MAT31 Transform Calculus Previous Year Solved question paper answers

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)=x² in (0,π).

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

-π, -π<x<0

x, 0<x<π

Hence deduce that π²/8=1/1²+1/3²+1/5²+—-.

11] Obtain the half-range cosine series for the function f(x)=(x-1)2, 0<=x<=1. Hence deduce that π²/6=1/1²+1/2²+1/3²+—–.

12] Find the Fourier series of the periodic function defined by f(x) = 2x – x², 0 < x < 3.

13] Show that the half range sine series for the function f(x) = lx – x² 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:	0o	30o	60o	90o	120o	150o	180o
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),y₀=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 u₀=u₁=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)=x²  as a Fourier series of period 2  in the interval 0<=x<=2π.

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

17] Obtasin the Fourier series of f(x)=π²/12 – x²/4 valid in the interval (-π π)

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

xo	0	60o	120o	180o	240o	300o
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=ax² + 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 xlog₁₀x=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 x³ – 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+xy²=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_e2.

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^xy, 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)+y²=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 d²y/dx² = x(dy/dx)² -y² 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 x²+y²=a², 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-yz² j-y² zk over the upper half surface of x²+y²+z²=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 y² = 4x and x² = 4y using Green’s theorem in a plane.

13] Verify Stoke’s theorem for the vector F = (x²+y²)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=x² .

16] If F=2xyi + yz²j + 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 .