Solve your IGNOU Doubts

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$
(b) 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:
$\quad$ (i) $\sqrt{x^2 + y^2}$
$\quad$ (ii) $\sqrt{4 - x^2 - y^2}$
$\quad$ (iii) x - y
$\quad$ (iv) y / x

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Question:

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

$x(t) = at^2, y(t) = 2at, 0 \le t \le 1$ .
(b) 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:

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}$

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

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Question:

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of $\mathbb{R}^2$ . Verify this theorem for the functions f and g, defined by

$$f(x, y) = \frac{y - x}{y + x}, \quad g(x, y) = \frac{x}{y}.$
(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that $g(1) = 2$ and $F(g(y), y) = 0$ in a neighbourhood of (2, 1), where

$$F(x, y) = x^5 + y^5 - 16xy^3 - 1 = 0$

defines the function F. Also find g'(y).
(c) Check the local invertibility of the function f defined by $f(x, y) = (x^2 - y^2, 2xy)$ at (1, -1). Find a domain for the function f in which f is invertible.

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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 $x = 4$ .
(b) Find the mass of the solid bounded by $z = 1$ and $z = x^2 + y^2$ , the density function being $\delta(x, y, z) = |x|$ .
5. (a) State Green's theorem, and apply it to evaluate

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

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

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).

(b) 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)$
(c) 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:

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.