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

   • State Rolle's theorem: conditions of continuity on $[a, b]$, differentiability on $(a, b)$, and $f(a) = f(b)$.
   • Show $f(x) = x^2 + 2x$ is continuous on $[-2, 0]$ as it is a polynomial.
   • Show $f'(x) = 2x + 2$ exists on $(-2, 0)$.
   • Verify $f(-2) = 0 = f(0)$.
   • Find $c \in (-2, 0)$ such that $f'(c) = 2c + 2 = 0 \implies c = -1 \in (-2, 0)$.
2. 7 Marks
   bFind the first six terms of the expansions of the function $e^x \log(1 + y)$ in a Taylor series in the neighbourhood of the point $(0, 0)$.
   How to answer

   • Recall standard expansions for $e^x = 1 + x + \frac{x^2}{2!} + \dots$ and $\log(1+y) = y - \frac{y^2}{2} + \frac{y^3}{3} - \dots$.
   • Multiply the two series: $(1 + x + \frac{x^2}{2} + \dots)(y - \frac{y^2}{2} + \frac{y^3}{3} - \dots)$.
   • Collect terms degree by degree: $y + xy - \frac{y^2}{2} + \frac{x^2 y}{2} - \frac{xy^2}{2} + \frac{y^3}{3}$.
   • State the first six non-zero terms clearly.
3. 7 Marks
   aThe temperature $u(x, y, z)$ at any point in space is $u = 400xyz^2$ find the highest temperature on surface of the sphere $x^2 + y^2 + z^2 = 1$.
   How to answer

   • Set up Lagrange function: $L(x,y,z,\lambda) = 400xyz^2 - \lambda(x^2 + y^2 + z^2 - 1)$.
   • Formulate partial derivatives: $\frac{\partial L}{\partial x} = 400yz^2 - 2\lambda x = 0$, $\frac{\partial L}{\partial y} = 400xz^2 - 2\lambda y = 0$, $\frac{\partial L}{\partial z} = 800xyz - 2\lambda z = 0$.
   • Solve for relations between variables: $x^2 = y^2 = \frac{z^2}{2}$.
   • Substitute into sphere equation: $x^2 + x^2 + 2x^2 = 1 \implies 4x^2 = 1 \implies x = \pm 1/2$, $y = \pm 1/2$, $z^2 = 1/2$.
   • Evaluate maximum $u = 400 \cdot (1/2) \cdot (1/2) \cdot (1/2) = 50$.
4. 7 Marks
   bIf $u = x^2 \tan^{-1}\frac{y}{x} - y^2 \tan^{-1}\frac{x}{y}$, find the value of $\frac{\partial^2 u}{\partial x \partial y}$.
   How to answer

   • Compute $\frac{\partial u}{\partial y}$: differentiate each term with respect to $y$ treating $x$ as constant.
   • Differentiate result with respect to $x$ to get $\frac{\partial^2 u}{\partial x \partial y}$.
   • Simplify algebraic expressions involving $x^2 + y^2$.
   • Obtain final simplified value: $\frac{x^2 - y^2}{x^2 + y^2}$.
5. 7 Marks
   aFind shortest distance from the origin to the curve $x^2 + 4xy + 6y^2 = 140$.
   How to answer

   • Define objective function $f(x, y) = x^2 + y^2$ subject to constraint $g(x, y) = x^2 + 4xy + 6y^2 - 140 = 0$.
   • Apply method of Lagrange multipliers: $2x = \lambda(2x + 4y)$ and $2y = \lambda(4x + 12y)$.
   • Eliminate $\lambda$ to find relation between $x$ and $y$: $\frac{x}{x + 2y} = \frac{y}{2x + 6y} \implies 2x^2 + 6xy = xy + 2y^2 \implies 2x^2 + 5xy - 2y^2 = 0$.
   • Solve for $(x, y)$ on the curve and determine corresponding distance $D = \sqrt{x^2 + y^2}$.
   • Identify the minimum value among the stationary points.
6. 7 Marks
   bFind $\frac{du}{dt}$ if $u = x^2 + y^2$, $x = a\cos t$, $y = b\sin t$.
   How to answer

   • State total derivative formula: $\frac{du}{dt} = \frac{\partial u}{\partial x}\frac{dx}{dt} + \frac{\partial u}{\partial y}\frac{dy}{dt}$.
   • Calculate partial derivatives: $\frac{\partial u}{\partial x} = 2x$, $\frac{\partial u}{\partial y} = 2y$.
   • Calculate ordinary derivatives: $\frac{dx}{dt} = -a\sin t$, $\frac{dy}{dt} = b\cos t$.
   • Substitute and simplify: $\frac{du}{dt} = 2(a\cos t)(-a\sin t) + 2(b\sin t)(b\cos t) = (b^2 - a^2)\sin 2t$.
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 $x = 0$, $x = 1$, $y = x^2$, and $y = 2 - x$.
   • Split region into two parts: from $y = 0$ to $y = 1$ ($x$ runs from $0$ to $\sqrt{y}$) and from $y = 1$ to $y = 2$ ($x$ runs from $0$ to $2 - y$).
   • Write equivalent sum of integrals: $\int_{0}^{1}\int_{0}^{\sqrt{y}} xy \, dx \, dy + \int_{1}^{2}\int_{0}^{2-y} xy \, dx \, dy$.
   • Integrate each term with respect to $x$, then with respect to $y$.
   • Sum the results to obtain the final value ($3/8$).
8. 7 Marks
   bEvaluate $\iint e^{2x+3y} dxdy$ over the triangle bounded by $x = 0, y = 0$ and $x + y = 1$.
   How to answer

   • Set up double integral: $\int_{0}^{1} e^{2x} \left[\int_{0}^{1-x} e^{3y} dy\right] dx$.
   • Evaluate inner integral: $\int_{0}^{1-x} e^{3y} dy = \frac{1}{3}(e^{3(1-x)} - 1)$.
   • Substitute back into outer integral: $\frac{1}{3}\int_{0}^{1} (e^{3-x} - e^{2x}) dx$.
   • Integrate with respect to $x$: $\frac{1}{3}\left[ -e^{3-x} - \frac{e^{2x}}{2} \right]_{0}^{1}$.
   • Simplify the result: $\frac{e^3}{3} - \frac{e^2}{2} + \frac{1}{6}$.
9. 7 Marks
   aTest the convergence of 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 general term $u_n = \frac{x^n}{n^2 + 1}$ and $u_{n+1} = \frac{x^{n+1}}{(n+1)^2 + 1}$.
   • Apply D'Alembert's ratio test: $\lim_{n \to \infty}\frac{u_{n+1}}{u_n} = x \lim_{n \to \infty}\frac{n^2+1}{n^2+2n+2} = x$.
   • Conclude convergent for $|x| < 1$, divergent for $|x| > 1$.
   • Test for $x = 1$: $u_n = \frac{1}{n^2 + 1} < \frac{1}{n^2}$, convergent by comparison test ($p=2 > 1$).
   • Test for $x = -1$: alternating series, converges by Leibniz's test. Hence convergent for $|x| \le 1$.
10. 7 Marks
    bExpand as a half range $f(x) = x$ sine series and cosine series for the interval $0 < x < 2$.
    How to answer

    • Sine series: $L = 2$, compute $b_n = \int_{0}^{2} x \sin\left(\frac{n\pi x}{2}\right) dx$.
    • Integrate by parts to obtain $b_n = -\frac{4}{n\pi}\cos(n\pi) = \frac{4(-1)^{n+1}}{n\pi}$; write sine series.
    • Cosine series: 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 full cosine series: $f(x) = 1 + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{2}\right)$.
11. 7 Marks
    aExpand $f(x) = x\sin x$, $0 < x < 2\pi$ as a Fourier series.
    How to answer

    • State general formula for Fourier series over $(0, 2\pi)$: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$.
    • Evaluate $a_0 = \frac{1}{\pi}\int_0^{2\pi} x \sin x dx = -2$.
    • Evaluate $a_1$ separately and $a_n$ for $n \neq 1$ using product-to-sum trigonometric identities and integration by parts.
    • Evaluate $b_1$ separately and $b_n$ for $n \neq 1$.
    • Assemble the final Fourier series.
12. 7 Marks
    bFind the $a_0$ and $a_n$ if the function $f(x) = x + x^2$ is expanded in Fourier series defined in $(-1, 1)$.
    How to answer

    • Recognize $L = 1$ and exploit even/odd properties: $x$ is odd, $x^2$ is even.
    • Calculate $a_0 = \int_{-1}^{1} (x + x^2) dx = 0 + 2\int_{0}^{1} x^2 dx = \frac{2}{3}$.
    • Calculate $a_n = \int_{-1}^{1} (x + x^2)\cos(n\pi x) dx = 2\int_{0}^{1} x^2 \cos(n\pi x) dx$.
    • Apply integration by parts twice to obtain $a_n = \frac{4(-1)^n}{n^2\pi^2}$.
13. 7 Marks
    ai) If $A$ is a skew symmetric matrix then show that $A^2$ is a symmetric matrix. ii) Find eigen values of the matrix $\begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix}$.
    How to answer

    • Part i: Definition of skew-symmetric matrix $A^T = -A$.
    • Calculate $(A^2)^T = (AA)^T = A^T A^T = (-A)(-A) = A^2$.
    • Conclude $A^2$ is symmetric.
    • Part ii: Form characteristic equation $|A - \lambda I| = 0 \implies (5-\lambda)(2-\lambda) - 4 = 0$.
    • Solve $\lambda^2 - 7\lambda + 6 = 0 \implies (\lambda - 6)(\lambda - 1) = 0 \implies \lambda = 1, 6$.
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 augmented matrix $[A | I]$.
    • Perform row swap $R_1 \leftrightarrow R_2$ to obtain leading 1.
    • Use row operations to eliminate entries in column 1 ($R_3 \to R_3 - 3R_1$).
    • Reduce column 2 and 3 using elementary operations until left side becomes identity matrix $I$.
    • Extract the right side as $A^{-1} = \frac{1}{2}\begin{bmatrix} 1 & 0 & -1 \\ 7 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$.
15. 7 Marks
    aVerify Cayley Hamilton theorem for the matrix $A$ and hence find $A^{-1}$ for $$\begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$$
    How to answer

    • Find characteristic polynomial: $\det(A - \lambda I) = 0 \implies -\lambda^3 + 5\lambda^2 - 7\lambda + 1 = 0$ (or $\lambda^3 - 5\lambda^2 + 7\lambda - 1 = 0$).
    • Compute $A^2$ and $A^3$.
    • Substitute $A$ into characteristic equation to show $A^3 - 5A^2 + 7A - I = O$.
    • Multiply by $A^{-1}$: $A^{-1} = A^2 - 5A + 7I$.
    • Compute and display resulting matrix $A^{-1}$.
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

    • Form augmented matrix $[A | B]$ of size $4 \times 4$.
    • Apply elementary row operations to reduce $[A | B]$ to row echelon form.
    • Compare $\text{rank}(A)$ and $\text{rank}([A | B])$ to determine consistency.
    • If ranks are equal to number of variables (3), back-substitute to find unique solution.
    • State values of $x$, $y$, and $z$.