Solve your IGNOU Doubts

Question:

Which of the following statements are True or False? Give reasons for your answer.
i) If $y=ax-b,$ then the correlation coefficient between x and y does not exist and if it exists it is equal to zero.
ii) For a normal distribution mean, median and standard deviation are all equal.
iii) If $x\geq y$ then $y-x$ assume only non-positive values and hence $E(x)\leq E(y).$
iv) If two unbiased dice are rolled, then the probability of their same score being 6 is $\frac{1}{6}.$

v) If the random variable x follows a normal distribution with known mean µ and unknown variance $\sigma ^2$ then $\frac{x-\mu }{\sigma }$ is a statistic but ) (x −µ is not.

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Question:

b) Estimate the eigenvalues of the matrix

$\begin{bmatrix} 1 &-2 &3 \\6 &-13 &18 \\4 &-10 &14 \end{bmatrix}$

using the Gershgorin bounds. Draw a rough sketch of the region where the eigenvalues lie.

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Question:

a) The equation $x^2+ax+b=0$ has two real roots p and q such that $|p|<|q|.$ If we use the fixed point iteration $x_{k+1}=\frac{-b}{x_k+a},$ to find a root then to which root does it converge?

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Question:

b) Find by Newton’s method the roots of the following equations correct to three places of decimals

i) $xlog_10x=4.772393$ near $x=6$

ii) $f(x)=x-2sin \,x=2$

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Question:

a) Using $x_0=-2$ as an initial approximation find an approximation to one of the zeros of $p(x)=2x^4-3x^2+3x-4$

by using Birge-Vieta method. Perform two iterations.

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Question:

Verify the second mean value theorem for the function $f(x)=x$ and $g(x)=cos\,x$ in the interval $\left [ 0,\frac{\pi}{2} \right ].$

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Question:

Show that the function $f:R\rightarrow R$ defined by $f(x)=2x+7$ has an inverse by applying the inverse function theorem. Find its inverse also.

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Question:

c) Solve $x^3-9x+1=0$ for the root lying between 2 and 4 by the method of false position. Perform two iterations

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Question:

b) Find the number of positive and negative roots of the polynomial

$p(x)=x^3-3x^3+4x-5.$ Find $p(2)$ and $p'(2)$ using synthetic division method.

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Question:

Apply Bonnet Mean Value Theorem for integrals to show that $\left | \int_{7}^{10}\frac{sin\,x}{x}dx \right |\leq \frac{2}{7}$

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Question:

a) Find the largest real root $a$ of $f(x)=x^6-x-1=0$ lying between 1 and 2. Perform three iterations by

i) bisection method

ii) secant method $(x_0=2,x_1=1).$

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Question:

Use the Fundamental Theorem of Integral Calculus to evaluate the integral $\int_{0}^1{}\left ( 2x\,sin\frac{1}{x}-cos\frac{1}{x} \right )dx.$

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Question:

Using Weiestrass M-test, show that the following series converges uniformly. $\sum_{n=1}^{\infty }n^3\,x^n,x\in \left [ -\frac{1}{3},\frac{1}{3} \right ].$