks
   bFind $f(0.18)$ from the following table using Newton's Forward interpolation formula.

   How to answer

   • Form forward difference table for given $x,f(x)$.
   • Compute $p=(0.18-0)/0.1=1.8$.
   • Apply Newton's forward interpolation formula.
   • Substitute differences to get $f(0.18)$.
7. 7 Marks
   aApply Gauss Elimination method to solve the following equations : $$2x - y + 3z = 9$$ $$x + y + z = 6$$ $$x - y + z = 2$$
   How to answer

   • Write augmented matrix for the 3 equations.
   • Eliminate $x$ then $y$ to get upper triangular form.
   • Back-substitute to find $z, y, x$.
   • Verify solution.
8. 7 Marks
   bFind missing values in the following table:

   How to answer

   • Assume equally spaced $x$ with $h=5$.
   • Use properties of finite differences / Newton's forward-backward.
   • Set up difference table with unknowns.
   • Solve for missing $y$ at $x=10$ and $x=20$.
9. 7 Marks
   aUse Runge-Kutta method to approximate $y$, when $x = 0.1$ and $x = 0.2$, given that $x = 0$ when $y = 1$ and $\frac{dy}{dx} = x + y$.
   How to answer

   • Identify $f(x,y)=x+y$, $h=0.1$, $x_0=0,y_0=1$.
   • Apply fourth-order Runge-Kutta formulas for $k_1,k_2,k_3,k_4$.
   • Compute $y(0.1)$ then repeat for $y(0.2)$.
   • Present results.
10. 7 Marks
    bUse Milne's method to solve $\frac{dy}{dx} = x + y$ with initial condition $y(0) = 1$, from $x = 0.20$ to $x = 0.30$.
    How to answer

    • Use RK values up to $x=0.20$ as starting values.
    • Apply Milne's predictor formula for $y_{n+1}$.
    • Apply Milne's corrector iteratively.
    • Obtain $y(0.30)$.
11. 7 Marks
    aFind the value of $f(5)$ from the following table.

    How to answer

    • Construct divided / forward difference table for X=1,2,3,4,7.
    • Apply Newton's forward or divided difference formula for $f(5)$.
    • Compute differences.
    • Evaluate at $x=5$.
12. 7 Marks
    bFind the real root of the equation $x \log_{10} x - 1.2 = 0$ correct to five places of decimal by Regula Falsi Method.
    How to answer

    • Define $f(x)=x\log_{10}x-1.2$ and locate root interval.
    • Write Regula Falsi formula $x_{n+1}=(a f(b)-b f(a))/(f(b)-f(a))$.
    • Iterate till 5 decimal stability.
    • Report root.
13. 7 Marks
    aFind the inverse Laplace transform of $\frac{2s+1}{s(s-1)}$.
    How to answer

    • Write $F(s)=(2s+1)/[s(s-1)]$ and split by partial fractions.
    • Recall $L^{-1}[1/s]=1$, $L^{-1}[1/(s-1)]=e^{t}$.
    • Invert termwise.
    • Simplify to $f(t)$.
14. 7 Marks
    bState convolution theorem and hence evaluate $$L^{-1}\left[\frac{s}{(s^2+1)(s^2+4)}\right].$$
    How to answer

    • State convolution theorem $L^{-1}[F(s)G(s)]=f*g$.
    • Write $s/[(s^2+1)(s^2+4)]$ as product and identify $f,g$.
    • Evaluate convolution integral $\int_0^t \sin u \sin 2(t-u)du$ or via partial fractions.
    • Simplify to result in $t$.
15. 7 Marks
    aA binomial variable $X$ satisfies the relation $9P(X = 4) = P(X = 2)$ when $n = 6$. Find the value of the parameter $p$ and $P(X = 1)$.
    How to answer

    • Write binomial pmf $P(X=r)=\binom{6}{r}p^r q^{6-r}$.
    • Use $9P(X=4)=P(X=2)$ to solve for $p,q$.
    • Obtain $p$.
    • Compute $P(X=1)=6pq^5$.
16. 7 Marks
    bA large number of measurement is normally distributed with a mean $65.5$" and S.D. of $6.2$". Find the percentage of measurements that fall between $54.8$" and $68.8$".
    How to answer

    • Standardize with $Z=(X-65.5)/6.2$.
    • Find $Z$ for 54.8 and 68.8.
    • Use normal table for $P(-1.73<Z<0.21)$.
    • Convert to percentage.