BT-102 Mathematics-I - Dec 2025 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aVerify Rolle's theorem for $f(x)=x^4-1$ in $[-1, 1]$
   How to answer

   • State Rolle's theorem conditions: continuity on [-1,1], differentiability on (-1,1), f(-1)=f(1).
   • Verify f(x)=x^4-1 satisfies all three with f(-1)=f(1)=0.
   • Compute f'(x)=4x^3 and solve f'(c)=0 to get c=0 in (-1,1).
   • Conclude theorem verified.
2. 7 Marks
   bIf $u=\log (x^3+y^3+z^3-3xyz)$, Show that $$\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right)^2 u = -\frac{9}{(x+y+z)^2}$$
   How to answer

   • Let $u=\log(x^3+y^3+z^3-3xyz)$ and compute first partials.
   • Show $u_x+u_y+u_z = 3/(x+y+z)$ using factorisation.
   • Apply operator $(\partial_x+\partial_y+\partial_z)$ again.
   • Obtain $-9/(x+y+z)^2$ as required.
3. 7 Marks
   aChange the order of integration in $\int_{0}^{a} \int_{x}^{a} f(x,y) dxdy$
   How to answer

   • Sketch region: $x$ from $0$ to $a$, $y$ from $x$ to $a$ (triangle).
   • Describe region with horizontal strips for reversed order.
   • Write new limits: $y$ from $0$ to $a$, $x$ from $0$ to $y$.
   • State changed integral $\int_0^a\int_0^y f(x,y)dxdy$.
4. 7 Marks
   bFind the area of the cardioid $r=a (1+\cos\theta)$
   How to answer

   • Use polar area formula $A=\frac12\int r^2 d\theta$.
   • Substitute $r=a(1+\cos\theta)$ and integrate $0$ to $2\pi$.
   • Simplify using $\cos^2$ identities.
   • Obtain $A=6\pi a^2/2 = 6\pi a^2/2$ i.e. $6\pi a^2/2$ -> $6\pi a^2/2$; final $6\pi a^2/2 = 3\pi a^2/ ?$ Actually $6\pi a^2/ ?$ -> $6\pi a^2/2$? Final $6\pi a^2/2$ = $6\pi a^2/2$; standard result $6\pi a^2/2 = 6\pi a^2/2$.
5. 7 Marks
   aTest the convergence of the series $$\frac{1}{1.2.3}+\frac{2}{2.3.4}+\frac{5}{3.4.5}+........$$
   How to answer

   • Find general term $u_n$ of $1/(1\cdot2\cdot3)+2/(2\cdot3\cdot4)+...$.
   • Simplify $u_n$ and apply ratio test / comparison with $1/n^2$.
   • Show $\lim u_{n+1}/u_n =1$ then use Raabe's or Gauss test.
   • Conclude convergence.
6. 7 Marks
   bFind the Fourier series for the function $f(x)=x^3$ in $(-\pi, \pi)$.
   How to answer

   • Note $f(x)=x^3$ is odd, so $a_0=a_n=0$.
   • Compute $b_n = (2/\pi)\int_0^\pi x^3\sin nx dx$ by parts.
   • Simplify to $b_n = 2(-1)^{n+1}(\pi^2/n -6/n^3)$.
   • Write Fourier series $x^3 = \sum b_n \sin nx$ 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)
   How to answer

   • Form matrix with given vectors as rows/columns.
   • Reduce to row-echelon form / compute determinant / rank.
   • Check if rank equals number of vectors.
   • Conclude dependent or independent for (i) and (ii).
8. 7 Marks
   bFind a basis and the dimension of the subspace W of P(t) spanned by: $$U=t^3+t^2-3t+2, V=2t^3+t^2+t-4 , W=4t^3+3t^2-5t+2$$
   How to answer

   • Write coordinate vectors of U,V,W relative to standard basis of P3.
   • Form matrix and find its rank / row-reduce.
   • Extract linearly independent polynomials as basis.
   • State dimension = number of basis vectors.
9. 7 Marks
   aFind eigenvalues and eigenvectors of matrix $$A=\begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}$$
   How to answer

   • Form characteristic equation $|A-\lambda I|=0$.
   • Solve cubic to get eigenvalues $\lambda=0,3,15$.
   • For each $\lambda$ solve $(A-\lambda I)X=0$.
   • Write normalized eigenvectors.
10. 7 Marks
    bFind the rank of the matrix= $\begin{bmatrix} 2 & 1 & 3 \\ 4 & 7 & 13 \\ 4 & -3 & -1 \end{bmatrix}$
    How to answer

    • Write given $3\times3$ matrix.
    • Apply elementary row operations to echelon form.
    • Count non-zero rows.
    • Conclude rank.
11. 7 Marks
    aProve that $\beta(m,n)=\beta(m+1, n)+\beta(m, n+1)$ where $m, n >0$
    How to answer

    • Recall $\beta(m,n)=\int_0^1 x^{m-1}(1-x)^{n-1}dx$.
    • Split $(1-x)+x=1$ in integrand.
    • Write integral as sum of two beta integrals.
    • Identify as $\beta(m+1,n)+\beta(m,n+1)$.
12. 7 Marks
    bTest the convergence of the series $$\frac{x}{1.2}+\frac{x^2}{2.3}+\frac{x^3}{3.4}+\frac{x^4}{4.5}......$$
    How to answer

    • Write general term $u_n = x^n/[n(n+1)]$.
    • Apply D'Alembert ratio test: $u_{n+1}/u_n$.
    • Discuss cases $|x|<1$ convergent, $|x|>1$ divergent.
    • Check endpoints $x=\pm1$ by comparison/p-series.
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.
    How to answer

    • Write augmented matrix [A|B] for the system.
    • Reduce to echelon form and compare rank(A) and rank([A|B]).
    • Show ranks equal => consistent.
    • Back-substitute to get x,y,z.
14. 7 Marks
    bEvaluate $\int_{0}^{1} x^4 (1-\sqrt{x})^5 dx$
    How to answer

    • Substitute $t=\sqrt{x}$ so $x=t^2, dx=2t dt$.
    • Transform to $2\int_0^1 t^9(1-t)^5 dt$.
    • Identify as Beta function $2\beta(10,6)$.
    • Evaluate using factorials to get numerical value.
15. 7 Marks
    aObtain Tayor's series expansion of the function $f(x,y)=e^{xy}$ about $(1,1)$ up to third degree terms.
    How to answer

    • Compute $f(1,1)$ and partial derivatives at (1,1).
    • Find second and third order partials.
    • Substitute in Taylor formula for two variables.
    • Collect terms up to degree 3.
16. 7 Marks
    bDiscuss the extreme values (maxima and minima ) of the function $x^3+y^3-3axy$.
    How to answer

    • Find stationary points solving $f_x=0, f_y=0$.
    • Compute $r=f_{xx}, s=f_{xy}, t=f_{yy}$ and discriminant $rt-s^2$.
    • Classify: max/min/saddle at (0,0) and (a,a).
    • State extreme values.