BT-102 Mathematics-I - Nov 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
   aProve that $\frac{\pi}{3} - \frac{1}{5\sqrt{3}} > \cos^{-1}\frac{3}{5} > \frac{\pi}{3} - \frac{1}{8}$ using Lagrange's mean value theorem.
   How to answer

   • State Lagrange's Mean Value Theorem: $\frac{f(b)-f(a)}{b-a} = f'(c)$ for some $c \in (a, b)$.
   • Choose $f(x) = \cos^{-1}x$ on the interval $[3/5, 1/2]$ or $[1/2, 3/5]$.
   • Compute derivative $f'(x) = -\frac{1}{\sqrt{1-x^2}}$.
   • Bound $f'(c)$ using the endpoints of the interval to obtain the desired inequality.
2. 7 Marks
   bFind the minimum and maximum value of $$f(x,y) = x^3 + 3xy^2 - 3x^2 - 3y^2 + 4$$.
   How to answer

   • Compute first partial derivatives: $f_x = 3x^2 + 3y^2 - 6x$ and $f_y = 6xy - 6y$.
   • Set $f_x = 0$ and $f_y = 0$ to find stationary/critical points.
   • Compute second partial derivatives $r = f_{xx}$, $s = f_{xy}$, $t = f_{yy}$.
   • Evaluate $rt - s^2$ at each critical point to test for local maxima, minima, or saddle points.
   • Calculate the corresponding function values to determine maximum and minimum values.
3. 7 Marks
   aFind C of Cauchy's Mean value theorem on $[a, b]$ for the function $f(x) = e^x$ and $g(x) = e^{-x}$, $(a, b > 0)$.
   How to answer

   • State Cauchy's Mean Value Theorem: $\frac{f'(c)}{g'(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}$.
   • Find $f'(x) = e^x$ and $g'(x) = -e^{-x}$.
   • Substitute into formula: $\frac{e^c}{-e^{-c}} = -e^{2c} = \frac{e^b - e^a}{e^{-b} - e^{-a}}$.
   • Simplify the right-hand side to $-e^{a+b}$.
   • Equate $e^{2c} = e^{a+b} \implies c = \frac{a+b}{2}$ and verify $c \in (a, b)$.
4. 7 Marks
   bProve that $\Gamma(n)\Gamma(1-n) = \frac{\pi}{\sin n\pi}$
   How to answer

   • Use relationship between Beta and Gamma functions: $B(n, 1-n) = \frac{\Gamma(n)\Gamma(1-n)}{\Gamma(1)} = \Gamma(n)\Gamma(1-n)$.
   • Express Beta function as an integral: $B(n, 1-n) = \int_0^\infty \frac{y^{n-1}}{1+y}\, dy$.
   • Evaluate the definite integral using contour integration or substitution to get $\frac{\pi}{\sin n\pi}$.
   • Conclude $\Gamma(n)\Gamma(1-n) = \frac{\pi}{\sin n\pi}$.
5. 7 Marks
   aBy Changing the order of integration, evaluate $$\int_0^1 \int_0^{\sqrt{1-x^2}} y^2\, dy\, dx$$
   How to answer

   • Identify given limits: $y$ from $0$ to $\sqrt{1-x^2}$ and $x$ from $0$ to $1$ (first quadrant of circle $x^2 + y^2 = 1$).
   • Sketch the region of integration in the first quadrant.
   • Determine new limits for changed order: $x$ goes from $0$ to $\sqrt{1-y^2}$, and $y$ goes from $0$ to $1$.
   • Set up integral: $\int_0^1 \left( \int_0^{\sqrt{1-y^2}} y^2\, dx \right) dy = \int_0^1 y^2 \sqrt{1-y^2}\, dy$.
   • Evaluate using substitution $y = \sin\theta$ to find the final value $\frac{\pi}{16}$.
6. 7 Marks
   bFind the area of a plane in the form of a quadrant of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
   How to answer

   • Set up double integral for area: $\text{Area} = \int_{x=0}^a \int_{y=0}^{\frac{b}{a}\sqrt{a^2-x^2}} dy\, dx$.
   • Integrate with respect to $y$: $\int_0^a \frac{b}{a} \sqrt{a^2 - x^2}\, dx$.
   • Apply standard integration formula: $\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a}$.
   • Evaluate limits from $0$ to $a$ to get $\frac{\pi a b}{4}$.
7. 14 Marks
   Let W be a subspace of a finite dimensional vector space V(F). Then $\dim(V/W) = \dim V - \dim W$.
   How to answer

   • Let $\dim W = m$ and let $S_1 = \{w_1, w_2, \dots, w_m\}$ be a basis for $W$.
   • Extend $S_1$ to form a basis for $V$: $S = \{w_1, \dots, w_m, v_1, \dots, v_k\}$, where $m + k = n = \dim V$.
   • Consider the set of cosets $S' = \{v_1 + W, v_2 + W, \dots, v_k + W\}$ in the quotient space $V/W$.
   • Prove that $S'$ is linearly independent in $V/W$.
   • Prove that $S'$ spans $V/W$.
   • Conclude that $\dim(V/W) = k = n - m = \dim V - \dim W$.
8. 14 Marks
   Obtain the Fourier series to represent $f(x) = x\sin x,\; 0 < x < 2\pi$.
   How to answer

   • Write Fourier series general formula: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$.
   • Calculate $a_0 = \frac{1}{\pi}\int_0^{2\pi} x \sin x\, dx = -2$.
   • Calculate $a_n = \frac{1}{\pi}\int_0^{2\pi} x\sin x \cos nx\, dx$ using product-to-sum identities and integration by parts (handle $n=1$ and $n \neq 1$).
   • Calculate $b_n = \frac{1}{\pi}\int_0^{2\pi} x\sin x \sin nx\, dx$ (handle $n=1$ and $n \neq 1$).
   • Combine coefficients into the final Fourier series expansion.
9. 7 Marks
   aShow that $T : V_2(\mathbb{R}) \rightarrow V_3(\mathbb{R})$ is defined as $T(a, b) = (a - b, b - a, -a)$ is linear transformation.
   How to answer

   • State conditions for a linear transformation: $T(u + v) = T(u) + T(v)$ and $T(c u) = c T(u)$.
   • Let $u = (a_1, b_1)$ and $v = (a_2, b_2) \in V_2(\mathbb{R})$.
   • Compute $T(u + v) = T(a_1+a_2, b_1+b_2) = (a_1+a_2 - (b_1+b_2), b_1+b_2 - (a_1+a_2), -(a_1+a_2))$.
   • Verify this equals $T(u) + T(v)$.
   • Compute $T(c u) = T(ca_1, cb_1)$ and verify it equals $c T(u)$.
   • Conclude $T$ is a linear transformation.
10. 7 Marks
    bTest the convergence of the series $$\sum_{n=1}^\infty (\sqrt{n+1} - \sqrt{n-1})$$
    How to answer

    • Rationalize the general term $u_n = \sqrt{n+1} - \sqrt{n-1} = \frac{(n+1)-(n-1)}{\sqrt{n+1} + \sqrt{n-1}} = \frac{2}{\sqrt{n+1} + \sqrt{n-1}}$.
    • Choose comparison series $v_n = \frac{1}{\sqrt{n}} = \frac{1}{n^{1/2}}$.
    • Evaluate $\lim_{n \to \infty} \frac{u_n}{v_n} = \lim_{n \to \infty} \frac{2\sqrt{n}}{\sqrt{n}(\sqrt{1+1/n} + \sqrt{1-1/n})} = 1 \neq 0$.
    • By $p$-series test, $\sum v_n = \sum \frac{1}{n^{1/2}}$ diverges since $p = 1/2 \le 1$.
    • By limit comparison test, the given series diverges.
11. 7 Marks
    aVerify Cayley-Hamilton theorem for the matrix $$A = \begin{bmatrix} 1 & -2 & 2 \\ 1 & -2 & 3 \\ 0 & -1 & 2 \end{bmatrix}$$. Hence find $A^{-1}$.
    How to answer

    • Find the characteristic equation $|A - \lambda I| = 0$.
    • Calculate trace, sum of principal minors, and determinant $|A|$.
    • Write characteristic polynomial: $\lambda^3 - \text{tr}(A)\lambda^2 + \dots = 0$.
    • Compute $A^2$ and $A^3$ and substitute $A$ into the characteristic equation to verify it equals the zero matrix $O$.
    • Multiply the matrix equation by $A^{-1}$ to express $A^{-1}$ in terms of $A^2$, $A$, and $I$, then evaluate.
12. 7 Marks
    bExamine the consistency of the system of the following equations. If consistent, solve the equation's. $$x + y + z = 3$$ $$x + 2y + 3z = 4$$ $$x + 4y + 9z = 6$$
    How to answer

    • Write the augmented matrix $[A|B] = \begin{bmatrix} 1 & 1 & 1 & | & 3 \\ 1 & 2 & 3 & | & 4 \\ 1 & 4 & 9 & | & 6 \end{bmatrix}$.
    • Perform row operations ($R_2 \to R_2 - R_1$, $R_3 \to R_3 - R_1$, etc.) to convert into row echelon form.
    • Compare $\text{rank}(A)$ and $\text{rank}([A|B])$; conclude whether the system is consistent with unique or infinite solutions.
    • Use back substitution to solve for $x, y, z$.
13. 14 Marks
    Diagonalize the matrix $A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
    How to answer

    • Find characteristic equation $|A - \lambda I| = 0$ to get eigenvalues $\lambda_1, \lambda_2, \lambda_3$.
    • For each eigenvalue, solve $(A - \lambda I)X = 0$ to determine linearly independent eigenvectors $X_1, X_2, X_3$.
    • Construct the modal matrix $P = [X_1 \quad X_2 \quad X_3]$.
    • Compute $P^{-1}$ and state the diagonalized matrix $D = P^{-1}AP = \text{diag}(\lambda_1, \lambda_2, \lambda_3)$.