Solve your IGNOU Doubts

Question:

Find the following limits:
$\quad$ (i) $\lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$
$\quad$ (ii) $\lim_{x \to 0^+} (\sin x)^{\sin x}$
$\quad$ (b) Find the third Taylor polynomial of the function $f(x, y) = 1 + 5xy + 3^2y$ at (1, 2).
$\quad$ (c) 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:

swers.
(i) $\lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$
(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:

Solve the following differential equations

(i) $[D^3 - DD'^2 - D^2 + DD'] \, z = 0$ .

(ii) $[D^4 - D'^4 - 2D^2 D'^2] \, z = 0$ .

(iii) $[D^2 - 2DD' + D'^2] \, z = 12xy$ .
b) Show that the wave equation $a^2 u_{xx} = u_{tt}$ can be reduced to the form $u_{\xi\eta} = 0$ by the change of variable $\xi = x - at, \eta = x + at$ .

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Question:

Using the method of separation of variables, solve $u_{xt} = e^{-t} \cos x$ when

$$u(x, 0) = 0, \frac{\partial u}{\partial t}(0, t) = 0.$
b) Find the temperature in a bar of length $\ell$ with both ends insulated and with initial temperature in the rod being $\sin \frac{\pi x}{\ell}$ .

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Question:

Solve the following differential equations

(i) $x^2p + y^2q = (x + y) z$ .

(ii) $\sqrt{p} - \sqrt{q} + 3x = 0$ .

b) Find the equation of the integral surface of the differential equation

$$(x^2 - yz) \, p + (y^2 - zx) \, q = z^2 - xy$

which passes through the line $x = 1, y = 0$ .

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Question:

Verify that the Pfaffian differential equation

$$yz \, dx + (x^2y - zx) \, dy + (x^2z - xy) \, dz = 0$

is integrable and hence find its integral.
b) Solve the following equation by Jacobi's method

$$x^2 \frac{\partial u}{\partial x} - \left( \frac{\partial u}{\partial y} \right)^2 - a \left( \frac{\partial u}{\partial z} \right)^2 = 0.$
c) Show that $2z = (ax + y)^2 + b$ , where a, b are arbitrary constants is a complete integral of $px + qy - q^2 = 0$ .

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Question:

Solve the following DEs
(i) $\left( \frac{dy}{dx} - 1 \right)^2 \left( \frac{d^2y}{dx^2} + 1 \right)^2 y = \sin^2(x/2) + e^x + x$ .
(ii) $2x^2y \left( \frac{d^2y}{dx^2} \right) + 4y^2 = x^2 \left( \frac{dy}{dx} \right)^2 + 2xy \left( \frac{dy}{dx} \right)$ .
b) The differential equation of a damped vibrating system under the action of an external periodic force is:

$$\frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2x = a \cos pt$

Show that, if n > m₀ > 0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.

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Question:

Solve: $\frac{d^2y}{dx^2} - 2 \tan x \frac{dy}{dx} + 5y = e^x \sec x$ .
b) Find the charge on the capacitor in an RLC circuit at $t = 0.01$ sec. when $L = 0.05$ Henry, $R = 2$ ohms, $C = 0.01$ Farad. $E(t) = 0, q(0) = 5 \text{ Columbus and } i(0) = 0$ .
c) Solve: $x \frac{dy}{dx} + y \ln y = x y e^x$ .

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Question:

Find the integrating factor of the differential equation

$$(6xy - 3y^2 + 2y) \, dx + 2(x - y) \, dy = 0$

and hence solve it.

b) Solve the equation $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = x^m$ , for all positive integer values of m.
c) Solve the following IVP

$$\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = -6 \sin 2x - 18 \cos 2x$

$$y(0) = 2, \quad y'(0) = 2.$

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Question:

Solve, using the method of variation of parameters

$$\frac{d^2y}{dx^2} - y = \frac{2}{1 + e^x}.$
b) Solve the following equation by changing the independent variable

$$(1 + x^2)^2 y'' + 2x(1 + x^2) y' + 4y = 0.$

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Question:

Solve $\frac{dy}{dx} + xy = y^2 e^{x^2/2} \sin x$ .
b) Write the ordinary differential equation

$$y \, dx + (xy + x - 3y) \, dy = 0$

in the linear form, and hence find its solution.
c) Given that $y_1(x) = x^{-1}$ is one solution of the differential equation

$$2x^2y'' + 3xy' - y = 0, \quad x > 0,$

find a second linearly independent solution of the equation.

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Question:

. State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.
i) The initial value problem

$$\frac{dy}{dx} = x^2 + y^2, \quad y(0) = 0$

has a unique solution in some interval of the form -h < x < h.
ii) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the x-axis is $x^2 + 2y^2 = c^2$ .
iii) The normal form of the differential equation

$$\frac{d^2v}{dx^2} + v = -3 \sin 2x, \text{ where } v = ye^{-x^2}.$
iv) The solution of the pde $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = z^2$ is $z = -[y + f(x - y)]^{-1}$ . (Note: The image appears to have a small typo in the solution provided, it should likely be to the power of -1 based on standard solutions, however, transcribing exactly as seen: $z = -[y + f(x - y)]$ .)
v) The pde $u_{xx} + x^2 u_{xy} - \left( \frac{x^2}{2} + \frac{1}{4} \right) u_{yy} = 0$ is hyperbolic in the entire xy-plane.

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Question:

Let X be a gamma variable with parameters $\alpha$ and $\lambda$ , having $E(X) = 6$ and $Var(X) = 3$ . Find $\alpha$ and $\lambda$ . Also, find the m.g.f. of a gamma variable, and hence verify that mean of X is 6 and variance of X is 3 using m.g.f.
b) For married couples living in a certain locality, the probability that the husband will vote in a school board election is 0.21, the probability that they both will vote is 0.15. What is the probability that
i) at least one of them will vote?
ii) neither of them will vote?

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Question:

Let X₁ be an observation from an exponential distribution with the p.d.f.

$$f(x) = \frac{1}{\theta}e^{-x/\theta}; x > 0$ $
Test the null hypothesis that the mean of the distribution is $\theta = 2$ against the alternative hypothesis that is $\theta = 5$ . The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. Find the probabilities of type-I and type-II errors.
b) The mean and standard deviation of 20 items is found to be 10 and 2 respectively. At the time of checking it was found that one item having value 8 was incorrect. Calculate the mean and standard deviation if the wrong item is omitted.

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Question:

For a mesokurtic distribution with standard deviation 5, find fourth central moment m₄.
b) The probability that a card will have a flat tyre while crossing a certain bridge is 0.00005. Find the probability that, among 10,000 cars crossing the bridge,
i) exactly two cars will have a flat tyre.
ii) at most two cards will have a flat tyre.

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Question:

Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations:
i) not more than one of E₁, E₂, E₃, E₄.
ii) one and only one of E₁, E₂, E₃, E₄.
iii) E₁ and at least one of E₂, E₃, E₄.
iv) none of E₂, E₃ and E₄ using E₁.
b) Let the probability density function of r.v. X be
$$f(x) = \begin{cases} 1 + x & ; -1 < x \le 0 \\ 1 - x & ; \phantom{-}0 < x < 1 \\ \phantom{+}0 & ; \text{otherwise} \end{cases}$

and if $u = X$ and $v = X^2$ , find Cov(u,v). Also check the independence of u and v.

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Question:

The mean and standard deviation of a variable x are m and $\sigma$ respectively. Obtain the mean and standard deviation of $\frac{(ax+b)}{c}$ , where a, b and c are constants.
b) If X is a random variable such that $E(X) = 3$ and $E(X^2) = 13$ , determine a lower bound for P(-2 < X < 8)

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Question:

Consider the joint probability density function
$$f(x, y) = y^2 e^{-y(x+1)} ; x \ge 0, y \ge 0.$ $
Are both x and y regressions linear? Give reasons for your answer.

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Question:

A die is thrown 60 times with the following results:

Face of die	1	2	3	4	5	6
Frequency	8	7	12	8	14	11

Test that the die is unbiased at 5% level of significance. Given that at 5, 6 and 7 d.f. the value of $\chi^2$ are 11.070, 15.592 and 14.067 respectively.

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Question:

6 observations on $(X, Y)$ yielded the following data:

\sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,
\sum X_i^2 = 200, \sum Y_i^2 = 5642.

i) Determine the correlation coefficient between X and Y.
ii) Given X = 10, what will be the predicted value of Y?
iii) Given Y = 15, what will be the predicted value of X?

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