BT-401 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
   aFind the real root of equation $f(x) = x^3 - x - 10$ by using Newton Raphson method correct up to 4 decimal places.
   How to answer

   • State Newton-Raphson iteration $x_{n+1}=x_n-f/f'$
   • Choose initial guess and iterate for $f(x)=x^3-x-10$
   • Continue till 4-decimal stability
   • Report real root
2. 7 Marks
   bFind solution $y(301)$ using Newton Divided difference interpolation formula

   How to answer

   • Form Newton divided-difference table for given data
   • Write Newton divided-difference interpolating polynomial
   • Substitute $x=301$
   • Evaluate $y(301)$
3. 7 Marks
   aEvaluate the following using Simpson's 3/8 rule.

   How to answer

   • Identify $h=0.2$ and $n=6$ intervals for Simpson 3/8
   • Apply Simpson's 3/8 formula with given $f(x)$ values
   • Compute weighted sum
   • Evaluate integral
4. 7 Marks
   bSolve the given set of equations by using Gauss elimination method: $$x + y + z = 4$$ $$x - 4y + 3z = 8$$ $$x + 6y + 2z = 6$$
   How to answer

   • Write augmented matrix for the system
   • Apply forward elimination to upper triangular form
   • Back-substitute for $z, y, x$
   • Verify solution
5. 7 Marks
   aUsing modified Euler's method, find an approximate value of $Y$, when $x = 0.3$. Given that $\frac{dy}{dx} = x + y$ and $y = 1$, when $x = 0$.
   How to answer

   • State modified Euler formula with $h=0.1$ or $0.15$
   • Iterate from $x=0, y=1$ to $x=0.3$
   • Apply predictor-corrector iterations
   • Report $y(0.3)$
6. 7 Marks
   bConsider an ordinary differential equation $\frac{dy}{dx} = x^2 + y^2, y(1) = 1.2$. Find $y(1.05)$ using the fourth order Runge-Kutta method.
   How to answer

   • Identify $x_0=1, y_0=1.2, h=0.05, f=x^2+y^2$
   • Compute $k_1,k_2,k_3,k_4$
   • Apply RK4 formula $y_1=y_0+h/6(k_1+2k_2+2k_3+k_4)$
   • Report $y(1.05)$
7. 7 Marks
   aFind the inverse Laplace transform of the following functions. $$F(s) = \frac{1}{(s+1)(s^2+1)}$$
   How to answer

   • Express $F(s)$ by partial fractions
   • Identify standard inverse transforms $e^{-t}, \sin t, \cos t$
   • Apply linearity of inverse Laplace
   • Write $f(t)$
8. 7 Marks
   bSuppose that the function $y(t)$ satisfies the DE $y'' - 2y' - y = 1$, with initial values, $y(0) = -1, y'(0) = 1$. Find the laplace transform of $y(t)$.
   How to answer

   • Take Laplace on both sides of $y''-2y'-y=1$
   • Substitute $y(0)=-1, y'(0)=1$
   • Solve for $Y(s)$
   • State Laplace transform of $y(t)$
9. 7 Marks
   aA device is used to measure the speed of cars on a highway. The speed is normally distributed with a $90$ km/hr mean and a standard deviation of $10$. What is the probability that the car picked was travelling at more than $100$ km/hr?
   How to answer

   • Standardize with $Z=(X-90)/10$
   • Find $P(X>100)=P(Z>1)$
   • Use normal table value
   • Report probability
10. 7 Marks
    bThe mean and the standard deviation of a binomial distribution are $100$ and $5$. Find the binomial distribution.
    How to answer

    • Use $mean=np=100$, $variance=npq=25$
    • Solve for $p,q$ then $n$
    • Write distribution $B(n,p)$
    • State parameters
11. 7 Marks
    aIn a normal distribution, $10.03\%$ of the items are under $25$ kg. weight and $89.97\%$ of the items are under $70$ kg. weight. What are the mean and standard deviation of the distribution?
    How to answer

    • Convert percentages to standard normal $z$ values
    • Set up $25=\mu+z_1\sigma$, $70=\mu+z_2\sigma$
    • Solve two equations for $\mu, \sigma$
    • Report mean and SD
12. 7 Marks
    bFind Solution using Lagrange's Interpolation formula.

    How to answer

    • Write Lagrange basis for $x=2,2.5,3$
    • Substitute $f$ values $0.69315,0.91629,1.09861$
    • Simplify interpolating polynomial
    • Evaluate required value
13. 7 Marks
    aFind the Laplace transform of the following function i) $L(t^2 e^{-t} \cosh 2t)$ ii) $L(e^{3t} \sin^2 4t)$
    How to answer

    • Use shifting/first translation theorem for $e^{-t}$
    • Apply $L\{t^n f(t)\}$ and $L\{\cosh 2t\}, L\{\sin^2 4t\}$
    • Simplify to rational function in $s$
    • Write final transforms
14. 7 Marks
    bUsing Milne's method find $y(4.4.)$ given $5xy + y^2 - 2 = 0$ Given $y(4) = 1, y(4.1) = 1.0049, y(4.2) = 1.0097$ and $y(4.3) = 1.0143$.
    How to answer

    • Rewrite ODE as $dy/dx=(2-5xy-y^2)/?$ from $5xy+y^2-2=0$
    • Use given $y(4)$ to $y(4.3)$ with $h=0.1$
    • Apply Milne predictor-corrector formula
    • Obtain $y(4.4)$
15. 7 Marks
    aFind mean of Poisson distribution.
    How to answer

    • Write Poisson pmf $P(X=r)=e^{-\lambda}\lambda^r/r!$
    • Compute expectation $E[X]=\sum rP(r)$
    • Simplify series to $\lambda$
    • Conclude mean = $\lambda$
16. 7 Marks
    bWrite short note on Exponential distribution.
    How to answer

    • Define exponential pdf with parameter $\lambda$
    • Give mean, variance and memoryless property
    • Mention applications
    • Write brief note