Solve your IGNOU Doubts

Question:

b) Use Simpson’s method to approximate $\int_{0}^{8}(x^2-x+3)$ with 8 sub-intervals.

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Question:

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

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Question:

Find the angle between the curves $y^{2}=a x\operatorname{and}\ a y^{2}=x^{3}(a>0),$ 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:

Trace the curve $y^2=x^2 (x=1)$ by the stating all the properties used to trace it.

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Question:

c) Find $\lim_{x\rightarrow1}\frac{x^3-3x+2}{x^2-5x+4}.$

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Question:

b) Evaluate $\int_{0}^\pi{}\frac{x\,sin\,x}{1+cos^2x}dx.$

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Question:

vii) Find the kernel of the linear transformation $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by $T(x,y)=(2x+3y,2x-3y).$

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Question:

find $\frac{\mathrm{d} }{\mathrm{d} x}$ if $y=xsin^{-1}x+\sqrt{1-x^2}.$

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Question:

vi) If $\left \{ v_1,v_2 \right \}$ is an ordered basis of $\mathbb{R}^2$ and $\left \{ f_1(v),f_2(v) \right \}$ is the corresponding dual basis find $f_1(2v_1+v_2)$ and $f_2(v_1-2v_2).$

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Question:

v) Check whether $T:\mathbb{R}^2\rightarrow \mathbb{R}^2,$ defined by $T(x,y)=(-y,x)$ is a linear transformation.

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Question:

vectors in $P^3,$ the vector space of polynomials of degree $\leq 3.$

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Question:

iv) Check whether the set of vector $\left \{ 1+x,x+x^2,1+x^3 \right \}$ is a linearly independent set of

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Question:

iii) Check whether $W=\left \{ \left ( x,y,z \right )\in \mathbb{R}^3|x+y-z=0 \right \}$

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Question:

ii) Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0).

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Question:

i) Find the angle between the vectors $\sqrt{2i+2j}+2k$ and i $\sqrt{2j}+\sqrt{2k}.$

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Question:

State whether the following statements are True or False? Justify your answers with the help of a short proof or a counter example: (10)
a) The function f, defined by $f(x)=cos\,x+sin\,x,$ is an odd function.
b) $\frac{\mathrm{d} }{\mathrm{d} x}\left [ \int_{2}^{e^x}1n\,t\,dt \right ]=x-1n\,2.$

c) The function f, defined by $f(x)=|x-2|,$ is differentiable in [0,1].
d) $y=x^2-3x^3$ has no points of inflection.
e) $y=-x^2$ is increasing in [− ,5 − 3].

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Question:

Give the characteristics bands observed in the IR spectra of carboxylic acids and carboxylic acid anhydrides

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Question:

i) List different factors which govern the position and intensity of bands appearing in the IR spectrum of an organic compound.

ii) The C–C stretching vibration is infrared inactive (or nearly inactive). Give reason. In which region, C–H stretching absorptions of alkanes are exhibited?

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Question:

Explain the degrees of freedom for polyatomic molecules.

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