BT-401 Mathematics III - Dec 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
   aUsing Newton-Raphson method find the real root of the equation $3x = \cos x + 1$ correct to five decimal places.
   How to answer

   • Define f(x)=3x-cos x-1 and f'(x)=3+sin x
   • Choose initial approximation and iterate x_{n+1}=x_n-f/f'
   • Iterate till stable to five decimal places
   • State real root value
2. 7 Marks
   bFind $f(g)$ from the following table

   How to answer

   • Identify unequal intervals - use Newton divided difference
   • Construct divided-difference table from given x, f(x)
   • Write Newton divided-difference interpolation polynomial
   • Substitute x = g (as printed) to get f(g)
3. 7 Marks
   aApply the Simpson's $\frac{3}{8}$ rule to evaluate the following integral for six digit $\int_{1.0}^{1.30} \sqrt{x} \cdot dx$.
   How to answer

   • State Simpson's 3/8 rule formula with h
   • Divide [1.0,1.30] into suitable multiple-of-3 subintervals and compute sqrt(x)
   • Apply weights 1,3,3,2,...1
   • Evaluate integral to six digits
4. 7 Marks
   bSolve the following system of equation using Gauss-Seidel method. $$27x + 6y - z = 85$$ $$6x + 15y + 2z = 72$$ $$x + y + 54z = 110$$
   How to answer

   • Rewrite equations in diagonally dominant iteration form
   • Assume initial values (0,0,0)
   • Iterate Gauss-Seidel formulas for x,y,z
   • Iterate to convergence and state solution
5. 7 Marks
   aUsing Runge-Kutta method of fourth order solve $\frac{dy}{dx} = \frac{y^2 - x^2}{y^2 + x^2}$, $y(0) = 1$ at $x = 0.4$ in step of $0.2$
   How to answer

   • Define f(x,y)=(y^2-x^2)/(y^2+x^2), h=0.2
   • Compute k1,k2,k3,k4 for each step from x=0 to 0.4
   • Apply y_{n+1}=y_n+(k1+2k2+2k3+k4)/6
   • Obtain y at x=0.4
6. 7 Marks
   bFind $y(2.2)$ using Euler's method for $\frac{dy}{dx} = -xy^2$ where $y(2) = 1$.
   How to answer

   • Write Euler formula y_{n+1}=y_n+h f(x_n,y_n)
   • Identify f=-x y^2, x0=2, y0=1, choose h to reach 2.2
   • Compute one/two steps
   • State y(2.2)
7. 7 Marks
   aFind the Inverse Laplace transform of $\frac{1}{p^3\left(p^2 + a^2\right)}$.
   How to answer

   • Use partial fractions for 1/[p^3(p^2+a^2)]
   • Recall L^{-1}{1/p^n} and L^{-1}{p/(p^2+a^2)}, L^{-1}{1/(p^2+a^2)}
   • Invert termwise
   • Simplify to expression in t and sin at, cos at
8. 7 Marks
   bUsing Laplace transform solve the differential equation $\frac{d^3y}{dt^3} + 2\frac{d^2y}{dt^2} - \frac{dy}{dt} - 2y = 0$ Where $y = 1, \frac{dy}{dt} = 2, \frac{d^2y}{dt^2} = 2$ at $t = 0$.
   How to answer

   • Take Laplace transform of ODE using initial conditions
   • Solve algebraically for Y(s)
   • Partial fractions and inverse Laplace
   • Obtain y(t)
9. 7 Marks
   aFit a Poisson distribution to the following.

   How to answer

   • Compute mean m = Σfx/Σf
   • Write Poisson probability P(x)=e^{-m} m^x/x!
   • Compute expected frequencies N·P(x)
   • Present fitted distribution
10. 7 Marks
    bCalculate the mean and standard deviation of Binomial Distribution.
    How to answer

    • Define Binomial P(x)=C(n,x) p^x q^{n-x}
    • Derive mean E(X)=np using summation/ mgf
    • Derive variance Var=npq hence SD=√(npq)
    • State final mean and standard deviation
11. 7 Marks
    aFind $\frac{dy}{dx}$ at $x = 1.5$ from the following table

    How to answer

    • Select numerical differentiation formula for equally spaced data
    • Construct forward/backward differences near x=1.5
    • Apply derivative formula at starting point
    • Compute dy/dx value
12. 7 Marks
    bEvaluate $\int_{0}^{\infty} \frac{e^{-t} \cdot \sin t}{t} dt$
    How to answer

    • Consider Laplace transform of sin t / t or parameter integral
    • Use property ∫0^∞ e^{-t} sin t /t dt = ∫1^∞ ... or L{sin t /t}
    • Evaluate resulting arctan / log integral
    • Obtain value π/4
13. 7 Marks
    aShow that Newton Raphson method is quadratic convergent.
    How to answer

    • Write Newton iteration x_{n+1}=x_n-f/f'
    • Expand f(x_n) about root α with Taylor series and error e_n
    • Show e_{n+1} ≈ f''(α)/2f'(α) e_n^2
    • Conclude second-order/quadratic convergence
14. 7 Marks
    bDrive Newton forward interpolation formula and use it to estimate the value of $f(1.25)$ from the following table

    How to answer

    • Derive Newton forward formula using binomial / difference operator
    • Build forward difference table for given data
    • Put p=(1.25-1.0)/0.5 and apply formula
    • Compute f(1.25)
15. 7 Marks
    aSolve the system of equations $$3x + y - z = 3$$ $$2x - 8y + z = -5$$ $$x - 2y + 9z = 8$$ Using Gauss elimination method.
    How to answer

    • Write augmented matrix for 3 equations
    • Forward elimination to upper triangular form
    • Back substitution for z,y,x
    • Verify solution
16. 7 Marks
    bFind Fourier sine transform of $\frac{1}{x}$.
    How to answer

    • Define Fourier sine transform Fs(s)=√(2/π)∫0^∞ f(x) sin sx dx
    • Substitute f=1/x and evaluate Dirichlet integral
    • Use ∫0^∞ sin sx /x dx = π/2
    • Obtain transform in s