Solve your IGNOU Doubts

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 $R^{n}.$

.
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.

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Question:

Evaluate $\int_{0}^{1}x^{2}e^{3}dx.$

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Question:

Is the function f , defined by

$f(x)=\frac{x^{2}-5x+4}{x^{2}-16},x\neq 4$

$f(4)=0,$ continuous at $x=4?$ Give reasons for your answer.

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Question:

Verify Rolle’s theorem for the function f , defined by $f(x)=x(x-2)e^{-x},$ on the interval $[0,2].$ .

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Question:

Expand $e^{2x}$ in powers of $(x-1),$ up to four terms.

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Question:

Obtain the largest possible domain, and corresponding range, of the function f , defined by $f(x)=\frac{x-2}{3-x}.$

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Question:

The curve $ay^{2}=x(x=a)^{2}\,,a>0$ has a loop between x = 0 and x = a. Find the area of this loop.

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Question:

Find the derivatives of $ln(1+x^{2})w.r.t.\,\,tan^{-1}\,x.$

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Question:

Use Simpson’s method to approximate $\int_0^{8}(x^{2}-x+3)dx$ with8 sub-intervals.

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Question:

Evaluate $\int \frac{x^{2}dx}{(x-3)(x-5)(x-7)}$

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Question:

Find the angle between the curves $y^{2}=ax\,and\,ay^{2}=x^{2}(a>0),$ at the points of intersection other than the origin.

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Question:

Find the length of the curve given by $x=t^{2}\,,y=2t^{2}\,in\,0\leq t\leq 2.$

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Question:

Give the consequences of Schottky and Frenkel defect.

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Question:

Explain close packing in two dimensions for a crystalline solid with the help of a suitable diagram.

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Question:

What are the elements of symmetry? Give the suitable diagrams for any one of them.

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Question:

Give the reasons for the striking contrast in the boiling points of ethanol (351 K) and dimethyl ether (249 K). Give suitable diagrams to illustrate this.

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Question:

For the reaction, $C1_{2}(g)+2NO(g)\rightarrow 2NOC1(g),$ the initial concentrations, $[c1_{2}]_{0}$ and $[NO]_{0}$ are given below along the corresponding with initial rates.