Solve your IGNOU Doubts

Question:

) Find the values of $a \in \mathbb{R}$ for which ai is a solution of $z^4 - 2z^3 + 7z^2 - 4z + 10 = 0$ . Also find all the roots of this equation.
b) Find all the $8^{\text{th}}$ roots of 3i - 3. Also show any one of them in an Argand diagram.

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Question:

) Consider the linear system
$2x - 3y + 4z = 20\frac{2}{3}$
$x + 2y - 3z + 13.5 = 0$
$-x - 2y + 5z = \frac{113}{6}$
Give the two reasons for Cramer's Rule being applicable for solving this system. Also use the rule to solve the linear system.

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Question:

Give the following:
i) a $2 \times 4$ matrix;
ii) the transpose of the matrix in (i) above;
iii) a system of linear equations represented by $AX = B$ , where A is the matrix in (ii) above.

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Question:

In the context of your IGNOU studies, give the following:

i) an example of an implication;

ii) the converse of your statement in (i) above;

iii) the contrapositive of your statement in (i) above;

iv) a statement using $\forall$ ;

v) a statement using $\exists$ .

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Question:

Express the following situation in a Venn diagram:

In a survey of 60 women, it is found that 25 have studied upto Class 12 only, 10 have studied till Class 10 only, 26 got scholarships, 9 of those studying till Class 12 got scholarships, 8 of those studying till Class 10 got scholarships, and 11 had completed their BA degree.

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Question:

Given any two sets C and D, under what conditions on them will $C \times D$ and $D \times C$ have the same number of elements? Give reasons for your answer.

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Question:

Find $A \times B$ , and the number of elements in it, where

$A = \{3n + 2 \mid 1 \le n \le 10\} \subseteq \mathbb{Z}$ , and

$B = \{n \in \mathbb{Z} \mid 1 \le n \le 15\} \cap \{m \in \mathbb{Z} \mid 2 \mid m\}$ .

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Question:

Find $A \times B$ , and the number of elements in it, where

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Question:

If A and B are the set of even integers and set of odd integers, respectively, find $A \cup B$ and $(A \cup B)^c$ .

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Question:

Using the discriminant, give the nature of the roots of $7x^3 + x^2 - 35x = 5$ . Also solve the equation.
b) Find the cubic equation whose roots are the cubes of the roots of $x^3 + ax^2 + bx + c = 0, a, b, c \in \mathbb{R}$ .
c) Obtain the resolvent cubics, by Descartes’ method and by Ferrari’s method, of the equation $x^4 + 4x^3 + 8 = 0$ . Are the cubics the same? Further, use either method to obtain the roots of this equation.

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Question:

Show that $1 + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} > 2\sqrt{n + 1} - 2, \forall n \in \mathbb{N}$ .
b) Let $a, b > 0, a + b = 1, n > 1$ . Show that $\left(a + \frac{1}{a}\right)^n + \left(b + \frac{1}{b}\right)^n \ge \frac{5^n}{2^{n-1}}$

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Question:

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) Eliminating z from $x + 2y + 3z = 2$ , $3x + 2y + 3z = 6$ and $2x + 3y = 5$ gives $x + 2y = 2$ .
ii) The roots of $x^3 - 8x - 3 = 0$ are given by $x = \frac{8 \pm \sqrt{64 + 12}}{2}$ .
iii) $(\sqrt{2}, 1, \frac{3}{5}) \in \mathbb{Q} \times \mathbb{Z} \times \mathbb{R}$ .
iv) Given any n positive numbers in $\mathbb{R}$ , the product of their harmonic mean and their arithmetic mean is 1.
v) If A and B are two sets such that $(A \cup B)^c$ is empty, then either $A = \emptyset$ or $B = \emptyset$ .
vi) For any $x, y \in \mathbb{R}, |x - y| \geq ||x| - |y||$ .
vii) The geometrical representation of the set $\{ix \mid x \in \mathbb{R}\}$ is a point.
viii) Any finite set is a subset of $\mathbb{Z}$ .
ix) Every biquadratic equation has at least one real root.
x) The converse of the statement, ‘Every student of MTE-04 has completed FST-01’, is ‘Every student of FST-01 has completed MTE-04’.

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Question:

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Question:

) Does the function

satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer.
$\quad$ (b) Locate and classify the stationary points of the following:
$\quad$ (i) $f(x, y) = 4xy + x^4 - y^4$
$\quad$ (ii) $f(x, y) = xy + \frac{2}{x} + \frac{4}{y}, x > 0, y > 0$

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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$
(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 $\mathbb{R}^3$ .