Solve your IGNOU Doubts

Question:

Which of the following statements are True or False? Give short proof or counter example in your answer.
i) If the correlation coefficient between X and Y is -0.8, then the correlation coefficient between 2X - 1 and -3Y - 1 is -0.48.
ii) If X and Y are independent binomial variates with parameters (n₁, p₁) and (n₂, p₂) respectively, then X + Y has binomial distribution with parameters (n₁ + n₂, p₁ + p₂).
iii) The function defined as
$$f(x) = \begin{cases} |x| ; & -1 < x < 1 \\ 0 ; & \text{otherwise} \end{cases}$
is a probability density function.
iv) For a normal distribution with mean $\mu$ and variance $\sigma^2$ , the hypotheses
$H_1 : \mu = \mu_0, \sigma^2 = 1$ and
$H_2 : \mu = \mu_0, \sigma^2 \ge 1$ are simple hypotheses.
v) In a problem of testing of a simple hypothesis against a simple alternative, if the probability of type-I error is known to be 0.06, then the power of the test will be 0.94.

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Question:

Evaluate $\displaystyle \iiint_S z^2 \, dx \, dy \, dz$ , where S is the solid region between the spheres $\rho = 1$ and
$\quad$ $\rho = 2$ , by using spherical coordinates.

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Question:

Check if the following integrals are independent of path and evaluate those which are independent.

$$\text{i) } \int_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$

$$\text{ii) } \int_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$

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Question:

Write $\displaystyle \int_0^1 \int_0^{\sqrt{1 - x^2}} (\sqrt{1 - y^2}) \, dy \, dx$ as an integral over a region D. Sketch the region D and show that it is of both types 1 and 2. Reverse the order of integration and evaluate it.

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Question:

Using polar coordinates, show that $\displaystyle \lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ . Also, find the two repeated limits.

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Question:

Suppose S and C are subsets of $\mathbb{R}^3$ . S is the unit open sphere with centre at the origin and C is the open cube $= \{P(x, y, z) \mid -1 < x < 1, -1 < y < 1, -1 < z < 1\}$ .

Which of the following is true. Justify your answer.
(i) $S \subset C$
(ii) $C \subset S$
(c) Identify the level curves of the following functions:
(i) $\sqrt{x^2 + y^2}$
(ii) $\sqrt{4 - x^2 - y^2}$

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Question:

Find the values of a and b, if

$$\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$

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Question:

Use double integration of find the volume of the ellipsoid

$$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1.$

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Question:

Evaluate $\displaystyle \int_C (2x^2 + 3y^2) dx$ , where C is the curve given by

$$x(t) = at^2, y(t) = 2at, 0 \leq t \leq 1.$

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Question:

Find the domain and range of the function f, defined by $\displaystyle f(x, y) = \frac{2xy}{x^2 + y^2}$ . Also find two level curves of this function. Give a rough sketch of them.

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Question:

Check the continuity and differentiability of the function at (0,0) where

$$f(x, y) = \begin{cases} \frac{2x^3 y}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

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Question:

Find the extreme values of the function

$f(x, y) = x^2 + y$ on the surface $x^2 + 2y^2 = 1$ .

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Question:

tate Green’s theorem, and apply it to evaluate

$$\int_C (3x^2 - 4y) \, dx - (2x + y^3) \, dy,$

Where C is the ellipse $4x^2 + 9y^2 = 36$ .

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Question:

Find the mass of the solid bounded by $z = 1$ and $z = x^2 + y^2$ , the density function being $\delta(x, y, z) = |x|$ .

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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) $\displaystyle \lim_{x \to 0^+} (\sin x)^{\sin x}$