Solve your IGNOU Doubts

Question:

If $y = e^{m \sin^{-1} x}$ , then show that $(1 - x^2)y_2 - xy_1 - m^2y = 0$ . Hence using Leibnitz's formula, find the value of (1 - x²)y_n+2 - (2n + 1)xy_n+1.
b) Find the largest subset of R on which the function $f : \mathbf{R} \to \mathbf{R}$ defined as:
$$f(x) = \begin{cases} 2x, & x > 5 \\ x + 5, & 1 \le x \le 5 \\ |x|, & x < 1 \end{cases}$
is continuous.

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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.
b) 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:

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.
b) 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.

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Question:

Let f and g be two functions defined on 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 f is continuous at $x = -3$ .

ii) Find all the roots of $f(x) = 0$ .

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Question:

For any two sets S and T, show that:

$$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Depict this situation in the Venn diagram.

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Question:

Find the least value of $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , where a > 0, b > 0.
b) Evaluate:

$$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} dx$

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Question:

If $|z - 1 + 2i| = 4$ , show that the point z + i describes a circle. Also draw this circle.
b) Express $\frac{x-1}{x^3 - x^2 - 2x}$ as a sum of partial fractions.

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Question:

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a <em> b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is () associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

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Question:

Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$ .

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Question:

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a <em> b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is () associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

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Question:

Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$

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Question:

Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.
a) The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.
b) The greatest interger function is continuous on $\mathbf{R}$ .
c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .
d) Every integrable function is monotonic.
e) $a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

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Question:

Verify Euler’s theorem for:
$f(x, y) = ax^2 + 2hxy + by^2$
b) The first and last term of a series are 4 and 76 respectively. The sum is given to be 1920. Find the number of terms in the series.

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Question:

Calculate the correlation coefficient between X and Y for the following data:

X	Y
1	9
2	8
3	10
4	12
5	11
6	13
7	14
8	16
9	15

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Question:

For $f(x) = \theta e^{-\theta x}, x \geq 0, \theta > 0$ , find the E(x) and V(x).

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Question:

Solve $(1 - \sin x \tan y)dx + (\cos x \sec^2 y)dy = 0$ .

b) Find the estimated value of $y = 70$ given the following data:
$\bar{x} = 67, \bar{y} = 65, \sigma_x = 3.5, \sigma_y = 2.5, r = 0.8$

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Question:

How many times the combination of 4 heads and 3 tails will appear when 7 coins are tossed 1000 times?
b) The position vectors of four points A, B, C and D are $2\hat{i} + 4\hat{k}$ , $5\hat{i} + 3\sqrt{3}\hat{j} + 4\hat{k}$ , $-2\sqrt{3}\hat{i} + \hat{k}$ , $2\hat{i} + \hat{k}$ respectively. Show that AB is parallel to CD and $CD = \frac{2}{3}AB$ .

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Question:

) Evaluate $\int \frac{(\sin^{-1} x)^2}{\sqrt{1 - x^2}} dx$ .
b) Find the equation of the plane through the intersection of the planes $x + 2y + 3z = 4$ and