BT-102 Mathematics-I - Dec 2024 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
   aFind the points where the function $x^3 + y^3 - 3axy$ has maximum or minimum value.
   How to answer

   • Calculate first partial derivatives: $f_x = 3x^2 - 3ay = 0$, $f_y = 3y^2 - 3ax = 0$.
   • Solve simultaneously to get critical points $(0,0)$ and $(a,a)$.
   • Calculate second partial derivatives $r = f_{xx} = 6x$, $s = f_{xy} = -3a$, $t = f_{yy} = 6y$.
   • Evaluate $rt - s^2$ at each point to test for local maximum, minimum, or saddle point.
2. 7 Marks
   bFind the Taylor's expansion of $y = \sin x$ about point $x = \pi/2$.
   How to answer

   • State Taylor series formula: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$ with $a = \pi/2$.
   • Evaluate derivatives of $f(x) = \sin x$ at $x = \pi/2$: $f(\pi/2) = 1$, $f'(\pi/2) = 0$, $f''(\pi/2) = -1$, $f'''(\pi/2) = 0$, etc.
   • Substitute into formula to get series: $1 - \frac{(x-\pi/2)^2}{2!} + \frac{(x-\pi/2)^4}{4!} - \dots$
3. 7 Marks
   aThe part of the parabola $y^2 = 4ax$ cut off by the latus rectum revolves about the tangent at the vertex. Find the volume of the reel thus generated.
   How to answer

   • Identify axis of rotation: tangent at vertex is the y-axis ($x = 0$).
   • Limits for $y$ at latus rectum $x = a$ are from $-2a$ to $2a$.
   • Formula for volume about y-axis: $V = \pi \int_{-2a}^{2a} x^2 dy = 2\pi \int_0^{2a} x^2 dy$.
   • Express $x$ in terms of $y$: $x = \frac{y^2}{4a}$.
   • Integrate: $V = 2\pi \int_0^{2a} \frac{y^4}{16a^2} dy = \frac{\pi}{8a^2} \left[\frac{y^5}{5}\right]_0^{2a} = \frac{4}{5}\pi a^3$.
4. 7 Marks
   bProve that: $$\int_0^1 \sqrt{(1 - x^4)} dx = \frac{\{\Gamma(1/4)\}^2}{6\sqrt{(2\pi)}}$$
   How to answer

   • Substitute $x^4 = t \implies x = t^{1/4}$, $dx = \frac{1}{4} t^{-3/4} dt$.
   • Transform integral to $\frac{1}{4} \int_0^1 t^{-3/4} (1-t)^{1/2} dt = \frac{1}{4} B(1/4, 3/2)$.
   • Use $B(m, n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$ so $I = \frac{1}{4} \frac{\Gamma(1/4)\Gamma(3/2)}{\Gamma(7/4)}$.
   • Simplify using $\Gamma(3/2) = \frac{1}{2}\sqrt{\pi}$, $\Gamma(7/4) = \frac{3}{4}\Gamma(3/4)$, and $\Gamma(1/4)\Gamma(3/4) = \frac{\pi}{\sin(\pi/4)} = \pi\sqrt{2}$ to obtain the result.
5. 7 Marks
   aShow that the following series is Convergent. $$\frac{1}{4} + \frac{1}{4^2} + \frac{1}{4^3} + \dots + \frac{1}{4^n} + \dots$$
   How to answer

   • Identify the series as an infinite geometric series with first term $a = 1/4$ and common ratio $r = 1/4$.
   • State condition for convergence of geometric series: $|r| < 1$.
   • Since $|1/4| < 1$, the series converges.
   • Calculate the sum $S = \frac{a}{1 - r} = \frac{1/4}{1 - 1/4} = \frac{1}{3}$.
6. 7 Marks
   bObtain Half Range Sine Series for $f(x) = e^x$ in $0 < x < 1$.
   How to answer

   • State half-range sine series expansion for $L=1$: $f(x) = \sum_{n=1}^\infty b_n \sin(n\pi x)$.
   • Formula for $b_n = 2\int_0^1 e^x \sin(n\pi x) dx$.
   • Use standard integral $\int e^{ax} \sin(bx) dx = \frac{e^{ax}}{a^2+b^2}(a\sin bx - b\cos bx)$.
   • Evaluate from $x=0$ to $1$ to find $b_n = \frac{2n\pi}{1 + n^2\pi^2} (1 - (-1)^n e)$.
   • Write the final series.
7. 7 Marks
   aShow that the set $w = \{(a, b, 0) : a, b \in \mathbb{R}\}$ is subspace of $\mathbb{R}^3$.
   How to answer

   • Show non-emptiness: the zero vector $(0,0,0) \in w$ since $a=0, b=0 \in \mathbb{R}$.
   • Verify closure under vector addition: let $u = (a_1, b_1, 0), v = (a_2, b_2, 0) \in w$, then $u+v = (a_1+a_2, b_1+b_2, 0) \in w$.
   • Verify closure under scalar multiplication: for $c \in \mathbb{R}$, $cu = (ca_1, cb_1, 0) \in w$.
   • Conclude that $w$ is a subspace of $\mathbb{R}^3$.
8. 7 Marks
   bAre the following vectors LD? If so express one of these as a LC of other two. $X_1 = (1, 3, 4, 2)$, $X_2 = (3, -5, 2, 2)$, $X_3 = (2, -1, 3, 2)$
   How to answer

   • Set up the vector equation $c_1 X_1 + c_2 X_2 + c_3 X_3 = 0$.
   • Formulate matrix and apply row echelon operations.
   • Show non-trivial solution exists (e.g., $X_1 + X_2 = (4, -2, 6, 4) = 2 X_3$).
   • Conclude vectors are linearly dependent (LD).
   • Express $X_3 = \frac{1}{2}X_1 + \frac{1}{2}X_2$.
9. 7 Marks
   aFind a similarity transformation that diagonalise the matrix. $$A = \begin{bmatrix} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix}$$
   How to answer

   • Find characteristic equation $|A - \lambda I| = 0$ and solve for eigenvalues $\lambda$.
   • Find linearly independent eigenvectors corresponding to each eigenvalue.
   • Construct modal matrix $P$ with eigenvectors as columns.
   • Verify/state that $P^{-1}AP = D$, the diagonal matrix of eigenvalues.
10. 7 Marks
    bFind the Eigen value and Corresponding Eigen Vectors of the following Matrix. $$\begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}$$
    How to answer

    • Formulate characteristic equation $\det(A - \lambda I) = 0$.
    • Expand determinant: $\lambda^3 - 18\lambda^2 + 45\lambda = 0 \implies \lambda(\lambda-3)(\lambda-15) = 0$.
    • Eigenvalues are $\lambda = 0, 3, 15$.
    • Solve $(A - \lambda I)X = 0$ for each $\lambda$ to determine the corresponding eigenvectors.
11. 7 Marks
    aDefine Beta and Gamma Function and show that relation $$B(m, n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m + n)}$$
    How to answer

    • Define Beta function: $B(m,n) = \int_0^1 x^{m-1}(1-x)^{n-1}dx$ for $m>0, n>0$.
    • Define Gamma function: $\Gamma(n) = \int_0^\infty e^{-x} x^{n-1}dx$ for $n>0$.
    • Use substitution $x = y^2$ to express $\Gamma(m)$ and $\Gamma(n)$ in terms of double integrals in polar coordinates.
    • Evaluate the product $\Gamma(m)\Gamma(n)$ and transform variables to show $\Gamma(m)\Gamma(n) = B(m,n)\Gamma(m+n)$.
12. 7 Marks
    bEvaluate: i) $\int_0^1 x^2 (1 - x)^3 dx$ ii) $\int_0^1 \sqrt{\left(\frac{1 - x}{x}\right)} dx$
    How to answer

    • Part i: Recognize as $B(3, 4) = \frac{\Gamma(3)\Gamma(4)}{\Gamma(7)} = \frac{2! \cdot 3!}{6!} = \frac{2 \times 6}{720} = \frac{1}{60}$.
    • Part ii: Rewrite as $\int_0^1 x^{-1/2} (1-x)^{1/2} dx = B(1/2, 3/2)$.
    • Compute $B(1/2, 3/2) = \frac{\Gamma(1/2)\Gamma(3/2)}{\Gamma(2)} = \frac{\sqrt{\pi} \cdot \frac{1}{2}\sqrt{\pi}}{1} = \frac{\pi}{2}$.
13. 7 Marks
    aProve that the surface area of the solid generated by the revolution of the ellipse $x^2/a^2 + y^2/b^2 = 1$ about the major axis is: $$2(\pi ab) \left\{\sqrt{(1 - e^2)} + \frac{\sin^{-1} e}{e}\right\}$$
    How to answer

    • Surface area of revolution formula about x-axis: $S = 2\pi \int_{-a}^a y \sqrt{1 + (dy/dx)^2} dx = 4\pi \int_0^a y \sqrt{1 + (dy/dx)^2} dx$.
    • For ellipse $x^2/a^2 + y^2/b^2 = 1$, find $\frac{dy}{dx} = -\frac{b^2 x}{a^2 y}$.
    • Show $y\sqrt{1 + (dy/dx)^2} = \frac{b}{a}\sqrt{a^2 - e^2 x^2}$ using $b^2 = a^2(1-e^2)$.
    • Integrate using standard form $\int \sqrt{a^2 - (ex)^2} dx$ from $0$ to $a$.
    • Substitute limits and express in terms of eccentricity $e$ to arrive at the given formula.
14. 7 Marks
    bShow that the sequence $\langle n^{1/n} \rangle$ converge to 1.
    How to answer

    • Let $y = n^{1/n}$, then $\ln y = \frac{\ln n}{n}$.
    • Take the limit $\lim_{n \to \infty} \ln y = \lim_{n \to \infty} \frac{\ln n}{n}$, which is of indeterminate form $\frac{\infty}{\infty}$.
    • Apply L'Hopital's rule: $\lim_{n \to \infty} \frac{1/n}{1} = 0$.
    • Conclude $\lim_{n \to \infty} y = e^0 = 1$.
15. 7 Marks
    aProve that a rectangular solid of maximum volume within a sphere is a cube.
    How to answer

    • Let the dimensions of the rectangular box inscribed in sphere of radius $R$ be $2x, 2y, 2z$, so $x^2 + y^2 + z^2 = R^2$.
    • Volume is $V = 8xyz$.
    • Form Lagrangian function $L(x,y,z,\lambda) = 8xyz + \lambda(x^2 + y^2 + z^2 - R^2)$ or substitute $z^2 = R^2 - x^2 - y^2$.
    • Differentiate and equate to zero: $yz + \frac{\lambda}{4} x = 0$, etc., leading to $x^2 = y^2 = z^2 = R^2/3$.
    • Conclude $x = y = z$, thus the rectangular solid of maximum volume is a cube.
16. 7 Marks
    bVerify Rolle's Theorem for the function $y = x^2 + 2$, $a = -2$ and $b = 2$.
    How to answer

    • Check continuity: $f(x) = x^2 + 2$ is a polynomial, continuous on $[-2, 2]$.
    • Check differentiability: $f'(x) = 2x$ exists on $(-2, 2)$, so $f(x)$ is differentiable on $(-2, 2)$.
    • Check boundary values: $f(-2) = (-2)^2 + 2 = 6$ and $f(2) = (2)^2 + 2 = 6$, so $f(-2) = f(2)$.
    • Find $c \in (-2, 2)$ such that $f'(c) = 0 \implies 2c = 0 \implies c = 0 \in (-2, 2)$.
    • Conclude Rolle's theorem is verified.