Reg. No. 2 o B TR Asos [18BS3MA01]
THIRD SEMESTER B.Tech EXAMINATION, DECEMBER-2021
MATHEMATICS-III
Time: 3 Hours Maximum Marks: 70
Instructions:
i. Answer FIVE full Questions. Question number 1 and 2 are compulsory. Each
question carries 14 marks.
i. Missing data may be suitably assumed.
1. a. Find the coefficient of correlation and regression lines to the following data:
X 5 7 8 10 11 13 16 07
33 30 28 20 18 16 9
b. Using least-squares method, fit a curve of the formy = ae" to the data
1 2 3 4 5 6 07
7.209 5.265 3.846 2.809 2.052 1.499
2. a. Given that f(x) = x +x for - T <X <T, find the Fourier expression of f(x). 07
b. Obtain the constant term and the coefficient of the first sine and cosine terms in
the Fourier series of S(x) as given in the following table.
1 2 07
S(x) 9 18 24 28 26 20
3. a.
Find the Fourier transform of the function f(x) = { x. sa 07
0 .>a
where 'a' is a positive constant.
b.
Find the Fourier sine transform of f(x)=e and hence evaluatee
07
XSin a where 'a' is a
positive constdi
0 1+x
OR
4. a. Find the Z-Transforms cosne and hence find Z(ncosn6). 07
b. Using Z-transform, solve 4 2 + 4u +3u,n =3" with uo =0, " =1 07
5. a.
Solve subject to the conditions z(x,0) = x and
Oxoy 07
z(1,y)=cosy.
Page 1 of 2
, Solve: ( - y a ) p +(y* - zxr)q = z* - xy. 07
OR
6. a. Derive one dimensional Wave equation. 07
b. Solve by the method of separation of variables the partial differential equation
07
7. a.
Use Newton-Raphson method, find the real root of Xlog10 X 1.2 correct to
07
correct to four decimal places near to 2.
b. Use suitable interpolation to find the equation of the cubic curve which passes
through the points ( 4, -43), ( 7, 83), ( 9, 327) and ( 12, 1053).
07
OR
8.
dx
Use Weddle's rule to evaluate and hence find approximate value of
1+ x 07
log,8.
Using the Runge-Kutta method, find the solution of the problem
dx y+x 07
y(0) =1 at the point x =0.1 with h=0.1.
Page 2 of 2
THIRD SEMESTER B.Tech EXAMINATION, DECEMBER-2021
MATHEMATICS-III
Time: 3 Hours Maximum Marks: 70
Instructions:
i. Answer FIVE full Questions. Question number 1 and 2 are compulsory. Each
question carries 14 marks.
i. Missing data may be suitably assumed.
1. a. Find the coefficient of correlation and regression lines to the following data:
X 5 7 8 10 11 13 16 07
33 30 28 20 18 16 9
b. Using least-squares method, fit a curve of the formy = ae" to the data
1 2 3 4 5 6 07
7.209 5.265 3.846 2.809 2.052 1.499
2. a. Given that f(x) = x +x for - T <X <T, find the Fourier expression of f(x). 07
b. Obtain the constant term and the coefficient of the first sine and cosine terms in
the Fourier series of S(x) as given in the following table.
1 2 07
S(x) 9 18 24 28 26 20
3. a.
Find the Fourier transform of the function f(x) = { x. sa 07
0 .>a
where 'a' is a positive constant.
b.
Find the Fourier sine transform of f(x)=e and hence evaluatee
07
XSin a where 'a' is a
positive constdi
0 1+x
OR
4. a. Find the Z-Transforms cosne and hence find Z(ncosn6). 07
b. Using Z-transform, solve 4 2 + 4u +3u,n =3" with uo =0, " =1 07
5. a.
Solve subject to the conditions z(x,0) = x and
Oxoy 07
z(1,y)=cosy.
Page 1 of 2
, Solve: ( - y a ) p +(y* - zxr)q = z* - xy. 07
OR
6. a. Derive one dimensional Wave equation. 07
b. Solve by the method of separation of variables the partial differential equation
07
7. a.
Use Newton-Raphson method, find the real root of Xlog10 X 1.2 correct to
07
correct to four decimal places near to 2.
b. Use suitable interpolation to find the equation of the cubic curve which passes
through the points ( 4, -43), ( 7, 83), ( 9, 327) and ( 12, 1053).
07
OR
8.
dx
Use Weddle's rule to evaluate and hence find approximate value of
1+ x 07
log,8.
Using the Runge-Kutta method, find the solution of the problem
dx y+x 07
y(0) =1 at the point x =0.1 with h=0.1.
Page 2 of 2
