BT-301 Mathematics-III - Dec 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
   aBy using Newton-Raphon's method find the root of A, which is nearer to 2, correct to three places of decimal.
   How to answer

   • Write Newton-Raphson iteration $x_{n+1}=x_n-f(x_n)/f'(x_n)$
   • Choose initial $x_0$ nearer to 2
   • Iterate to 3-decimal accuracy
   • State final root
2. 7 Marks
   bFind the first term of the series whose second and subsequent terms are 8,3,0,-1,0.
   How to answer

   • Form difference table for 8,3,0,-1,0
   • Use Newton forward/backward relation
   • Extrapolate to first term
   • Verify result
3. 7 Marks
   aSolve $27x+6y-z=85,\ 6x+15y+2z=72,\ x+y+54z=110$ by Gauss-Seidel iteration method.
   How to answer

   • Check diagonal dominance and rewrite in iteration form
   • Assume initial $x_0,y_0,z_0$
   • Apply Gauss-Seidel successive displacement
   • Iterate to convergence and state $x,y,z$
4. 7 Marks
   bCalculate by Simpson's 1/3 rule (up to 3 places of decimal) $\int_{2}^{10} \frac{dx}{1+x}$ by dividing the range in to eight equal parts.
   How to answer

   • Compute $h=(10-2)/8$ and 9 ordinates $y=1/(1+x)$
   • Apply Simpson's 1/3 formula $h/3[(y_0+y_8)+4\sum odd+2\sum even]$
   • Evaluate sum
   • Round to 3 decimal places
5. 7 Marks
   aSolve $\frac{dy}{dx} = 1-2xy$ given that $y(0)=0$, by Taylor's method.
   How to answer

   • Compute $y',y'',...$ from $dy/dx=1-2xy$ at $x_0=0$
   • Write Taylor series $y(x)=y_0+xy'+x^2y''/2!+...$
   • Substitute derivatives
   • State series solution
6. 7 Marks
   bSolve the equation $\frac{dy}{dx} = x+y$ , with initial condition $y(0)=1$ by Runge-Kutta method, from $x=0$ to $x=0.2$ with $h=0.1$.
   How to answer

   • Write $f(x,y)=x+y$ with $x_0=0,y_0=1,h=0.1$
   • Compute $k_1,k_2,k_3,k_4$ for each step
   • Advance to $x=0.1$ then $x=0.2$
   • State $y(0.1),y(0.2)$
7. 7 Marks
   aShow that $L\left\{\frac{\cos \sqrt{t}}{\sqrt{t}}\right\} = \sqrt{\frac{\pi}{s}} e^{-\frac{1}{4s}}$
   How to answer

   • Expand $\cos\sqrt{t}$ as power series
   • Take Laplace termwise using $L\{t^{n-1/2}\}$
   • Sum to $\sqrt{\pi/s}\,e^{-1/4s}$
   • Conclude proof
8. 7 Marks
   bEvaluate $L^{-1}\left\{\frac{6s^{2}+22s+18}{s^{3}+6s^{2}+11s+6}\right\}$.
   How to answer

   • Factor denominator $(s+1)(s+2)(s+3)$
   • Resolve into partial fractions
   • Invert each term $e^{-at}$
   • Combine result
9. 7 Marks
   aIf $f(x) = cx^{2}, 0 < x < 1$ , find the value of c and determine the probability that $\frac{1}{3} < x < \frac{1}{2}$.
   How to answer

   • Use normalisation $\int_0^1 cx^2 dx=1$ to find $c$
   • Set up $P(1/3<x<1/2)=\int_{1/3}^{1/2}cx^2dx$
   • Evaluate integral
   • State probability
10. 7 Marks
    bIn sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2 out of 1000 such samples, how many would be expected to contain (i) at least 3 defective Parts? (ii) none defective?
    How to answer

    • Estimate $p$ with mean $np=2$ for $n=20$
    • Use Poisson $P(r)=e^{-m}m^r/r!$ with $m=2$
    • Compute $P(r\ge 3)$ and $P(0)$ times 1000
    • State expected numbers
11. 7 Marks
    aFrom the following table, estimate the number of students who obtained marks between 40 and 45.

    

    How to answer

    • Form cumulative frequencies for <40,<50,...
    • Apply Newton forward/backward interpolation for 45
    • Subtract to get 40-45 frequency
    • State estimated number
12. 7 Marks
    bBy means of Newton's Divided difference formula, find the values of $f(8)$, $f(15)$ from the following table:

    

    How to answer

    • Construct divided difference table for given x,f(x)
    • Write Newton divided difference polynomial
    • Substitute $x=8$ and $x=15$
    • State $f(8), f(15)$
13. 7 Marks
    aState Convolution theorem and hence evaluate $L^{-1}\left\{\frac{s}{(s^{2}+a^{2})^{2}}\right\}$.
    How to answer

    • State convolution $L^{-1}\{F G\}=f*g$
    • Identify $F,G$ for $s/(s^2+a^2)^2$
    • Evaluate convolution integral
    • Simplify to $(t\sin at)/(2a)$
14. 7 Marks
    bSolve $(D^{2}+6D+9)y = \sin x$ given that $y=1,y=0$ when $x=0$.
    How to answer

    • Write auxiliary equation $(m+3)^2=0$
    • Find C.F. $(C_1+C_2x)e^{-3x}$
    • Find P.I. for $\sin x$ by $1/(D^2+6D+9)$
    • Apply $y(0)$ conditions for constants
15. 7 Marks
    aFind mean of Poisson distribution.
    How to answer

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

    • Define pdf $f(x)=\lambda e^{-\lambda x}, x>0$
    • Give mean $1/\lambda$ and variance
    • Mention memoryless property
    • Give applications