Solve your IGNOU Doubts

Question:

Using Lagrange’s method, solve the differential equation:

$(x^{2}-y^{2}-z^{2})p+2xq=2xz$

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Question:

Find the envelope and the characteristic curves of the family of curves:

$x^{2}(y-c)^{2}+z^{2}=c^{2}\, cos^{2}\, \alpha ,$

c and αare constants.

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Question:

Transform the given equation to Clairaut’s form and hence find its general solution:

$xy(y-px)=x-py$

Also find its singular solution, if it exists

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Question:

Find the differential equations of the space curve in which the two families of surfaces:

$u=x^{2}+y^{2}+3z=c_{1}$

and $v= zx+x^{2}+y^{2}=c_{2}$

intersect.

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Question:

The rate of change of the price of a commodity is proportional to the difference between the demand D and the supply S. If D = α − bP and , S = c sin βt where a,b,cand β are constants, determine ). P(t It is given that at $t=0,p=p_{0}$

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Question:

Solve the differential equation:

$2x\frac{dy}{dx}+y(6y^{2}-x-1)=0$

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Question:

If cos and α,cosβ cos γ are the direction cosines of a line, then show that:

$sin^{2}\alpha +sin^{2}\beta +sin^{2}\,\gamma =2$

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Question:

Find the limit of

$f(x,y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$

$as\, (x,y)\rightarrow (0,0) \, along:$

i) y = 3x

ii) y = 5x

What can you conclude about $\lim_{(x,y)\rightarrow (0,0)}f(x,y)?$ Justify your answer.

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Question:

Find all the first order partial derivatives of the following function:

$h(x,y,t)=e^{x-1}\, cos(y+t)$

What is the value of $\frac{\partial h}{\partial t}\, \, at\, \left ( 0,\frac{\pi }{2,} ,0\right )?$

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Question:

Using Charpit’s method, find the complete integral of the differential equation:

$p(1+q^{2})+(b-z)q=0,$

b being a constant

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Question:

Solve the differential equation:

$x^{2}\frac{\mathrm{d^{2}} y}{\mathrm{d} x}-2x\frac{\mathrm{d} y}{\mathrm{d} x}-4y=x^{2}+2\, ln\, x$

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Question:

Show that the differential equation:

$(y^{2}+yx)dx+(z^{2}+zx)dy+(y^{2}-xy)dz=0$

is integrable and find its integral.

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Question:

Solve the differential equation:

$\frac{\mathrm{dy} }{\mathrm{d} x}\frac{-tan\, y}{\left ( 1+x \right )}=\left ( l+x \right )e^{x}sex\, y$

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Question:

Solve the differential equation:

$\frac{\mathrm{d} y}{\mathrm{d} x}\frac{-tany}{(1+x)} = (1+x)e^{x}secy$

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Question:

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 $f\left ( x \right )=ln\left ( \frac{x+y}{x} \right )$ is not homogeneous function.

c) The cylindrical coordinates of the point whose spherical coordinates is $\left ( 8,\frac{\pi }{6},\frac{\pi }{2} \right )\: is\: \! \! \left ( 8,\frac{\pi }{6},0 \right ).$

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

$\frac{dy}{dx}=$ $\frac{dy}{dx}= \left\{\begin{matrix} 0, for \,\, \, x< 0 & \\ 1, for\, \, x\geq 0 & \end{matrix}\right.$

exists $\forall$ $x\in \mathbb{R}.$

e) The differential equation $\left [ 1+\left ( y \right )^{2} \right ]^{\frac{5}{3}}=y$ is a second order differential equation of degree 3.

f) Differential equation: $cos\: x\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+xy^{2}=0$

in $]$ $0,\pi$ $[$ is a linear, homogeneous equation.

g) The total differential equation corresponding to the family of surfaces , $x^{3}z+x^{2}y=c,$ where c is a parameter is . $3x^{2}dz+x\left ( ydx+xdy \right )=0.$

h) Differential equation:

$5x^{2}y^{2}z^{2}=2px^{2}y^{2}+5qx^{2}y^{3}+2pz^{2}+9x^{2}y^{2}$

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

i) The differential equation:

$v\frac{du}{dy}=e^{2v}+uv-u$

has an integrating factor v exp(−v) .

j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is: $\frac{dx}{y}=\frac{dy}{-w^{2}x}=y_{0}$

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Question:

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 $f\left ( x \right )=ln\left ( \frac{x+y}{x} \right )$ is not homogeneous function.

c) The cylindrical coordinates of the point whose spherical coordinates is $\left ( 8,\frac{\pi }{6},\frac{\pi }{2} \right )\: is\: \! \! \left ( 8,\frac{\pi }{6},0 \right ).$

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

$\frac{dy}{dx}=$ $\frac{dy}{dx}= \left\{\begin{matrix} 0, for \,\, \, x< 0 & \\ 1, for\, \, x\geq 0 & \end{matrix}\right.$

exists $\forall$ $x\in \mathbb{R}.$

e) The differential equation $\left [ 1+\left ( y \right )^{2} \right ]^{\frac{5}{3}}=y$ is a second order differential equation of degree 3.

f) Differential equation: $cos\: x\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+xy^{2}=0$

in $]$ $0,\pi$ $[$ is a linear, homogeneous equation.

g) The total differential equation corresponding to the family of surfaces , $x^{3}z+x^{2}y=c,$ where c is a parameter is . $3x^{2}dz+x\left ( ydx+xdy \right )=0.$

h) Differential equation:

$5x^{2}y^{2}z^{2}=2px^{2}y^{2}+5qx^{2}y^{3}+2pz^{2}+9x^{2}y^{2}$

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

i) The differential equation:

$v\frac{du}{dy}=e^{2v}+uv-u$

has an integrating factor v exp(−v) .

j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is: $\frac{dx}{y}=\frac{dy}{-w^{2}x}=y_{0}$

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Question:

1. State whether the following statements are True or False? Justify your answers with the 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" />                                                 (10) 
 
 a) The function f, defined by <img alt="f(x) = cosx + sinx" src="https://latex.codecogs.com/gif.latex?f%28x%29%20%3D%20cosx%20&plus;%20sinx" /> is an odd function. 
 b)  <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" /> 
c) The function f, defined by  <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" />  is differentiable in [0,1].

d) y = <img alt="x^{2}" src="https://latex.codecogs.com/gif.latex?x%5E%7B2%7D" /> − 3<img alt="x^{3}" src="https://latex.codecogs.com/gif.latex?x%5E%7B3%7D" /> has no points of inflection. 
e) y = −<img alt="x^{2}" src="https://latex.codecogs.com/gif.latex?x%5E%7B2%7D" /> is increasing in <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" />

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&plus;%5Csqrt%7B1-x%5E%7B2%20%7D%20%2C" />                                    (3)

b) Evaluate <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&plus;cos%5E%7B2%5C%2C%20%7Dx%7Ddx." />                                                                                    (5)

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&plus;2%7D%7Bx%5E%7B2%7D-5x&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&plus;2%7D%7Bx%5E%7B2%7D-5x&plus;4%7D." />                                                 (2)

3. Trace the curve <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&plus;1%20%5Cright%20%29" /> by the stating all the properties used to trace it.                                                                                                                                (10)

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." />     (4)

b) Find the angle between the curves  <img alt="y^{2}=ax\: and\: ay^{2}=x^{3}\left ( a> 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.                                                                 (6)

5. a) Evaluate    <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" />                                                    (4)

b) Use Simpson’s method to approximate  <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&plus;3%20%5Cright%20%29dx" />  with 8 sub-intervals.                                                                                                                      (3)

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&plus;x%5E%7B2%7D%20%5Cright%20%29%20w.r.t.%5C%3A%20tan%5E%7B-1%7D%20%5C%3A%20x." />                                         (3)

6. a) The curve  <img alt="ay^{2}" src="https://latex.codecogs.com/gif.latex?ay%5E%7B2%7D" /> <img alt="ay^{2}=x\left ( x-a \right )^{2},a> 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" />  has a loop between x = 0 and x = a. Find the area of this loop.                                                                                     (4)

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." />                                                                                   (2)

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

7. a) Verify Rolle’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" /> on the interval ].2,0[                                                                                                    (5)

b) Is the function <img alt="f" src="https://latex.codecogs.com/gif.latex?f" /> , defined by <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&plus;4%7D%7Bx%5E%7B2%7D-16%7D%2Cx%5Cneq%204" />

<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.         (2)   .

c) Evaluate  <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." />                                                                                      (3)

8. a) Using Trapezoidal rule, calculate <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&plus;x%5E%7B2%7D%7D" />  by dividing the interval ]1,0[ in 5 equal subintervals. Hence evaluate <img alt="\pi" src="https://latex.codecogs.com/gif.latex?%5Cpi" />.                                                                       (4)

b) Find , <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&plus;%5Cleft%20%28%20sin%5C%2C%20x%20%5Cright%20%29%5E%7Bx%5C%3A%20%2C%7D" />                                                           (3)

c) Find the perimeter of the cardioid r = a (<img alt="1" src="https://latex.codecogs.com/gif.latex?1" /> − cosθ).                                                (3)

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

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

c) Evaluate <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&plus;x%20%5Cright%20%29%5E%7B1/x%7D" />                                                                                   (2)

10. a) Find the slope of the normal to the curve   <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." />               (2)

b) Find the points of inflexion of the curve   <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&plus;a%5E%7B2%7D%7D." />   Also, show that they lie on a straight line.                                                                                                         (5)

c) Evaluate <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&plus;1%7D%7B%5Cleft%20%28%201&plus;x%5E%7B2%7D%20%5Cright%20%29%5E%7B3/2%7D%7Ddx" />                                                                          (3)

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Question:

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

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Question:

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

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Question:

Generation of error matrix

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