Solve your IGNOU Doubts

Question:

Find the centre of gravity of a thin sheet with density $\delta(x, y) = y$ , bounded by the curves $y = 4x^2$ and $x = 4$ .

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Question:

Let $a = (1, 2, 3), b = (-5, 3, -2), c = (2, -4, 1)$ be three points in $\mathbb{R}^3$ .

Find |2b - a + 3c|.

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Question:

Let $x = e^r \cos \theta, y = e^r \sin \theta$ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$$\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial \theta^2} = (x^2 + y^2) \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right)$

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Question:

Let the function f be defined by
$$f(x, y) = \begin{cases} \frac{3x^2 y^4}{x^4 + y^8}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}$
$\quad$ Show that f has directional derivatives in all directions at (0, 0).

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Question:

Using only the definitions, find f_xy(0, 0) and f_yx(0, 0), if they exists, for the function
$$f(x, y) = \begin{cases} \frac{x^2 y}{\sqrt{x^2 + y^2}}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

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Question:

Find the following limits:
$\quad$ (i) $\displaystyle \lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$
$\quad$ (ii) $\displaystyle \lim_{x \to 0^+} (\sin x)^{\sin x}$

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Question:

State whether the following statements are true or false. Give reasons for your answers.
(i) $\displaystyle \lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$
(ii) A real-valued function of three variables which is continuous everywhere is differentiable.
(iii) The function $F : \mathbb{R}^2 \to \mathbb{R}^2$ , defined by $F(x, y) = (y + 2, x + y)$ , is locally invertible at any $(x, y) \in \mathbb{R}^2$ .
(iv) $f : [-1,1] \times [-2,2] \to \mathbb{R}$ , defined by
$$f(x, y) = \begin{cases} x, & \text{if } y \text{ is rational} \\ 0, & \text{if } y \text{ is not rational} \end{cases}$
is integrable.
(v) The function $f : \mathbb{R}^2 \to \mathbb{R}$ , defined by $f(x, y) = 1 - y^2 + x^2$ , has an extremum at (0, 0).

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Question:

Verigy Lagrange's mean value theorem for the function f defined by

$f(x) = 2x^2 - 7x - 10$ over [2, 5].

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Question:

If $I_{m,n} = \int x^m (\log x)^n dx$ , show that:

$$(m+1)I_{m,n} = x^{m+1}(\log x)^n - n I_{m,n-1}.$
Hence find the value of $\int x^4 (\log x)^3 dx$ .

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Question:

Evaluate:

$$\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$

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Question:

Solve the equation:

$$x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$

given that its roots are in G.P.

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Question:

Find the largest subset of $\mathbf{R}$ on which the function $f : \mathbf{R} \to \mathbf{R}$ defined as:
$$f(x) = \begin{cases} 2x &, x > 5 \\ x + 5 &, 1 \leq x \leq 5 \\ |x| &, x < 1 \end{cases}$
is continuous.

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Question:

If $y = e^{m \sin^{-1} x}$ , then show that $(1 - x^2)y_2 - xy_1 - m^2y = 0$ . Hence using Leibniz's formula, find the value of (1 - x²)y_n+2 - (2n + 1)xy_n+1.

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Question:

) Find the condition for the curves, $ax^2 + by^2 = 1$ and $a'x^2 + b'y^2 = 1$ intersecting orthogonally.

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Question:

Find the length of the cycloid $x = \alpha(\theta - \sin \theta), y = \alpha(1 - \cos \theta)$ and show that the line $\theta = \frac{2\pi}{3}$ divides it in the ratio 1 : 3.

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Question:

Trace the curve $y^2(x + 1) = x^2(3 - x)$ , clearly stating all the properties used for tracing it.

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Question:

Find the area between the curve $y^2(4 - x) = x^3$ and its asymptote parallel to y-axis. If the revenue function is given by $\frac{dR}{dx} = 15 + 2x - x^2$ , x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

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Question:

Find the area between the curve $y^2(4 - x) = x^3$ and its asymptote parallel to y-axis.

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Question:

) Let f and g be two functions defined on $\mathbf{R}$ by:

$$f(x) = x^3 - x^2 - 8x + 12$

and $g(x) = \begin{cases} \frac{f(x)}{x + 3}, & \text{when } x \neq -3 \\ \alpha, & \text{when } x = -3 \end{cases}$
i) Find the value of $\alpha$ for which g is continuous at $x = -3$ .
ii) Find all the roots of $f(x) = 0$ .