BT-102 Mathematics-I - Jun 2022 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 the function $f(x) = \frac{\sin x}{e^x}$.
   How to answer

   • State conditions of Rolle's theorem (continuity, differentiability, $f(a)=f(b)$)
   • Identify interval typically $[0, \pi]$ where $\sin 0 = \sin \pi = 0$
   • Compute $f'(x) = e^{-x}(\cos x - \sin x)$
   • Solve $f'(c) = 0 \implies \tan c = 1 \implies c = \pi/4 \in (0, \pi)$
   • Conclude Rolle's theorem is verified
2. 7 Marks
   bFind the minimum value of $x^2 y z^3$, subject to the condition $2x + y + 3z = a$.
   How to answer

   • Set up Lagrange function: $L(x, y, z, \lambda) = x^2 y z^3 - \lambda(2x + y + 3z - a)$ (or take $\ln u$)
   • Differentiate with respect to $x, y, z$ and equate to zero
   • Obtain relations: $\frac{2}{x} = 2\lambda', \frac{1}{y} = \lambda', \frac{3}{z} = 3\lambda'$ leading to $x = y = z$
   • Substitute into constraint $2x + x + 3x = a \implies 6x = a \implies x = y = z = a/6$
   • Calculate the extreme value $(a/6)^6$
3. 7 Marks
   aVerify Cauchy's Mean value theorem for the function $\sin x$ and $\cos x$ in $\left[0, \frac{\pi}{2}\right]$.
   How to answer

   • State Cauchy's Mean Value Theorem: $\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}$ for $c \in (a, b)$
   • Calculate $f(\pi/2) - f(0) = 1 - 0 = 1$ and $g(\pi/2) - g(0) = 0 - 1 = -1$
   • RHS $= \frac{1}{-1} = -1$
   • Calculate derivatives: $f'(c) = \cos c$, $g'(c) = -\sin c$, ratio $= -\cot c$
   • Solve $-\cot c = -1 \implies \cot c = 1 \implies c = \pi/4$
   • Check that $c = \pi/4 \in (0, \pi/2)$ and conclude verification
4. 7 Marks
   bState and prove relationship between Beta and Gamma function.
   How to answer

   • Define Beta function $B(m,n)$ and Gamma function $\Gamma(n)$
   • Express $\Gamma(m)$ and $\Gamma(n)$ in terms of integrals with substitution $t = x^2, t = y^2$
   • Take product $\Gamma(m)\Gamma(n)$ as a double integral over the first quadrant
   • Convert double integral to polar coordinates ($x = r\cos\theta, y = r\sin\theta$)
   • Separate integrals to yield $\Gamma(m+n) \cdot B(m,n)$
   • Conclude $B(m, n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$
5. 7 Marks
   aChange the order of integration and evaluate $\int_0^{4a} \int_{\frac{x^2}{4a}}^{2\sqrt{ax}} dy\,dx$.
   How to answer

   • Identify given region bounded by $y = \frac{x^2}{4a} \implies x^2 = 4ay$ and $y = 2\sqrt{ax} \implies y^2 = 4ax$
   • Determine points of intersection: $(0,0)$ and $(4a, 4a)$
   • Change order: for horizontal strips, $x$ varies from $\frac{y^2}{4a}$ to $2\sqrt{ay}$, and $y$ varies from $0$ to $4a$
   • Set up new integral: $\int_0^{4a} \left[ \int_{y^2/(4a)}^{2\sqrt{ay}} dx \right] dy$
   • Evaluate inner integral: $2\sqrt{a}y^{1/2} - \frac{y^2}{4a}$
   • Integrate with respect to $y$ from $0$ to $4a$ to get final answer $\frac{16 a^2}{3}$
6. 7 Marks
   bUsing Double integration, find the volume of the tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.
   How to answer

   • Express $z$ as $z = c\left(1 - \frac{x}{a} - \frac{y}{b}\right)$
   • Identify projection region $R$ in $xy$-plane bounded by $x=0, y=0$, and $\frac{x}{a} + \frac{y}{b} = 1$
   • Set up double integral: $V = c \int_0^a \int_0^{b(1 - x/a)} \left(1 - \frac{x}{a} - \frac{y}{b}\right) dy\,dx$
   • Evaluate inner integral with respect to $y$
   • Evaluate outer integral with respect to $x$
   • Obtain volume $V = \frac{1}{6}abc$
7. 14 Marks
   Let $W_1$ and $W_2$ be two subspace of a finite dimensional vector space $V(F)$. Then $\dim(W_1 + W_2) = \dim W_1 + \dim W_2 - \dim(W_1 \cap W_2)$.
   How to answer

   • Let $\dim(W_1 \cap W_2) = k$ with basis $S = \{v_1, v_2, \dots, v_k\}$
   • Extend $S$ to a basis of $W_1$: $S_1 = \{v_1, \dots, v_k, u_1, \dots, u_m\}$, so $\dim W_1 = k + m$
   • Extend $S$ to a basis of $W_2$: $S_2 = \{v_1, \dots, v_k, w_1, \dots, w_p\}$, so $\dim W_2 = k + p$
   • Consider set $B = \{v_1, \dots, v_k, u_1, \dots, u_m, w_1, \dots, w_p\}$ and show it spans $W_1 + W_2$
   • Prove $B$ is linearly independent
   • Conclude $\dim(W_1 + W_2) = k + m + p = (k+m) + (k+p) - k = \dim W_1 + \dim W_2 - \dim(W_1 \cap W_2)$
8. 14 Marks
   Obtain the Fourier series to represent $$f(x) = \frac{1}{4}(\pi - x^2) \quad \text{in } 0 < x < 2\pi.$$
   How to answer

   • State the standard Fourier series formula on $(0, 2\pi)$
   • Compute $a_0 = \frac{1}{\pi} \int_0^{2\pi} \frac{1}{4}(\pi - x^2) dx$
   • Compute $a_n = \frac{1}{\pi} \int_0^{2\pi} \frac{1}{4}(\pi - x^2) \cos(nx) dx$ using integration by parts
   • Compute $b_n = \frac{1}{\pi} \int_0^{2\pi} \frac{1}{4}(\pi - x^2) \sin(nx) dx$ using integration by parts
   • Substitute coefficients $a_0, a_n, b_n$ into the Fourier series expansion
9. 7 Marks
   aShow that $T : V_3(\mathbb{R}) \to V_2(\mathbb{R})$ is defined as $T(a_1, a_2, a_3) = (a_1 - a_2, a_1 - a_3)$ is linear transformation.
   How to answer

   • Let $u = (a_1, a_2, a_3)$ and $v = (b_1, b_2, b_3) \in V_3(\mathbb{R})$, and scalars $\alpha, \beta \in \mathbb{R}$
   • Compute $\alpha u + \beta v = (\alpha a_1 + \beta b_1, \alpha a_2 + \beta b_2, \alpha a_3 + \beta b_3)$
   • Apply transformation $T(\alpha u + \beta v)$
   • Simplify components to separate $\alpha$ and $\beta$ terms
   • Show $T(\alpha u + \beta v) = \alpha T(u) + \beta T(v)$
   • Conclude that $T$ is a linear transformation
10. 7 Marks
    bTest the convergence of the series $$\sum_{n=1}^\infty \left(\sqrt{n^4+1} - \sqrt{n^4-1}\right).$$
    How to answer

    • Rationalize the general term $u_n = \frac{(n^4+1) - (n^4-1)}{\sqrt{n^4+1} + \sqrt{n^4-1}} = \frac{2}{\sqrt{n^4+1} + \sqrt{n^4-1}}$
    • For large $n$, approximate $u_n \approx \frac{2}{2n^2} = \frac{1}{n^2}$
    • Choose auxiliary series $\sum v_n = \sum \frac{1}{n^2}$
    • Evaluate $\lim_{n\to\infty} \frac{u_n}{v_n} = \lim_{n\to\infty} \frac{2 n^2}{\sqrt{n^4+1} + \sqrt{n^4-1}} = 1$ (finite and non-zero)
    • State that $\sum \frac{1}{n^2}$ converges by p-series test ($p=2 > 1$)
    • Conclude by Limit Comparison Test that the given series converges
11. 7 Marks
    aVerify Cayley-Hamilton theorem for the matrix $$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$
    How to answer

    • Formulate characteristic equation: $|A - \lambda I| = 0$
    • Calculate the polynomial: $\lambda^3 - \text{tr}(A)\lambda^2 + M\lambda - \det(A) = 0$
    • Trace $= 1 + 4 + 6 = 11$; compute sum of principal minors and determinant
    • Obtain characteristic equation $\lambda^3 - 11\lambda^2 - 4\lambda + 1 = 0$ (or respective coefficients)
    • Compute $A^2$ and $A^3$
    • Substitute $A$ into characteristic equation and show $A^3 - 11A^2 - 4A + I = O$
12. 7 Marks
    bExamine the consistency of the system and if consistent, solve the equations $$4x - 2y + 6z = 8, \quad x - y - 3z = -1, \quad 15x - 3y + 9z = 21.$$
    How to answer

    • Write system in matrix form $AX = B$ and construct augmented matrix $[A|B]$
    • Apply elementary row operations to reduce $[A|B]$ to row echelon form
    • Compare $\text{rank}(A)$ with $\text{rank}([A|B])$ to determine consistency
    • Identify whether system has unique solution or infinitely many solutions
    • Express variables in terms of free parameter(s) if dependent, or find unique values
13. 14 Marks
    Diagonalize the matrix $A = \begin{bmatrix} 8 & -8 & -2 \\ 4 & -3 & -2 \\ 3 & -4 & 1 \end{bmatrix}$.
    How to answer

    • Find characteristic equation $|A - \lambda I| = 0$ to get eigenvalues $\lambda$
    • Determine eigenvectors corresponding to each eigenvalue by solving $(A - \lambda I)X = 0$
    • Construct modal matrix $P = [X_1 \; X_2 \; X_3]$ from linearly independent eigenvectors
    • Compute $P^{-1}$
    • Calculate $D = P^{-1}AP$ to obtain the diagonal form displaying the eigenvalues