21MAT31, 15MAT31, 18MAT31 Transform Calculus, Fourier Series and numerical Technique Previous Year Solved question paper with answer.

**Module 1**

1] Find the Laplace transform of:

i)\left( \frac{4t+5}{e^{2t}} \right)^{2}

ii)\left( \frac{\sin2t}{\sqrt{t}} \right)^{2}

iii)tcosat

2] The square wave of the function f(t) with period 2a defined by f(t) =

i) 1 0<=t<a

ii) -1 a<=t<2a

show that \left( \frac{1}{s} \right)\tanh\left( \frac{as}{2} \right)

3] Employ Laplace transform to solve \frac{d^{2}y }{dt^{2}}-\frac{dy }{dt}=0, y(0)= yt(0)= 3.

4] Find

i) L^{-1}\left{ \frac{s^{2}-3s+4}{s^{3}} \right}\right}

ii) \cos^{-1}\left( \frac{s}{2} \right)

iii) L^{-1}\left{ \frac{s}{(s+2)(s+3)} \right}

5]Find the inverse Laplace transform of , \frac{1}{s(s^{2}+1)}

6] Express f(t)=

i) 2 if 0<t<1

ii) t2/2 if 1<t<π/2

iii) cost if t>π/2

in terms of units step function and hence find its Laplace transformation.

7] Express f(x)=(π-x)2 as a Fourier series of period 2π in the interval 0<x<2π. Hence deduce the sum of the series 1+1/22+1/32+……..

8] Find the Fourier Series expansion of the periodic function, f(x)=

i) l+x, -l<=X<=0

ii) l-x, 0<=X<=1

9] Obtain a half-range cosine series for f(x)=x^{2} in (0,π).

10] Obtain the Fourier series for the function: f(x)=

-π, -π<x<0

x, 0<x<π

Hence deduce that π^{2}/8=1/1^{2}+1/3^{2}+1/5^{2}+—-.

11] Obtain the half-range cosine series for the function f(x)=(x-1)2, 0<=x<=1. Hence deduce that π^{2}/6=1/1^{2}+1/2^{2}+1/3^{2}+—–.

12] Find the Fourier series of the periodic function defined by f(x) = 2x – x^{2}, 0 < x < 3.

13] Show that the half range sine series for the function f(x) = lx – x^{2} in 0 < x < l is \frac{8l^2}{π^3}\sum_{0}^{\infty }\frac{1}{(2n+1)^{3}}\sin\left( \frac{2n+1}{l} \right)πx .

14] The turning moment T units of the Crank shaft of a steam engine is a series of values of the crank angle θ in degrees. Find the first four terms in a series of sines to represent T. Also calculate T when θ = 75°.

θ: | 0^{o} | 30^{o} | 60^{o} | 90^{o} | 120^{o} | 150^{o} | 180^{o} |

T: | 0 | 5224 | 8097 | 7850 | 5499 | 2626 | 0 |

15] The following table gives the variations of periodic current over a period:

t sec: | 0 | T/6 | T/3 | T/2 | 2T/3 | 5T/6 |

A amp: | 1.98 | 1.30 | 1.05 | 1.30 | -0.88 | -0.25 |

Show that there is a direct current part 0.75 amp in the variable current and obtain the amplitude of the first harmonic.

16] Express y as a Fourier series up to 1st harmonic given :

x | 0 | 1 | 2 | 3 | 4 | 5 |

y | 4 | 8 | 15 | 7 | 6 | 2 |

**Module 2**

1] Obtain the Fourier series of f(x)=

i) 2 -2<x<0

ii) X 0<x<2

2] Find the Fourier transform of f(x)=

i) 1 for |x|<1

ii) 0 for |x|>1

and evaluate \int_{0}^{\infty }(\frac{sin x}{x})dx

3] Find the Fourier cosine transform of, f(x)=

i) x for 0<x<1

ii) 2-x for 1<x<2

iii) 0 for x>2

4] Find the Fourier transform of f(x)=

i) 1-|x|, |x|<=1

ii) 0, |x|>1

and hence deduce that \int_{0}^{\infty }\frac{\sin^{2}t}{t^{2}}dt=\frac{\Pi}{2}

5] Find the Fourier Sine and Cosine transforms of f(x)= e^{-\alpha\text{x}},\alpha\gt 0 .

6] Solve by using z-transforms y_{n+1} + 1/4 y_{n} = (1/4)^{n} (n>=0),y_{0}=0.

7] Find the Fourier transform of f(x)=e^{-|x|} .

8] Obtain the inverse Z-transform of the following function, \frac{z}{(z-1)(z-2)}

9] Find the Z-transform of sin(3n+5).

10] Obtain the inverse Z-transform of the following function, \frac{z}{(z-2)(z-3)}

11] Find the Z-transform of \cos\left( \frac{n\Pi}{2} +\alpha\right)

12] Solve u_{n+2}-5u_{n+1}+6u_{n} = 36 with u_{0}=u_{1}=0, using Z-transforms.

13] If Fourier sine transform of f(x) is \frac{e^{-a\alpha}}{\alpha}, \alpha\neq 0 . Find f(x) and hence obtain the inverse Fourier sine transform of \frac{1}{\alpha} .

14] Find the half-range cosine series of, f(x)=(x+1) in the interval 0<=x<=1.

15] Express f(x)=x^{2}_{ } as a Fourier series of period 2^{ }in the interval 0<=x<=2*π*.

16] Find the half-range size series of e^{x} in the interval 0<=x<=1.

17] Obtasin the Fourier series of f(x)=*π*^{2}/12 – x^{2}/4 valid in the interval (*-π* *π)*

18] Compute the first two harmonics of the f(x) given the following table:

x^{o} | 0 | 60^{o} | 120^{o} | 180^{o} | 240^{o} | 300^{o} |

y | 7.9 | 7.2 | 3.6 | 03.5 | 0.9 | 6.8 |

**Module 3**

1] Find the Infinite Fourier transform of e^{|-X|}

2] Find the Fourier cosine transform of f(x)= e^{-2x}+4e^{-3x}

3] Solve u_{n+2}-3u_{n+1}+2u_{n}=3^{n} , given u_{0}=u_{1}=0

4] Obtain the Z – transform of \cosh n\theta and \sinh n\theta .

5] Find the inverse Z – transform of \frac{4z^{2}-2z}{z^{3}-5z^{2}+8z-4}

6] Calculate the Karl Pearson’s co-efficient for the following ages of husbands and wives:

Husband’s age X : | 23 | 27 | 28 | 28 | 29 | 30 | 31 | 33 | 35 | 36 |

Wife’s age y: | 18 | 20 | 22 | 27 | 21 | 29 | 27 | 29 | 28 | 29 |

7] By the method of least square, find the parabola y=ax^{2} + bx + c that best fits the following data :

x: | 10 | 12 | 15 | 23 | 20 |

y: | 14 | 17 | 23 | 25 | 21 |

8] Using the Newton-Raphson method, find the real root that lies near x= 4.5 of the equation tanx=x correct to four decimal places.(Here x is in radians)

9] Find the correlation coefficient and the equation of the line of regression for the following values of x and y.

x | 1 | 2 | 3 | 4 | 5 |

y | 2 | 5 | 3 | 8 | 7 |

10] Find the equation of the best-fitting straight line for the data:

x | 0 | 1 | 2 | 3 | 4 | 5 |

y | 9 | 8 | 24 | 28 | 26 | 20 |

11] Use Newton – Raphson method to find a real root of the equation xlog_{10}x=1.2 (carryout 3 iterations) .

12] Obtain the lines of regression and hence find the coefficient of correlation for the data :

x | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

y | 9 | 8 | 10 | 12 | 11 | 13 | 14 |

13] Fit a second-degree parabola to the following data :

x | 1 | 2 | 3 | 4 | 5 |

y | 10 | 12 | 13 | 16 | 19 |

14] Use the Regula – Falsi method to find a real root of the equation x^{3} – 2x – 5 =0, correct to 3 decimal places.

15] In a partially destroyed laboratory record, only the lines of regression of y on x and x on y are available as 4x – 5y + 33 =0 and 20x – 9y = 107 respectively. Calculate x, y and the coefficient of correlation between x and y.

16] Find the curve of best fit of the type y = ae^{bx} to the following data by the method of least squares :

X : | 1 | 5 | 7 | 9 | 12 |

Y : | 10 | 15 | 12 | 15 | 21 |

17] Find the real root of the equation xe^{x} -3 =0 by the Regula Falsi method, correct to three decimal places.

**Module-4**

1] Solve dy/dx = ex – y, y(0)=2 using Taylor’s Series method up to 4th-degree terms and find the value of y(1.1).

2] Use Runge-Kutta method of fourth order to solve dy/dx + y=2x at x=1.1 given y(1)=3 (Take h=0.1)

3] Given dy/dx =x+siny; y(0)=1. Compute y(0.4) with h=0.2 using Euler’s modified method.

4] Apply Runge-Kutta fourth order method, to find y(0.1) with h=0.1 given dy/dx +y+xy^{2}=0; y(0)=1.

5] Using Simpsons (1/3)^{rd} rule to find \int_{0}^{0.6}e^-x^2 dx by taking seven ordinates.

6] Compute the value of \int_{0.2}^{1.4}(sinx-log_{e}x+e^x) dx using simpsons (3/8)^{th} rule.

7] Use Simpsons 1/3^{rd} rule with 7 ordinates to evaluate \int_{2}^{8}\frac{dx}{log_{10}x} .

8] Given f(40) = 184, f(50) = 204, f(60) = 226, f(70) = 250, f(80) = 276, f(90) = 304, find f(38) using Newton’s forward interpolation formula.

9] Evaluate \int_{0}^{1}\frac{x}{1+x^2}dx by weddle’s rule taking seven ordinates and hence find log_{e}2.

10] From the following table of half-yearly premiums for policies maturing at different ages, estimate the premium for policies maturing at age of 46:

Age: | 45 | 50 | 55 | 60 | 65 |

Premium (in Rupees): | 114.84 | 96.16 | 83.32 | 74.48 | 68.48 |

11] Using Newton’s divided difference interpolation, find the polynomial of the given data:

x | 3 | 7 | 9 | 10 |

f(x) | 168 | 120 | 72 | 63 |

12] Apply Milne’s predictor-corrector formula to compute y(0.4) given dy/dx=2e^{x}y, with

x | 0 | 0.1 | 0.2 | 0.3 |

y | 2.4 | 2.473 | 3.129 | 4.059 |

13] Using Adams- Bashforth method, find y(4.4) given 5x(dx/dy)+y^{2}=2 with

x | 4 | 4.1 | 4.2 | 4.3 |

y | 1 | 1.0049 | 1.0097 | 1.0143 |

14] Find the number of men getting wages below ` 35 from the following data:

Wages in Rs.: | 0-10 | 10-20 | 20-30 | 30-40 |

Frequency : | 9 | 30 | 35 | 42 |

15] Find the Polynomial f(x) by using Lagrange’s formula from the following data:

x: | 0 | 1 | 2 | 5 |

f(x): | 2 | 3 | 12 | 147 |

16] Construct the interpolation polynomial for the data given below using Newton’s divided difference formula :

x | 2 | 4 | 5 | 6 | 8 | 10 |

y | 10 | 96 | 196 | 350 | 868 | 1746 |

17] Use Lagrange’s interpolation formula to fit a polynomial for the data :

x | 0 | 1 | 3 | 4 |

y | -12 | 0 | 6 | 12 |

**Module-5**

1] Solve by Ruge Kutta method d^{2}y/dx^{2} = x(dy/dx)^{2} -y^{2} for x=0.2 correct 4 decimal places, using initial conditions y(0)=1, y'(0)=0, h=0.2.

2] Derive Euler’s equation in the standard form, \frac{\partial f}{\partial y}-\frac{d }{dx}\left[ \frac{\partial f}{\partial y'} \right]=0 .

3] Find the extramal of the functional, \int_{x1}^{x2}y^2+(y')^2+2ye^x dx .

4] Find the extramal for the functional, \int_{0}^{\frac{π}{2}}\left[ y^2-y'^2-2y sinx \right]dx; y(0)=0; y(\frac{π}{2})=1 .

5] Prove that the geodesics of a plane surface are straight lines.

6] A vector field is given by F=sinyi + x(1+cosy)j. Evaluate the line integral over a circular path given by x^{2}+y^{2}=a^{2}, z=0.

7] If C is a simple closed curve in the xy-plane not enclosing the origin. Shown that \int_{c}^{}F.dR=0, where F = \frac{yi-xj}{x^2+y^2} .

8] Derive Euler’s equation in the standard form viz., \frac{\partial f}{\partial x}-\frac{d }{dx}\left[ \frac{\partial f}{\partial y'} \right]=0 .

9] Use Stoke’s theorem to evaluate \int_{c}^{}F.dR where F=(2x-y)i-yz^{2} j-y^{2} zk over the upper half surface of x^{2}+y^{2}+z^{2}=1 , bounded by its projection on the xy-plane.

10] Show that the geodesics on a plane are straight lines.

11] Find the curves on which the functional \int_{0}^{1}\left( \left( y' \right)^2+12xy \right) dx with y(0)=0 and y(1)=1 can be extermized.

12] Find the area between the parabolas y^{2} = 4x and x^{2} = 4y using Green’s theorem in a plane.

13] Verify Stoke’s theorem for the vector F = (x^{2}+y^{2})i-2xyj taken round the rectangle bounded by x = 0, x = a, y = 0, y = b.

14] Find the extremal of the functional : \int_{x1}^{x2}\left[ y'+x^2(y')^2 \right]dx .

15] Verify Green’s theorem in a plane for \oint_{c}^{}(3x^2-8y^2)dx+(4y-6xy)dy is the boundary of the region enclosed by y= \sqrt{x} and y=x^{2} .

16] If F=2xyi + yz^{2}j + xzk and S is the rectangular parallelepiped bounded by x=0, y=0, z=0, x=1, z=3 evaluate \int_{s}^{}\int_{}^{}F.nds .

17] Find the geodesics on a surface given that the arc length on the surface is S=\int_{x1}^{x2}\sqrt{x[1+(y')^2]}dx .