1. State whether the following statements are true or false. Give a short proof or a counterexample in support of your answer:                                                  

a) A real-valued function of three variables which is continuous everywhere is differentiable. 

b) The function  is not homogeneous function.

c) The cylindrical coordinates of the point whose spherical coordinates is 

d) The unique solution y(x) of an ordinary differential equation:

    exists  

e) The differential equation     is a second order differential equation of degree 3.

f) Differential equation:

           in   is a linear, homogeneous equation.

g) The total differential equation corresponding to the family of surfaces ,  where c is a parameter is .

h) Differential equation:

     is a semi- linear partial differential equation of  first order.

i) The differential equation:

has an integrating factor v exp(−v) .

j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is:

1. State whether the following statements are true or false. Give a short proof or a counterexample in support of your answer:                                                  

a) A real-valued function of three variables which is continuous everywhere is differentiable. 

b) The function  is not homogeneous function.

c) The cylindrical coordinates of the point whose spherical coordinates is 

d) The unique solution y(x) of an ordinary differential equation:

    exists  

e) The differential equation     is a second order differential equation of degree 3.

f) Differential equation:

           in   is a linear, homogeneous equation.

g) The total differential equation corresponding to the family of surfaces ,  where c is a parameter is .

h) Differential equation:

     is a semi- linear partial differential equation of  first order.

i) The differential equation:

has an integrating factor v exp(−v) .

j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is:

<p>1.&nbsp;State whether the following statements are True or False? Justify your answers with the&nbsp;help of a short proof or a counter example:<img alt="\: \: \: \:" src="https://latex.codecogs.com/gif.latex?%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; (10)<br />
<br />
&nbsp;a) The function f, defined by&nbsp;<img alt="f(x) = cosx + sinx" src="https://latex.codecogs.com/gif.latex?f%28x%29%20%3D%20cosx%20&amp;plus;%20sinx" /> is an odd function.<br />
&nbsp;b)&nbsp;&nbsp;<img alt="\frac{\mathrm{d} }{\mathrm{d} x}\left [ \int_{2}^{e^{x}} ln \: t \: dt \right ] = x - ln2" src="https://latex.codecogs.com/gif.latex?%5Cfrac%7B%5Cmathrm%7Bd%7D%20%7D%7B%5Cmathrm%7Bd%7D%20x%7D%5Cleft%20%5B%20%5Cint_%7B2%7D%5E%7Be%5E%7Bx%7D%7D%20ln%20%5C%3A%20t%20%5C%3A%20dt%20%5Cright%20%5D%20%3D%20x%20-%20ln2" /><br />
c) The function f, defined by&nbsp;&nbsp;<img alt="f\left ( x \right )=\left | x-2 \right |," src="https://latex.codecogs.com/gif.latex?f%5Cleft%20%28%20x%20%5Cright%20%29%3D%5Cleft%20%7C%20x-2%20%5Cright%20%7C%2C" />&nbsp; is differentiable in [0,1].</p>

<p>d)&nbsp;y = <img alt="x^{2}" src="https://latex.codecogs.com/gif.latex?x%5E%7B2%7D" />&nbsp;&minus; 3<img alt="x^{3}" src="https://latex.codecogs.com/gif.latex?x%5E%7B3%7D" />&nbsp;has no points of inflection.<br />
e) y = &minus;<img alt="x^{2}" src="https://latex.codecogs.com/gif.latex?x%5E%7B2%7D" />&nbsp;is increasing in&nbsp;<img alt="\left [ -5,-3 \right ]," src="https://latex.codecogs.com/gif.latex?%5Cleft%20%5B%20-5%2C-3%20%5Cright%20%5D%2C" /><br />
&nbsp;</p>

<p>2. a) Find , <img alt=".\frac{\mathrm{d}y }{\mathrm{d} x}," src="https://latex.codecogs.com/gif.latex?.%5Cfrac%7B%5Cmathrm%7Bd%7Dy%20%7D%7B%5Cmathrm%7Bd%7D%20x%7D%2C" /><img alt=".\frac{\mathrm{d}y }{\mathrm{d} x},if \: y=xsin^{-1}" src="https://latex.codecogs.com/gif.latex?.%5Cfrac%7B%5Cmathrm%7Bd%7Dy%20%7D%7B%5Cmathrm%7Bd%7D%20x%7D%2Cif%20%5C%3A%20y%3Dxsin%5E%7B-1%7D" /><img alt="x+\sqrt{1-x^{2 } ," src="https://latex.codecogs.com/gif.latex?x&amp;plus;%5Csqrt%7B1-x%5E%7B2%20%7D%20%2C" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)</p>

<p>b) Evaluate&nbsp;<img alt="\frac{x sin \: x}{1+cos^{2\, }x}dx." src="https://latex.codecogs.com/gif.latex?%5Cfrac%7Bx%20sin%20%5C%3A%20x%7D%7B1&amp;plus;cos%5E%7B2%5C%2C%20%7Dx%7Ddx." />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;(5)</p>

<p>c) Find .<img alt="\lim_{x\rightarrow 1}\frac{x^{2}-3x+2}{x^{2}-5x+4}" src="https://latex.codecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%201%7D%5Cfrac%7Bx%5E%7B2%7D-3x&amp;plus;2%7D%7Bx%5E%7B2%7D-5x&amp;plus;4%7D" /><img alt="\lim_{x\rightarrow 1}\frac{x^{2}-3x+2}{x^{2}-5x+4}." src="https://latex.codecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%201%7D%5Cfrac%7Bx%5E%7B2%7D-3x&amp;plus;2%7D%7Bx%5E%7B2%7D-5x&amp;plus;4%7D." />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)&nbsp;</p>

<p>3. Trace the curve&nbsp;<img alt="y^{2}=x^{2}\left ( x+1 \right )" src="https://latex.codecogs.com/gif.latex?y%5E%7B2%7D%3Dx%5E%7B2%7D%5Cleft%20%28%20x&amp;plus;1%20%5Cright%20%29" /> by the stating all the properties used to trace it.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;(10)</p>

<p>4. a) Find the length of the curve given by <img alt="x=t^{2}\: ,y=2t^{2}in \: 0\leq t\leq 2." src="https://latex.codecogs.com/gif.latex?x%3Dt%5E%7B2%7D%5C%3A%20%2Cy%3D2t%5E%7B2%7Din%20%5C%3A%200%5Cleq%20t%5Cleq%202." />&nbsp; &nbsp; &nbsp;(4)&nbsp;</p>

<p>b) Find the angle between the curves&nbsp;&nbsp;<img alt="y^{2}=ax\: and\: ay^{2}=x^{3}\left ( a&gt; 0 \right )" src="https://latex.codecogs.com/gif.latex?y%5E%7B2%7D%3Dax%5C%3A%20and%5C%3A%20ay%5E%7B2%7D%3Dx%5E%7B3%7D%5Cleft%20%28%20a%3E%200%20%5Cright%20%29" /> , at the points of intersection other than the origin.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(6)</p>

<p>5. a) Evaluate&nbsp; &nbsp;&nbsp;<img alt="\int \frac{x^{2}dx}{\left ( x-3 \right )\left ( x-5 \right )\left ( x-7 \right )}" src="https://latex.codecogs.com/gif.latex?%5Cint%20%5Cfrac%7Bx%5E%7B2%7Ddx%7D%7B%5Cleft%20%28%20x-3%20%5Cright%20%29%5Cleft%20%28%20x-5%20%5Cright%20%29%5Cleft%20%28%20x-7%20%5Cright%20%29%7D" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;(4)</p>

<p>b) Use Simpson&rsquo;s method to approximate&nbsp;&nbsp;<img alt="\int \left ( x^{2}-x+3 \right )dx" src="https://latex.codecogs.com/gif.latex?%5Cint%20%5Cleft%20%28%20x%5E%7B2%7D-x&amp;plus;3%20%5Cright%20%29dx" /> &nbsp;with 8 sub-intervals.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;(3)</p>

<p>c) Find the derivatives of ln<img alt="\left ( 1+x^{2} \right ) w.r.t.\: tan^{-1} \: x." src="https://latex.codecogs.com/gif.latex?%5Cleft%20%28%201&amp;plus;x%5E%7B2%7D%20%5Cright%20%29%20w.r.t.%5C%3A%20tan%5E%7B-1%7D%20%5C%3A%20x." />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(3)</p>

<p>6. a) The curve&nbsp;&nbsp;<img alt="ay^{2}" src="https://latex.codecogs.com/gif.latex?ay%5E%7B2%7D" />&nbsp;<img alt="ay^{2}=x\left ( x-a \right )^{2},a&gt; 0" src="https://latex.codecogs.com/gif.latex?ay%5E%7B2%7D%3Dx%5Cleft%20%28%20x-a%20%5Cright%20%29%5E%7B2%7D%2Ca%3E%200" />&nbsp; has a loop between x = 0 and x = a. Find the area of this loop.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(4)</p>

<p>b) Obtain the largest possible domain, and corresponding range, of the function f , defined by .<img alt="f\left ( x \right )=\frac{x-2}{3-2}." src="https://latex.codecogs.com/gif.latex?f%5Cleft%20%28%20x%20%5Cright%20%29%3D%5Cfrac%7Bx-2%7D%7B3-2%7D." />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)</p>

<p>c) Expand&nbsp;<img alt="e^{2x}" src="https://latex.codecogs.com/gif.latex?e%5E%7B2x%7D" /> in powers of (x &minus; ),&nbsp;up to four terms.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(4)</p>

<p>7. a) Verify Rolle&rsquo;s theorem for the function f , defined by <img alt="f\left ( x \right )=x\left ( x-2 \right )e^{-x}," src="https://latex.codecogs.com/gif.latex?f%5Cleft%20%28%20x%20%5Cright%20%29%3Dx%5Cleft%20%28%20x-2%20%5Cright%20%29e%5E%7B-x%7D%2C" />&nbsp;on the interval ].2,0[&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (5)</p>

<p>b) Is the function <img alt="f" src="https://latex.codecogs.com/gif.latex?f" />&nbsp;, defined by&nbsp;<img alt="f\left ( x \right )=\frac{x^{2}-5x+4}{x^{2}-16},x\neq 4" src="https://latex.codecogs.com/gif.latex?f%5Cleft%20%28%20x%20%5Cright%20%29%3D%5Cfrac%7Bx%5E%7B2%7D-5x&amp;plus;4%7D%7Bx%5E%7B2%7D-16%7D%2Cx%5Cneq%204" /></p>

<p><img alt="f\left ( x=0 \right )" src="https://latex.codecogs.com/gif.latex?f%5Cleft%20%28%20x%3D0%20%5Cright%20%29" /><img alt="f\left ( x=0 \right )," src="https://latex.codecogs.com/gif.latex?f%5Cleft%20%28%20x%3D0%20%5Cright%20%29%2C" /> continuous at x = 4? Give reasons for your answer.&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;(2) &nbsp; .</p>

<p>c) Evaluate&nbsp;&nbsp;<img alt="\int_{0}^{1}x^{2}e^{3x}dx." src="https://latex.codecogs.com/gif.latex?%5Cint_%7B0%7D%5E%7B1%7Dx%5E%7B2%7De%5E%7B3x%7Ddx." />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)&nbsp;</p>

<p>8. a) Using Trapezoidal rule, calculate&nbsp;<img alt="\int_{0}^{1}\frac{dx}{1+x^{2}}" src="https://latex.codecogs.com/gif.latex?%5Cint_%7B0%7D%5E%7B1%7D%5Cfrac%7Bdx%7D%7B1&amp;plus;x%5E%7B2%7D%7D" />&nbsp; by dividing the interval ]1,0[ in 5 equal subintervals. Hence evaluate <img alt="\pi" src="https://latex.codecogs.com/gif.latex?%5Cpi" />.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(4)</p>

<p>b) Find ,&nbsp;<img alt="\frac{\mathrm{} dy}{\mathrm{d} x},if \: y=x^{sin \, x}+\left ( sin\, x \right )^{x\: ,}" src="https://latex.codecogs.com/gif.latex?%5Cfrac%7B%5Cmathrm%7B%7D%20dy%7D%7B%5Cmathrm%7Bd%7D%20x%7D%2Cif%20%5C%3A%20y%3Dx%5E%7Bsin%20%5C%2C%20x%7D&amp;plus;%5Cleft%20%28%20sin%5C%2C%20x%20%5Cright%20%29%5E%7Bx%5C%3A%20%2C%7D" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)&nbsp;</p>

<p>c) Find the perimeter of the cardioid r = a (<img alt="1" src="https://latex.codecogs.com/gif.latex?1" />&nbsp;&minus; cos&theta;).&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)</p>

<p>9. a) If the first three non-zero terms of Maclaurin&rsquo;s series for sin x are used to approximate sin <img alt="\pi /2" src="https://latex.codecogs.com/gif.latex?%5Cpi%20/2" />&nbsp;&nbsp;show that the error is less than 1/50.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (4) &nbsp; &nbsp;</p>

<p>b) Find the least value of&nbsp;<img alt="a^{2 } sec^{2} \: x+b^{2} \: cosec^{2\: }x, where\: a&gt; 0,b&gt; 0." src="https://latex.codecogs.com/gif.latex?a%5E%7B2%20%7D%20sec%5E%7B2%7D%20%5C%3A%20x&amp;plus;b%5E%7B2%7D%20%5C%3A%20cosec%5E%7B2%5C%3A%20%7Dx%2C%20where%5C%3A%20a%3E%200%2Cb%3E%200." />&nbsp; &nbsp;(4)</p>

<p>c) Evaluate&nbsp;<img alt=".\lim_{x\rightarrow 0}\left ( 1+x \right )^{1/x}" src="https://latex.codecogs.com/gif.latex?.%5Clim_%7Bx%5Crightarrow%200%7D%5Cleft%20%28%201&amp;plus;x%20%5Cright%20%29%5E%7B1/x%7D" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)</p>

<p>10. a) Find the slope of the normal to the curve&nbsp;&nbsp;&nbsp;<img alt="y=x^{3}\, at\, \left ( \frac{1}{2},\frac{1}{8} \right )." src="https://latex.codecogs.com/gif.latex?y%3Dx%5E%7B3%7D%5C%2C%20at%5C%2C%20%5Cleft%20%28%20%5Cfrac%7B1%7D%7B2%7D%2C%5Cfrac%7B1%7D%7B8%7D%20%5Cright%20%29." />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(2)</p>

<p>b) Find the points of inflexion of the curve&nbsp; &nbsp;<img alt="y=\frac{a^{2}x}{x^{2}+a^{2}}." src="https://latex.codecogs.com/gif.latex?y%3D%5Cfrac%7Ba%5E%7B2%7Dx%7D%7Bx%5E%7B2%7D&amp;plus;a%5E%7B2%7D%7D." /> &nbsp;&nbsp;Also, show that they lie on a straight line.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(5)</p>

<p>c) Evaluate&nbsp;<img alt="\int e^{x}\frac{x^{2}-x+1}{\left ( 1+x^{2} \right )^{3/2}}dx" src="https://latex.codecogs.com/gif.latex?%5Cint%20e%5E%7Bx%7D%5Cfrac%7Bx%5E%7B2%7D-x&amp;plus;1%7D%7B%5Cleft%20%28%201&amp;plus;x%5E%7B2%7D%20%5Cright%20%29%5E%7B3/2%7D%7Ddx" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (3)</p>
 

Discuss R programming and its uses for data exploration and image processing.

What is supervised classification? Describe the various steps involved in supervised classification.

Generation of error matrix

Unsupervised classification

Change detection techniques

Give an overview of various image fusion techniques.

What are radiometric errors? Give a detailed account of radiometric image corrections techniques.

Linear image enhancement techniques

UAV based remote sensing

LiDAR data products

Give an account of the active microwave remote sensing and its application potential.

Describe principles of thermal remote sensing. Also discuss its application potential. Support your answer with the help of neat well labelled diagrams, wherever necessary

Imaging spectroscopy

Photogrammetry

The following data comprising the number of customers (in hundred) and monthly sales (in thousand Rupees):

 Calculate the residuals and determine the standardised residuals for the model

Use graphical method to minimise the time added to process the following jobs on the machines shown:

 Calculate the total time elapsed to complete both the jobs

A department head has four subordinates, and four tasks to be performed. The subordinates differ in efficiency, and the tasks differ in their intrinsic difficulty. His estimate, of the time each man would take to perform each task, is given in the table below:

 How should the tasks be allocated, one to a subordinate, so as to minimise the total man hour?