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BT-102 · Mathematics I/Unsolved PYQ Paper

BT-102 Mathematics-I - Nov 2022 Question Paper

  1. 7 Marks
    aProve that $\frac{\pi}{3} - \frac{1}{5\sqrt{3}} > \cos^{-1}\frac{3}{5} > \frac{\pi}{3} - \frac{1}{8}$ using Lagrange's mean value theorem.
  2. 7 Marks
    bFind the minimum and maximum value of $$f(x,y) = x^3 + 3xy^2 - 3x^2 - 3y^2 + 4$$.
  3. 7 Marks
    aFind C of Cauchy's Mean value theorem on $[a, b]$ for the function $f(x) = e^x$ and $g(x) = e^{-x}$, $(a, b > 0)$.
  4. 7 Marks
    bProve that $\Gamma(n)\Gamma(1-n) = \frac{\pi}{\sin n\pi}$
  5. 7 Marks
    aBy Changing the order of integration, evaluate $$\int_0^1 \int_0^{\sqrt{1-x^2}} y^2\, dy\, dx$$
  6. 7 Marks
    bFind the area of a plane in the form of a quadrant of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
  7. 14 Marks
    Let W be a subspace of a finite dimensional vector space V(F). Then $\dim(V/W) = \dim V - \dim W$.
  8. 14 Marks
    Obtain the Fourier series to represent $f(x) = x\sin x,\; 0 < x < 2\pi$.
  9. 7 Marks
    aShow that $T : V_2(\mathbb{R}) \rightarrow V_3(\mathbb{R})$ is defined as $T(a, b) = (a - b, b - a, -a)$ is linear transformation.
  10. 7 Marks
    bTest the convergence of the series $$\sum_{n=1}^\infty (\sqrt{n+1} - \sqrt{n-1})$$
  11. 7 Marks
    aVerify Cayley-Hamilton theorem for the matrix $$A = \begin{bmatrix} 1 & -2 & 2 \\ 1 & -2 & 3 \\ 0 & -1 & 2 \end{bmatrix}$$. Hence find $A^{-1}$.
  12. 7 Marks
    bExamine the consistency of the system of the following equations. If consistent, solve the equation's. $$x + y + z = 3$$ $$x + 2y + 3z = 4$$ $$x + 4y + 9z = 6$$
  13. 14 Marks
    Diagonalize the matrix $A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
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