Skip to content
CampusPrep
BT-102 · Mathematics I/Unsolved PYQ Paper

BT-102 Mathematics-I - Dec 2023 Question Paper

  1. 7 Marks
    aState Lagrange'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 $e^x \cos y$ in a Taylor series in the neighbourhood of the point $(0, 0)$.
  3. 7 Marks
    aEstimate the extreme values of the function $x^3 + y^3 - 63(x + y) + 12xy$.
  4. 7 Marks
    bIf $u = u\left(\frac{y - x}{xy}, \frac{z - x}{xz}\right)$ find the value of $x^2 u_x + y^2 u_y + z^2 u_z$.
  5. 7 Marks
    aShow that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube.
  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
    bi) Find the value of $\sqrt{-\frac{3}{2}}$. ii) Evaluate $\int_0^1 x^3 (1-x)^5 \, dx$.
  9. 7 Marks
    aTest 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 $f(x) = x \sin x, 0 < x < 2\pi$ as a Fourier series.
  11. 7 Marks
    aShow that $$\beta(l, m) = \frac{\Gamma(l) \Gamma(m)}{\Gamma(l + m)}$$
  12. 7 Marks
    bExpand as a half range $f(x) = x$ sine series and cosine series for the interval $0 < x < 2$.
  13. 7 Marks
    aTransform the following matrix into normal form and hence find its rank $$\begin{bmatrix} 5 & 3 & 14 & 4 \\ 0 & 1 & 2 & 1 \\ 1 & -1 & 2 & 0 \end{bmatrix}$$.
  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
    aFind the eigen values and eigen vectors of matrix $$\begin{bmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{bmatrix}$$.
  16. 7 Marks
    bTest the consistency and hence, solve the following set of equations $$x + 2y - z = 3, 3x - y + 2z = 1, 2x - 2y + 3z = 2, x - y + z = -1$$.
Go to where you left off?

Quick Add to Notes

Save questions, your own notes and screenshots into notes filed by unit. It takes a free account.

Create free account

Have an account? Log in