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BT-102 · Mathematics I/Unsolved PYQ Paper

BT-102 Mathematics-I - Jun 2023 Question Paper

  1. 7 Marks
    aState Rolle's theorem hence verify for $f(x) = x^2 + 2x$ defined in the interval $[-2, 0]$.
  2. 7 Marks
    bFind the first six terms of the expansions of the function $e^x \log(1 + y)$ in a Taylor series in the neighbourhood of the point $(0, 0)$.
  3. 7 Marks
    aThe temperature $u(x, y, z)$ at any point in space is $u = 400xyz^2$ find the highest temperature on surface of the sphere $x^2 + y^2 + z^2 = 1$.
  4. 7 Marks
    bIf $u = x^2 \tan^{-1}\frac{y}{x} - y^2 \tan^{-1}\frac{x}{y}$, find the value of $\frac{\partial^2 u}{\partial x \partial y}$.
  5. 7 Marks
    aFind shortest distance from the origin to the curve $x^2 + 4xy + 6y^2 = 140$.
  6. 7 Marks
    bFind $\frac{du}{dt}$ if $u = x^2 + y^2$, $x = a\cos t$, $y = b\sin t$.
  7. 7 Marks
    aChange the order of integration in $\int_{0}^{1} \int_{x^2}^{2-x} xy \, dy \, dx$ and hence evaluate.
  8. 7 Marks
    bEvaluate $\iint e^{2x+3y} dxdy$ over the triangle bounded by $x = 0, y = 0$ and $x + y = 1$.
  9. 7 Marks
    aTest the convergence of the series $$1 + \frac{x}{2} + \frac{x^2}{5} + \frac{x^3}{10} + \dots + \frac{x^n}{n^2 + 1} + \dots$$
  10. 7 Marks
    bExpand as a half range $f(x) = x$ sine series and cosine series for the interval $0 < x < 2$.
  11. 7 Marks
    aExpand $f(x) = x\sin x$, $0 < x < 2\pi$ as a Fourier series.
  12. 7 Marks
    bFind the $a_0$ and $a_n$ if the function $f(x) = x + x^2$ is expanded in Fourier series defined in $(-1, 1)$.
  13. 7 Marks
    ai) If $A$ is a skew symmetric matrix then show that $A^2$ is a symmetric matrix. ii) Find eigen values of the matrix $\begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix}$.
  14. 7 Marks
    bFind the inverse of $\begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 2 \end{bmatrix}$ by using elementary row transformations.
  15. 7 Marks
    aVerify Cayley Hamilton theorem for the matrix $A$ and hence find $A^{-1}$ for $$\begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$$
  16. 7 Marks
    bTest the consistency and hence, solve the following set of equations. $$x + 2y - z = 3$$ $$3x - y + 2z = 1$$ $$2x - 2y + 3z = 2$$ $$x - y + z = -1$$
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