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BT-401 · Mathematics-III/Unsolved PYQ Paper

BT-401 Mathematics-III - Jun 2026 Question Paper

  1. 7 Marks
    aProve that :
  2. 7 Marks
    bApply Newton-Raphson method to evaluate approximately $\sqrt{12}$.
  3. 7 Marks
    aFit a cubic polynomial which takes the following value : | x | 0 | 1 | 2 | 3 | | y | 1 | 0 | 1 | 10 | Hence find $y(4)$.
  4. 7 Marks
    bCalculate $\log_e 7$ by Simpson's 1/3 rule correct to four decimal places.
  5. 7 Marks
    aSolve system of Linear equations by Jacobi's method : $$4x+y+z=7,\quad x+5y+z=-8,\quad x+y+6z=6$$
  6. 7 Marks
    bExpand $\frac{e^z}{z(z+1)}$ as Taylor's series about $z = 2$.
  7. 7 Marks
    aTabulated by Milne's method the numerical solution of $\frac{dy}{dx} = x + y$ with $x_0 = 0, y_0 = 1$ from $x = 0.2$ to $x = 0.3$.
  8. 7 Marks
    bSolve : $\nabla^2 u = -81xy, 0 < x, y < 1, h = \frac{1}{3}$. Given, $$u(0,y) = u(x,0) = 0, u(1,y) = u(x,1) = 100.$$
  9. 7 Marks
    aObtain y for x=0.1 from the differential equation $\frac{dy}{dx} = x^2 + y$, given $y = -1$ when $x = 0$ by using Runge-Kutta formula of fourth order.
  10. 7 Marks
    bSolve the equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ in $0 < x < 5, t \ge 5$ by Crank-Nicholson method. Given that $u(x,0) = 20, u(0,t) = 0, u(5,t) = 100$ and $h = 1$.
  11. 7 Marks
    aProve that : $\int_{t=0}^{\infty} \int_{u=0}^{t} e^{-t} \frac{\sin u}{u} du dt = \frac{\pi}{4}$ and also find the Laplace Transform of $\cos h$ $at$ $\sin bt$.
  12. 7 Marks
    bUsing Convolution theorem, prove that $$L^{-1}\left[\frac{1}{p^3(p^2+1)}\right] = \frac{t^2}{2} + \cos t -1$$
  13. 7 Marks
    aSolve the differential equation by Laplace Transform : $$(D^2 - 2D + 1)y = e^t,\quad y(0) = 2, y'(0) = -1.$$
  14. 7 Marks
    bFind the Fourier cosine transform of $e^{-x^2}$.
  15. 7 Marks
    aIf the probability of hitting a object is 10% and 10 shots are fixed independently. What is the probability that the object will be hit at least once?
  16. 7 Marks
    bIn a distribution exactly normal, 7% of the items are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution?
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