BT-301 Mathematics - III - Jun 2025 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 $x^3 - 2x - 5 = 0$. Considering the initial approximation at $x = 2$ then find the value of next approximation correct upto 2 decimal places.
   How to answer

   • Compute f(2) = 2^3 - 2(2) - 5 = -1
   • Compute f'(x) = 3x^2 - 2, hence f'(2) = 3(4) - 2 = 10
   • Apply Newton-Raphson formula: x_1 = 2 - (-1)/10 = 2.1
   • State result correct to 2 decimal places: 2.10
2. 7 Marks
   bWhat is the rate of convergence of Newton Raphson method?
   How to answer

   • State the order of convergence (p = 2, quadratic convergence)
   • Write the mathematical relation: |e_{n+1}| <= c |e_n|^2
   • Explain briefly the significance of quadratic convergence
3. 7 Marks
   aFind i) $\Delta \tan x$ ii) $\Delta^2 e^{2x}$ iii) $\Delta \log x$
   How to answer

   • For i): Delta tan x = tan(x+h) - tan x = sin(h) / (cos(x+h) cos x)
   • For ii): Delta e^(2x) = e^(2(x+h)) - e^(2x) = e^(2x)(e^(2h) - 1); Delta^2 e^(2x) = e^(2x)(e^(2h) - 1)^2
   • For iii): Delta log x = log(x+h) - log x = log((x+h)/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

   • Express operators in terms of shift operator E or expand RHS terms directly
   • RHS = f(3) + [f(3) - f(2)] + [f(3) - 2f(2) + f(1)] + [f(4) - 3f(3) + 3f(2) - f(1)] or alternatively rewrite terms starting from Delta^3 f(1)
   • Simplify the algebraic sum of function values
   • Verify LHS = RHS = f(4)
5. 7 Marks
   aUsing following table, by Lagrange's Method find $f(x)$ as a polynomial in $x$: $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 3 & 6 & 7 \\ \hline f(x) & 3 & -6 & 39 & 822 & 1611 \\ \hline \end{array} $$
   How to answer

   • State Lagrange's interpolation formula for 5 points
   • Calculate Lagrange basis polynomials l_0(x), l_1(x), l_2(x), l_3(x), l_4(x)
   • Multiply each basis polynomial by its respective f(x_i) value
   • Sum and simplify terms to obtain the polynomial in x
6. 7 Marks
   bFind $f(3.5)$ from the following table using Newton's Forward interpolation formula. $$ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 \\ \hline f(x) & 2.625 & 3.454 & 4.784 & 6.986 \\ \hline \end{array} $$
   How to answer

   • Construct the forward difference table up to Delta^3 y
   • Determine x_0 = 2, h = 1, u = (3.5 - 2)/1 = 1.5
   • Substitute values into Newton's forward difference interpolation formula
   • Calculate the numerical value of f(3.5)
7. 7 Marks
   aApply Gauss Elimination method to solve the following equations. $$\begin{aligned} 2x - y - 3z &= 9 \\ x - y - 2z &= 6 \\ x - y - z &= 2 \end{aligned}$$
   How to answer

   • Write the augmented matrix [A|B]
   • Perform elementary row operations to reduce to upper triangular form
   • Perform back substitution to find values of z, y, and x
   • Check consistency and state final solution
8. 7 Marks
   bFind missing values in the following table $$ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 5 & 10 & 15 & 20 & 25 \\ \hline y & 6 & 10 & - & 17 & - & 31 \\ \hline \end{array} $$
   How to answer

   • Identify 4 known values, implying 3rd degree polynomial, so Delta^4 y_0 = 0 and Delta^4 y_1 = 0
   • Expand (E - 1)^4 y_0 = 0 to get y_4 - 4y_3 + 6y_2 - 4y_1 + y_0 = 0
   • Expand (E - 1)^4 y_1 = 0 to get y_5 - 4y_4 + 6y_3 - 4y_2 + y_1 = 0
   • Solve the two simultaneous equations for y_2 and y_4
9. 7 Marks
   aFind $\int_0^6 \frac{e^x}{1 + x} \, dx$ approximately using Simpson's $3/8^{\text{th}}$ rule on integration.
   How to answer

   • Divide interval [0, 6] into 6 equal parts with h = 1
   • Tabulate values of f(x) = e^x / (1 + x) for x = 0, 1, 2, 3, 4, 5, 6
   • State Simpson's 3/8th rule formula: (3h/8)*[(y_0 + y_6) + 3(y_1 + y_2 + y_4 + y_5) + 2*y_3]
   • Substitute the calculated ordinate values and evaluate the final result
10. 7 Marks
    bSolve the equation $\frac{dy}{dx} = 1 - y$ with initial condition $y(0) = 0$ using Euler's modified method and tabulate the solution at $x = 0.1, 0.2, 0.3$.
    How to answer

    • State Euler's modified predictor-corrector formulas
    • Step 1: calculate y(0.1) using h = 0.1 through predictor and corrector iterations until convergence
    • Step 2: calculate y(0.2) similarly from y(0.1)
    • Step 3: calculate y(0.3) similarly from y(0.2)
    • Present the final results in a table for x = 0.1, 0.2, 0.3
11. 7 Marks
    aUsing Lagrange's interpolation formula to fit a polynomial to the data: $$ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 2 & 3 \\ \hline f(x) & -8 & 3 & 1 & 12 \\ \hline \end{array} $$
    How to answer

    • Write Lagrange interpolation formula for four points
    • Formulate individual terms l_0(x), l_1(x), l_2(x), l_3(x)
    • Multiply by y_0=-8, y_1=3, y_2=1, y_3=12 respectively
    • Combine and simplify terms into standard polynomial form a x^3 + b x^2 + c x + d
12. 7 Marks
    bUsing Regula-Falsi method, find the real root of $x \log_{10} x = 1.2$, correct to four decimal places.
    How to answer

    • Define f(x) = x log_10(x) - 1.2
    • Find initial interval [a, b] such that f(a)*f(b) < 0 (e.g. f(2) < 0 and f(3) > 0)
    • Iterate using Regula-Falsi formula: x = [a f(b) - b f(a)] / [f(b) - f(a)]
    • Update interval and continue iterations until successive approximations agree to 4 decimal places
13. 7 Marks
    aFind the inverse Laplace transform of $\frac{4s + 12}{s^2 + 8s + 16}$.
    How to answer

    • Rewrite denominator as (s + 4)^2
    • Rewrite numerator in terms of (s + 4): 4s + 12 = 4(s + 4) - 4
    • Separate terms: F(s) = 4/(s+4) - 4/(s+4)^2
    • Apply first shifting property to get L^{-1}{F(s)} = 4e^{-4t} - 4t e^{-4t} = 4e^{-4t}(1 - t)
14. 7 Marks
    bState convolution theorem and hence evaluate $$L^{-1}\left[\frac{1}{s(s^2 - a^2)}\right]$$
    How to answer

    • State convolution theorem: L^{-1}{F(s)G(s)} = f(t) * g(t) = \int_0^t f(u) g(t-u) du
    • Identify F(s) = 1/s => f(t) = 1, and G(s) = 1/(s^2 - a^2) => g(t) = (1/a) sinh(at)
    • Set up integral: \int_0^t 1 * \frac{1}{a} \sinh(au) du
    • Evaluate: [\frac{1}{a^2} \cosh(au)]_0^t = \frac{1}{a^2} (\cosh(at) - 1)
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 expressions: P(X=4) = binom(6,4) p^4 q^2 and P(X=2) = binom(6,2) p^2 q^4
    • Note binom(6,4) = binom(6,2) = 15, yielding 9 p^4 q^2 = p^2 q^4
    • Simplify to 9 p^2 = q^2 => 3p = q = 1 - p, solving for p = 1/4 and q = 3/4
    • Compute P(X = 1) = binom(6,1) p^1 q^5 = 6 * (1/4) * (3/4)^5
16. 7 Marks
    bA manufacturer knows from experience that the resistance of resistors he produces is with mean $\mu = 100\Omega$ and s.d. $\sigma = 2\,\Omega$. What percentage of resistors will have resistance between $98\Omega$ and $102\Omega$?
    How to answer

    • Calculate standard normal variates: z_1 = (98 - 100)/2 = -1, z_2 = (102 - 100)/2 = 1
    • Identify area P(-1 <= Z <= 1) = 2 * P(0 <= Z <= 1)
    • Use standard normal area value for Z = 1 (approx 0.3413, so total probability = 0.6826)
    • Convert to percentage: 68.26%