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BT-102 · Mathematics I/Unsolved PYQ Paper

BT-102 Mathematics-I - Dec 2024 Question Paper

  1. 7 Marks
    aFind the points where the function $x^3 + y^3 - 3axy$ has maximum or minimum value.
  2. 7 Marks
    bFind the Taylor's expansion of $y = \sin x$ about point $x = \pi/2$.
  3. 7 Marks
    aThe part of the parabola $y^2 = 4ax$ cut off by the latus rectum revolves about the tangent at the vertex. Find the volume of the reel thus generated.
  4. 7 Marks
    bProve that: $$\int_0^1 \sqrt{(1 - x^4)} dx = \frac{\{\Gamma(1/4)\}^2}{6\sqrt{(2\pi)}}$$
  5. 7 Marks
    aShow that the following series is Convergent. $$\frac{1}{4} + \frac{1}{4^2} + \frac{1}{4^3} + \dots + \frac{1}{4^n} + \dots$$
  6. 7 Marks
    bObtain Half Range Sine Series for $f(x) = e^x$ in $0 < x < 1$.
  7. 7 Marks
    aShow that the set $w = \{(a, b, 0) : a, b \in \mathbb{R}\}$ is subspace of $\mathbb{R}^3$.
  8. 7 Marks
    bAre the following vectors LD? If so express one of these as a LC of other two. $X_1 = (1, 3, 4, 2)$, $X_2 = (3, -5, 2, 2)$, $X_3 = (2, -1, 3, 2)$
  9. 7 Marks
    aFind a similarity transformation that diagonalise the matrix. $$A = \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix}$$
  10. 7 Marks
    bFind the Eigen value and Corresponding Eigen Vectors of the following Matrix. $$\begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}$$
  11. 7 Marks
    aDefine Beta and Gamma Function and show that relation $$B(m, n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m + n)}$$
  12. 7 Marks
    bEvaluate: i) $\int_0^1 x^2 (1 - x)^3 dx$ ii) $\int_0^1 \sqrt{\left(\frac{1 - x}{x}\right)} dx$
  13. 7 Marks
    aProve that the surface area of the solid generated by the revolution of the ellipse $x^2/a^2 + y^2/b^2 = 1$ about the major axis is: $$2(\pi ab) \left\{\sqrt{(1 - e^2)} + \frac{\sin^{-1} e}{e}\right\}$$
  14. 7 Marks
    bShow that the sequence $\langle n^{1/n} \rangle$ converge to 1.
  15. 7 Marks
    aProve that a rectangular solid of maximum volume within a sphere is a cube.
  16. 7 Marks
    bVerify Rolle's Theorem for the function $y = x^2 + 2$, $a = -2$ and $b = 2$.
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