# IGNOU MTE-07 Solved Assignment 2023 | MTE

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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## IGNOU MTE-07 Assignment Question Paper 2023

Course Code: MTE-07

Assignment Code: MTE-07/TMA/2023

Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers.

(a) The function $$f(x, y)=\frac{(x+2)(y-2)}{x+y}$$ is homogeneous on its domain.

(b) $$\left\{3 x+\frac{1}{2 x} \mid 0<x<1\right\}$$ is bounded above.

(c) $$\mathrm{f}: \mathbf{R}^{2} \rightarrow \mathbf{R}, \mathrm{f}(\mathrm{x}, \mathrm{y})=\frac{1+\mathrm{x}}{\mathrm{x}}, \mathrm{x} \neq 0$$ is continuous on $$[5,10] \times \mathbf{R}$$.

(d) The function $$\mathrm{f}: \mathbf{R}^{3} \rightarrow \mathbf{R}, \mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{e}^{2 \cos \pi}$$ is integrable on the region bounded by the sphere, $$x^{2}+y^{2}+z^{2}=1$$

(e) If $$\mathrm{f}: \mathbf{R}^{2} \rightarrow \mathbf{R}, \mathrm{f}(\mathrm{x}, \mathrm{y})=10$$ and $$D=[3,10] \times[-1,5]$$, then $$\iint_{\mathrm{D}} \mathrm{f} \mathrm{dx} \mathrm{dy}=420$$.

2) (a) Show that the following limits do not exist:
i) $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{x^{2}+y^{2}}$$
ii) $$\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{|y|}$$

(b) Show that $$\mathrm{f}(\mathrm{x}, \mathrm{y})=4 \mathrm{xy}-\mathrm{x}^{2}-\mathrm{y}^{4}$$ has a saddle point at $$(0,0)$$.

3) (a) Find the directional derivation of

$$f(x, y)= \begin{cases}\frac{x^{3}-y^{3}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{cases}$$

at $$(0,0)$$ in the direction $$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$.

(b) Verify the chain rule for the Jacobians for the functions,

\begin{aligned} & x=e^{3 u}, y=2 u+5 v-w, z=u+v \\ & u=p+6, v=q^{3}, w=3 r \end{aligned}

4. (a) Identify the intermediate forma and evaluate:
i) $$\quad \lim _{x \rightarrow \pi / 2} \frac{\ln \sin x}{1-\sin x}$$
ii) $$\quad \lim _{x \rightarrow \infty} 2 x(\ln (x+1)-\ln x)$$

(b) Show that

$f(x, y)= \begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ y \sin \frac{1}{y}, & y \neq 0 \\ 0, & x=0, y=0\end{cases}$

is continuous at $$(0,0)$$, but $$\mathrm{f}_{\mathrm{x}}(0,0)$$ does not exist.

5. (a) Show that the equation $$\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{y}^{2}-\mathrm{yx} \mathrm{x}^{2}-2 \mathrm{x}^{5}=0$$ defines a continuously differentiable function $$y=g(x)$$, in the neighbourhood of $$(1,-1)$$. Also find the derivative of $$g$$.

(b) Show that the following functions are functionally dependent:

\begin{aligned} & u=3 x+2 y-z \\ & v=x-2 y+z \\ & w=x(x+2 y+z) \end{aligned}

(c) Find the values of $$a$$ and $$b$$, if

$\lim _{x \rightarrow 0} \frac{x(1+a \cos x)-b \sin x}{x^{3}}=1$

6. (a) Evaluate the integral by converting to polar coordinates:

\begin{aligned} & \iint_{D}\left(4-x^{2}-y^{2}\right) d x d y \text {, where } D \text { is the region in the } 1^{\text {st }} \text { quadrant and bounded by } \\ & x^{2}+y^{2}=2 x . \end{aligned}

(b) Find the work done by a force $$\overline{\mathrm{F}}=\left(x^{2} y, x y^{2}\right)$$ in moving a particle from $$(0,0)$$ to $$(1,1)$$ along $$\mathrm{y}=\mathrm{x}^{2}$$, and then from $$(1,1)$$ to $$(2,1)$$ along $$\mathrm{y}=1$$.

7. (a) Find the surface area of the portion of the paraboloid $$z=4-x^{2}-y^{2}$$, that lies above the xy-plane.

(b) Show that the following integral is independent of path and evaluate it:

$\int_{(0, \pi)}^{(1, \pi / 2)}\left(e^{x} \cos y d x-e^{x} \sin y d y\right)$

8. (a) What are the domain and range of $$f: \mathbf{R}^{2} \rightarrow \mathbf{R}$$ defined by $$f(x, y)=\ln x+\frac{1}{y}$$. (b) Evaluate $$f_{x y}$$ at a point $$(x, y)$$ for the function $$f$$ defined by $$f(x, y)=x \tan ^{-1} y$$. Using Schwarz’s Theorem, evaluate $$f_{y x}$$ at the point $$(x, y)$$.

(c) Let $$\mathrm{f}$$ be a function defined by

$f(x, y)=\left(\frac{|x|}{1+|x|}, \frac{|y|}{1+|y|}\right)$

Check whether the composition gof exists, where $$\mathrm{g}:\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\} \rightarrow \mathbf{R}$$ is defined by $$g(x, y)=x y$$. Find gof.

9. (a) Find the volume below the plane $$\mathrm{z}=1-\mathrm{y}$$ and inside the cylinder $$\mathrm{x}^{2}+\mathrm{y}^{2}=1$$, $$0 \leq \mathrm{z} \leq 1$$.

(b) Reverse the order of integration and integrate:

$\int_{0}^{2} \int_{y / 2}(x+y)^{2} d x d y$

10. (a) Find the second Taylor polynomial for $$f(x, y)=e^{x y} \cos x$$ about $$(0, \pi / 2)$$.

(b) If $$f(x, y)=\left\{\begin{array}{l}x \sin \frac{1}{x}+y \sin \frac{1}{y}, x y \neq 0 \\ 0, x y=0\end{array}\right.$$

$$sin\:3\theta =3\:sin\:\theta -4\:sin^3\:\theta$$

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