Solve your IGNOU Doubts

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

$4x + 3y + 2z + 1 = 0$ and passing through the origin.

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Question:

Given $x + y = 6$ , find the least value of x² + y²

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Question:

) For a given data, the mean and S.D. of 100 observations were obtained are 40 and 5.1 respectively. Later it was found that an observation was wrongly written as 50 instead of 40. Find the true mean and S.D.
b) If the p.d.f. of x is $f(x) = 2x^k, 0 < x < 1$ and zero elsewhere. Find k and S.D. of x.

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Question:

Differentiate $\sin x$ w.r.t. $\tan x$ .

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Question:

) Find $\lim_{x \to 2} \frac{x^2 + 6}{x - 1}$ .

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Question:

Which of the following statements are true or false? Give a short proof or a counter-example in support of your answer.
i) The mean of a binomial distribution, when $n=6$ and $P(X=4) = P(X=2)$ is $\frac{3}{2}$ .
ii) Mean deviation is minimum about median.
iii) The domain of 6x³ - 7y³ + 4xy, where it is continuous is $-\infty < x < \infty, 0 < y < \infty$ .
iv) $\int_{1}^{2} \frac{1}{x^4} dx = \frac{7}{23}$ .
v) The CDF of any distribution satisfies $F(-\infty) = 0, F(\infty) = 1$ and non-decreasing.