BT-102 Mathematics-I - Dec 2023 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
   aState Lagrange's theorem hence verify for $f(x) = x^2 + 2x$ defined in the interval $[-2, 0]$.
   How to answer

   • State Lagrange's Mean Value Theorem: If $f(x)$ is continuous in $[a,b]$ and differentiable in $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.
   • Check continuity of $f(x) = x^2 + 2x$ on $[-2, 0]$ and differentiability on $(-2, 0)$.
   • Compute $f(-2) = 0$ and $f(0) = 0$.
   • Compute $f'(x) = 2x + 2$ and equate $f'(c) = \frac{0 - 0}{0 - (-2)} = 0$.
   • Solve $2c + 2 = 0 \implies c = -1 \in (-2, 0)$, verifying the theorem.
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 $(0, 0)$.
   How to answer

   • Write formula for two-variable Taylor/Maclaurin series around $(0,0)$.
   • Calculate partial derivatives at $(0,0)$ up to second and third order: $f(0,0)=1$, $f_x=1$, $f_y=0$, $f_{xx}=1$, $f_{xy}=0$, $f_{yy}=-1$, $f_{xxx}=1$, $f_{xxy}=0$, $f_{xyy}=-1$, $f_{yyy}=0$.
   • Write terms: $1 + x + \frac{1}{2}(x^2 - y^2) + \frac{1}{6}(x^3 - 3xy^2) + \dots$
   • Collect and specify the first six non-zero terms.
3. 7 Marks
   aEstimate the extreme values of the function $x^3 + y^3 - 63(x + y) + 12xy$.
   How to answer

   • Find first partial derivatives: $f_x = 3x^2 - 63 + 12y$, $f_y = 3y^2 - 63 + 12x$.
   • Set $f_x = 0$ and $f_y = 0$ to solve for critical points $(x, y)$.
   • Calculate second partial derivatives $r = f_{xx} = 6x$, $s = f_{xy} = 12$, $t = f_{yy} = 6y$.
   • Evaluate discriminant $rt - s^2 = 36xy - 144$ at each stationary point.
   • Classify points into local maxima, local minima, or saddle points, and find the corresponding extreme values.
4. 7 Marks
   bIf $u = u\left(\frac{y - x}{xy}, \frac{z - x}{xz}\right)$ find the value of $x^2 u_x + y^2 u_y + z^2 u_z$.
   How to answer

   • Let $X = \frac{y - x}{xy} = \frac{1}{x} - \frac{1}{y}$ and $Y = \frac{z - x}{xz} = \frac{1}{x} - \frac{1}{z}$.
   • Compute partial derivatives: $\frac{\partial X}{\partial x} = -\frac{1}{x^2}$, $\frac{\partial X}{\partial y} = \frac{1}{y^2}$, $\frac{\partial X}{\partial z} = 0$, and $\frac{\partial Y}{\partial x} = -\frac{1}{x^2}$, $\frac{\partial Y}{\partial y} = 0$, $\frac{\partial Y}{\partial z} = \frac{1}{z^2}$.
   • By chain rule, express $u_x, u_y, u_z$ in terms of $u_X$ and $u_Y$.
   • Form expressions for $x^2 u_x$, $y^2 u_y$, and $z^2 u_z$.
   • Add the three terms to obtain the result: 0.
5. 7 Marks
   aShow that the rectangular solid of maximum volume that can be inscribed in a given sphere is a cube.
   How to answer

   • Let dimensions of rectangular solid be $2x, 2y, 2z$. Volume $V = 8xyz$.
   • Sphere constraint: $x^2 + y^2 + z^2 = R^2$.
   • Formulate using Lagrange's method of undetermined multipliers: $F(x,y,z) = 8xyz - \lambda(x^2 + y^2 + z^2 - R^2)$.
   • Take partial derivatives with respect to $x, y, z$ and equate to zero.
   • Show that $x^2 = y^2 = z^2 = \frac{R^2}{3} \implies x = y = z$, thus proving the solid is a cube.
6. 7 Marks
   bFind $\frac{du}{dt}$ if $u = x^2 + y^2, x = a \cos t, y = b \sin t$.
   How to answer

   • Use total derivative formula: $\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt}$.
   • Compute $\frac{\partial u}{\partial x} = 2x$, $\frac{\partial u}{\partial y} = 2y$.
   • Compute $\frac{dx}{dt} = -a \sin t$, $\frac{dy}{dt} = b \cos t$.
   • Substitute $x = a \cos t$ and $y = b \sin t$: $\frac{du}{dt} = 2(a \cos t)(-a \sin t) + 2(b \sin t)(b \cos t)$.
   • Simplify to $(b^2 - a^2) \sin 2t$.
7. 7 Marks
   aChange the order of integration in $\int_0^1 \int_{x^2}^{2-x} xy \, dy \, dx$ and hence evaluate.
   How to answer

   • Sketch the region bounded by $y = x^2$, $y = 2 - x$, $x = 0$, and $x = 1$.
   • Identify intersection points: $(0,0)$, $(1,1)$, $(0,2)$.
   • Split the region into two sub-regions with horizontal strips: from $y = 0$ to $y = 1$ ($x$ from $0$ to $\sqrt{y}$) and from $y = 1$ to $y = 2$ ($x$ from $0$ to $2-y$).
   • Write the integral as $\int_0^1 \int_0^{\sqrt{y}} xy \, dx \, dy + \int_1^2 \int_0^{2-y} xy \, dx \, dy$.
   • Evaluate both integrals and sum them up to get the final numerical result.
8. 7 Marks
   bi) Find the value of $\sqrt{-\frac{3}{2}}$. ii) Evaluate $\int_0^1 x^3 (1-x)^5 \, dx$.
   How to answer

   • For part (i): Note that symbol $\sqrt{\dots}$ represents $\Gamma(n)$. Use $\Gamma(n) = \frac{\Gamma(n+1)}{n}$.
   • Compute $\Gamma(-3/2) = \frac{\Gamma(-1/2)}{-3/2} = \frac{\Gamma(1/2)}{(-3/2)(-1/2)} = \frac{4}{3}\sqrt{\pi}$.
   • For part (ii): Express in beta function form $B(4, 6) = \int_0^1 x^{4-1} (1-x)^{6-1} dx$.
   • Calculate $B(4, 6) = \frac{\Gamma(4)\Gamma(6)}{\Gamma(10)} = \frac{3! \times 5!}{9!} = \frac{6}{9 \times 8 \times 7 \times 6} = \frac{1}{504}$.
9. 7 Marks
   aTest the series $1 + \frac{x}{2} + \frac{x^2}{5} + \frac{x^3}{10} + \dots + \frac{x^n}{n^2+1} + \dots$
   How to answer

   • Identify $n$-th term $u_n = \frac{x^n}{n^2+1}$.
   • Apply D'Alembert's ratio test: $\lim_{n\to\infty} |\frac{u_{n+1}}{u_n}| = |x|$.
   • Conclude convergent for $|x| < 1$, divergent for $|x| > 1$.
   • Test for $x = 1$: $\sum \frac{1}{n^2+1}$, compare with $\sum \frac{1}{n^2}$ ($p=2 > 1$, convergent).
   • Test for $x = -1$: alternating series converges absolutely. State overall convergence for $|x| \le 1$.
10. 7 Marks
    bExpand $f(x) = x \sin x, 0 < x < 2\pi$ as a Fourier series.
    How to answer

    • Write standard Fourier expansion formula on $(0, 2\pi)$: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$.
    • Calculate $a_0 = \frac{1}{\pi} \int_0^{2\pi} x \sin x \, dx = -2$.
    • Calculate $a_n = \frac{1}{\pi} \int_0^{2\pi} x \sin x \cos nx \, dx$ for $n = 1$ and $n \neq 1$.
    • Calculate $b_n = \frac{1}{\pi} \int_0^{2\pi} x \sin x \sin nx \, dx$ for $n = 1$ and $n \neq 1$.
    • Substitute all coefficients into the Fourier expansion.
11. 7 Marks
    aShow that $$\beta(l, m) = \frac{\Gamma(l) \Gamma(m)}{\Gamma(l + m)}$$
    How to answer

    • Express Gamma function in polar/substitution form: $\Gamma(l) = 2 \int_0^\infty e^{-x^2} x^{2l-1} dx$.
    • Multiply $\Gamma(l) \Gamma(m) = 4 \int_0^\infty \int_0^\infty e^{-(x^2+y^2)} x^{2l-1} y^{2m-1} dx \, dy$.
    • Transform double integral into polar coordinates: $x = r \cos\theta$, $y = r \sin\theta$, $dx \, dy = r \, dr \, d\theta$.
    • Separate integrals: $2 \int_0^{\pi/2} \cos^{2l-1}\theta \sin^{2m-1}\theta \, d\theta \cdot 2 \int_0^\infty e^{-r^2} r^{2(l+m)-1} dr$.
    • Identify terms as $\beta(l, m) \cdot \Gamma(l+m)$, establishing the relation.
12. 7 Marks
    bExpand as a half range $f(x) = x$ sine series and cosine series for the interval $0 < x < 2$.
    How to answer

    • Half-range Sine Series: $L = 2$, $f(x) = \sum_{n=1}^\infty b_n \sin\left(\frac{n\pi x}{2}\right)$.
    • Compute $b_n = \int_0^2 x \sin\left(\frac{n\pi x}{2}\right) dx = -\frac{4}{n\pi} (-1)^n$.
    • Half-range Cosine Series: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos\left(\frac{n\pi x}{2}\right)$.
    • Compute $a_0 = \int_0^2 x \, dx = 2$.
    • Compute $a_n = \int_0^2 x \cos\left(\frac{n\pi x}{2}\right) dx = \frac{4}{n^2\pi^2} [(-1)^n - 1]$.
    • Write both resulting Fourier series.
13. 7 Marks
    aTransform the following matrix into normal form and hence find its rank $$\begin{bmatrix} 5 & 3 & 14 & 4 \\ 0 & 1 & 2 & 1 \\ 1 & -1 & 2 & 0 \end{bmatrix}$$.
    How to answer

    • Swap row 1 and row 3 ($R_1 \leftrightarrow R_3$) to get leading 1 at position $(1,1)$.
    • Apply row operations to make all other elements in the first column zero ($R_3 \to R_3 - 5R_1$).
    • Apply column operations to make all other elements in the first row zero.
    • Perform row and column operations on the remaining $2 \times 3$ submatrix to reduce to identity matrix $I_r$.
    • Obtain normal form $\begin{bmatrix} I_2 & 0 \\ 0 & 0 \end{bmatrix}$ or $\begin{bmatrix} I_3 & 0 \end{bmatrix}$ and conclude rank $r$.
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.
    How to answer

    • Set up the augmented matrix $[A | I]$ or write $A = IA$.
    • Swap $R_1 \leftrightarrow R_2$ to obtain leading 1 in row 1.
    • Use row operations: $R_3 \to R_3 - 3R_1$.
    • Use $R_2$ to clear elements in column 2.
    • Normalize $R_3$ and clear remaining elements in column 3 to transform $A$ into $I$.
    • The resulting right-hand side is $A^{-1} = \frac{1}{2}\begin{bmatrix} 1 & 0 & -1 \\ 7 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$.
15. 7 Marks
    aFind the eigen values and eigen vectors of matrix $$\begin{bmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{bmatrix}$$.
    How to answer

    • Set up the characteristic equation: $|A - \lambda I| = 0$.
    • Expand determinant to get the cubic polynomial in $\lambda$ and solve for eigenvalues $\lambda_1, \lambda_2, \lambda_3$.
    • For each eigenvalue $\lambda$, set up the system $(A - \lambda I)X = 0$.
    • Solve for non-trivial eigenvector solutions $X_1, X_2, X_3$ using row reduction or cofactor methods.
    • State the eigenvalues and their corresponding linearly independent eigenvectors.
16. 7 Marks
    bTest the consistency and hence, solve the following set of equations $$x + 2y - z = 3, 3x - y + 2z = 1, 2x - 2y + 3z = 2, x - y + z = -1$$.
    How to answer

    • Write the system in matrix form $AX = B$ with augmented matrix $[A | B]$.
    • Reduce $[A | B]$ to row echelon form using elementary row operations.
    • Determine the rank of coefficient matrix $\text{rank}(A)$ and augmented matrix $\text{rank}(A|B)$.
    • Check consistency: consistent if $\text{rank}(A) = \text{rank}(A|B)$, otherwise inconsistent.
    • If consistent, back-substitute to find the values of $x, y, z$.