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BT-102 · Mathematics I/Unsolved PYQ Paper

BT-102 Mathematics-I - Jun 2024 Question Paper

  1. 7 Marks
    aExpand $(1+x)^m$ by Maclaurin's Theorem.
  2. 7 Marks
    bFind the Maximum value of $u = \sin x \sin y \sin(x+y)$.
  3. 7 Marks
    aFind the volume of the solid generated by the revolution of the Cardioid $r = a(1+\cos\theta)$ about the initial line.
  4. 7 Marks
    bProve that $$\int_0^{\infty} \cos(x^2)dx = \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
  5. 7 Marks
    aShow that Sequence $\langle x^n \rangle$ where $|x| < 1$ converge to 0.
  6. 7 Marks
    bFind the Fourier Series for the function $f(x) = x\sin x$, $(-\pi < x < \pi)$.
  7. 7 Marks
    aShow that the following equations are consistent and solve them. $$x - y + 2z = 4$$ $$3x + y + 4z = 6$$ $$x + y + z = 1$$
  8. 7 Marks
    bIf $w_1$ and $w_2$ be two subspace of V(F) then Show that $w_1 \cap w_2$ also subspace of V(F).
  9. 7 Marks
    aFind the Characteristic equation of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & -4 \\ 1 & 0 & -1 \end{bmatrix}$ and hence find the Eigen values and Eigen vectors.
  10. 7 Marks
    bShow that the following matrix A is Diagonalizable. $$A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{bmatrix}$$
  11. 7 Marks
    aFind the Maximum and Minimum value of $u = a^2x^2 + b^2y^2 + c^2z^2$, where $x^2 + y^2 + z^2 = 1$ and $lx + my + nz = 0$.
  12. 7 Marks
    bFind the Fourier Series for the function $f(x) = x + x^2$ $(-\pi < x < \pi)$.
  13. 7 Marks
    aShow that the surfaces area of the solid generated by revolution of the loop of the curve $x = t^2$, $y = t - 1/3t^3$ about the $x$ axis is $3\pi$.
  14. 7 Marks
    bInvestigate for what values of $\lambda$ and $\mu$ the simultaneous equations. $$X + Y + Z = 6$$ $$X + 2Y + 3Z = 10$$ $$X + 2Y + \lambda Z = \mu$$
  15. 7 Marks
    aExpand $\log x$ in power $(x-1)$ by Taylor's theorem and hence find the value $\log 1.1$.
  16. 7 Marks
    bEvaluate $\iint xy \, dx \, dy$ where the region of integration is $x + y < 1$ in the positive quadrant.
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