Solve your IGNOU Doubts

Question:

b) Find the term independent of a in the binomial expansion of $\left ( 3a-\frac{4}{a^6} \right )^7$ $\left ( 3a-\frac{4}{a^6} \right )^7.$

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Question:

a) Find asymptotes of the graph of the function $y=\frac{x^2-8}{x-1}.$

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Question:

Eliminating z from $x+2y+3z=2,3x+2y+3z=6$ and $2x+3y=5$ gives $x+2y=2.$

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Question:

c) Compute the correlation co-efficient between X and Y for the following data:

x	9	7	6	1	3	9	4
y	1	3	5	6	9	6	4

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Question:

b) A bag contains 3 white balls and 2 red balls. Another bag contains 5 white and 3 red balls. A bag is chosen at random and a ball is drawn from it. Find the probability that it is white.

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Question:

a) Define $f:R:f(x)=\frac{x^2}{1+x^2}.$ Is f injective, surjective, monotone?

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Question:

State whether the following statements are true or false giving reasons in support of your answer.

i) A binomial distribution has mean 3 and variance 4.

ii) The function $f(x)=x^3$ has no maxima or minima.

iii) The line of regression of x on y is the same as the line of regression of y on x .

iv) The plane $x+2_y-z=5$ is parallel to the line $\frac{x}{1}=\frac{y-5}{-1}=\frac{z+1}{-1}.$

v) A continuous random variable can have probability density function

$f(x)=\left\{\begin{matrix} 4x^2, &0<x\leq 1 \\ 0, & otherwise \end{matrix}\right.$

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Question:

Evaluate $\int e^{x}\,.\frac{x^{2}-x+1}{(1+x^{2})^{3/2}}\,dx$

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Question:

Find the points of inflexion of the curve $y=\frac{a^{2}x}{x^{2}+a^{2}}.$ Also, show that they lie on a straight line.

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Question:

Find the slope of the normal to the curve $y=x^{3}\,at\,\left ( \frac{1}{2},\frac{1}{8} \right ).$

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Question:

Evaluate $\lim_{x\rightarrow 0}(1+x)^{1/x}$

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Question:

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

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Question:

If the first three non-zero terms of Maclaurin’s series for sin x are used to approximate sinπ ,2/ show that the error is less than 1/50.

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Question:

Find the perimeter of the cardioid $r=a(1-cos\theta ).$

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Question:

Find $\frac{\mathrm{d} y}{\mathrm{d} x},if\,y=x^{sin\,x}+(sin\,x)^{x}$

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Question:

Using Trapezoidal rule, calculate $\int_{0}^{1}\frac{dx}{1+x^{2}}$ by dividing the interval [0,1] equal subintervals. Hence evaluate π.

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Question:

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) The function f : R → R defined by f(x) = cos x is 1-1.
ii) The operation ∗ defined by x ∗ y = log(xy) is a binary operation on S, where S is the
set {x ∈ R|x > 0}.
iii) The set $\left \{ \left ( x_1.x_2,....,x_n \right )|x_1,x_2,...,x_n \right \}\epsilon R,x_1=2x\}$ is a subspace of $R^{n}.$

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iv) There is no 7×5 matrix of rank 6.
v) If $V{}'$ and V 0 are vector spaces and T : V → V 0 is a linear transformation, then
whenever u1,,...,uk are linearly independent, Tu1, Tu2, ..., Tuk are also linearly
independent.
vi) If V is a vector space and T : V → V is a linear operator with det(T) = 0, then T is
not diagonalisable.
vii) The degree of the minimal polynomial of a 3×3 matrix is at most 2.
viii) For any 2×2 matrix $A, Adj(A^t)=(Adj(A))^t.$
.
ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.
x) There is no co-ordinate transformation that transforms the quadratic form $x^2+y^2+z^2$ to the quadratic form xz+yz.