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BT-102 · Mathematics I/Unsolved PYQ Paper

BT-102 Mathematics-I - Dec 2025 Question Paper

  1. 7 Marks
    aVerify Rolle's theorem for $f(x)=x^4-1$ in $[-1, 1]$
  2. 7 Marks
    bIf $u=\log (x^3+y^3+z^3-3xyz)$, Show that $$\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right)^2 u = -\frac{9}{(x+y+z)^2}$$
  3. 7 Marks
    aChange the order of integration in $\int_{0}^{a} \int_{x}^{a} f(x,y) dxdy$
  4. 7 Marks
    bFind the area of the cardioid $r=a (1+\cos\theta)$
  5. 7 Marks
    aTest the convergence of the series $$\frac{1}{1.2.3}+\frac{2}{2.3.4}+\frac{5}{3.4.5}+........$$
  6. 7 Marks
    bFind the Fourier series for the function $f(x)=x^3$ in $(-\pi, \pi)$.
  7. 7 Marks
    aDetermine whether the following vectors in $R^4$ are linearly dependent or independent: i) (1,2,-3,1), (3,7,1,-2) (1,3,7,-4) ii) (1,3,1,-2), (2,5,-1,3), (1,3,7,-2)
  8. 7 Marks
    bFind a basis and the dimension of the subspace W of P(t) spanned by: $$U=t^3+t^2-3t+2, V=2t^3+t^2+t-4 , W=4t^3+3t^2-5t+2$$
  9. 7 Marks
    aFind eigenvalues and eigenvectors of matrix $$A=\begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}$$
  10. 7 Marks
    bFind the rank of the matrix= $\begin{bmatrix} 2 & 1 & 3 \\ 4 & 7 & 13 \\ 4 & -3 & -1 \end{bmatrix}$
  11. 7 Marks
    aProve that $\beta(m,n)=\beta(m+1, n)+\beta(m, n+1)$ where $m, n >0$
  12. 7 Marks
    bTest the convergence of the series $$\frac{x}{1.2}+\frac{x^2}{2.3}+\frac{x^3}{3.4}+\frac{x^4}{4.5}......$$
  13. 7 Marks
    aShow that the given system of equations $x+y+z=6,\; x+2y+3z=14,\; x+4y+7z=30$ are consistent and solve them.
  14. 7 Marks
    bEvaluate $\int_{0}^{1} x^4 (1-\sqrt{x})^5 dx$
  15. 7 Marks
    aObtain Tayor's series expansion of the function $f(x,y)=e^{xy}$ about $(1,1)$ up to third degree terms.
  16. 7 Marks
    bDiscuss the extreme values (maxima and minima ) of the function $x^3+y^3-3axy$.
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