Mathematics I (BT-102) - Important Questions

Prove that a rectangular solid of maximum volume within a sphere is a cube.

If $u = \tan^{-1}\frac{x^3+y^3}{x-y}$, then prove that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = \sin 2u$.

Show that the surface area of solid generated by revolution of the loop of curve $x = t^2, y = t - t^3/3$ about the x-axis is $3\pi$.

Change the order of integration in the following integral and then evaluate $$\int_0^1 \int_{e^x}^e \frac{dx dy}{\log y}$$

Find the Fourier Series for $f(x) = x + x^2$ in $(-\pi, \pi)$

Find the half range sine series for $f(x) = x(\pi - x)$ in $(0, \pi)$. Hence deduce that $$\frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \cdots = \frac{\pi^3}{32}$$

Let $W_1$ and $W_2$ be subspaces of a vector space $V$ and assume that $W_1 \cap W_2 = \{0\}$. Let $w_1 \in W_1$ and $w_2 \in W_2$ be such that $w_1 \neq 0$ and $w_2 \neq 0$. Prove that $\{w_1, w_2\}$ is linearly independent.

Let $W_1$ and $W_2$ be subspaces of a vector space $V$. i) Prove that $W_1 \cap W_2$ is a subspace of $V$. ii) Give an example to show that $W_1 \cup W_2$ need not be a subspace of $V$. iii) Is $W_1 \setminus W_2$ a subspace of $V$? Justify.

Find Eigen values and Eigen vectors of $\begin{bmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{bmatrix}$

Diagonalize the matrix $\begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix}$

Reduce the matrix $$\begin{bmatrix} 3 & 2 & -1 \\ 4 & 2 & 6 \\ 7 & 4 & 5 \end{bmatrix}$$ to the normal form, hence find its rank.

Investigate for what values of $\lambda$ and $\mu$, the simultaneous equations $$\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + \lambda z &= \mu \end{aligned}$$ have i) no solution ii) a unique solution iii) infinite solutions.

Find the first 3 terms in the Maclaurin series for i) $\sin^2 x$ ii) $x e^{-x}$

Find the Taylor series for the function $x^4 + x - 2$ centered at $a = 1$.

Evaluate $\int_0^\infty \int_x^\infty \frac{e^{-y}}{y} dydx$

If $x^x y^y z^z = C$ then show that $\frac{\partial^2 z}{\partial x \partial y} = -(x \log ex)^{-1}$ at $x = y = z$.