Ques 1.

<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>