R5100204-mathematical-methods

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Code No: R5100204

I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICAL METHODS (Common to Electrical & Electronics Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Find a real root of the equation x3 –x–11=0 by bisection method. (b) Construct difference table for the following data: [8+8] x 0.1 0.3 0.5 0.7 0.9 1.1 1.3 F(x) 0.003 0.067 0.148 0.248 0.370 0.518 0.697 And find F(0.6) using a cube that fits at x=0.3, 0.5, 0.7 and 0.9 using Newton’s forward formula. 2. Fit a curve of the form y = A1 eλ1 x + A2 eλ2 x for the following data x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 y 1.175 1.336 1.51 1.698 1.904 2.129 2.376 2.646 2.942 3. Find y(.1), y(.2) and y(.3) using Taylor’s series method that 4. (a) Find  1  1  2 3 (b) Find

dy dx

[16]

= l − y, y(0) = 0 [16]

the rank of the matrix by reducing it to the normal form.  2 1 2 3 2 2  4 3 4  7 5 6 whether the following set of equations are consistent if so, solve them. 3x + y + z = 8, −x + y − 2z = −5 2x + 2y + 2z = 12, −2x + 2y − 3z = −7. [8+8]

5. Verify that the sum ofeigen values is equal to the trace of A for the matrix 3 −1 1 A =  −1 5 −1  and find the corresponding eigen vectors. 1 −1 3

[16]

6. (a) Prove that the eigen vectors corresponding to two different eigen values of a symmetric matrix are orthogonal. (b) Reduce the matrix 9x2 + 2y 2 + 2z 2 + 6xy + 2yz − 2zx to a canonical form and find the rank, index and the signature. [6+10] 1 − 7. (a) Obtain the Fourier expansion of x sin x as cosine series in (0, π) and show that 1.3 1 π−2 7.9 + ........ = 4 . ∞ 2(b2 −a2 ) R λ sin λxdλ (b) Using Fourier integral show that e−ax − e−bx = , a, b > 0 π (λ2 +a2 )(λ2 +b2 ) 0

1 3.5

+

1 5.7

−

[8+8]

8. (a) Form the partial differential equation by eliminating the arbitrary function z = f1 (y + 2x) + f2 (y − 3x). (b) Solve the partial differential equation p tan x+q tan y=tanz. (c) State and prove final value theorem. [5+5+6] ?????

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Code No: R5100204

I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICAL METHODS (Common to Electrical & Electronics Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Find a real root of the equation f(x)=x+logx–2 using Newton Raphson method. (b) Find f(22) from the following data using Newton’s Backward formula x y

20 354

25 30 332 291

35 260

40 45 231 204

[8+8]

2. (a) Fit a straight line for the following data. x 6 7 7 8 8 8 9 9 10 y 5 5 4 5 4 3 4 3 3 (b) Evaluate the following integrals by Simpson’s one-third rule

R3

cos2 xdx,(n=6)

0

3. Tabulate the values of y(.1),y(.2), y(.3) ad y(4) using Taylor’s series given that y,y(0)=1.

[8+8] dy dx

= x2 − [16]

4. (a) For  what value of  K the matrix 4 4 −3 1  1 1 −1 0   k 2 2 2 has rank 3. 9 9 K 3 (b) Solve the following tridiagonal system x1 +2x2 =5,x1 +3x2 +x3 =6, x2 -2x3 +x4 =3, 3x3 -5x4 =2 by expressing the coefficient matrix as a product of a lower triangular and upper triangular matrices. [8+8] " # 8 −6 2 5. Diagonalize the matrix −6 7 −4 [16] 2 −4 3 6. (a) Prove that the product of two orthogonal matrices is orthogonal. (b) Reduce the quadratic form 8x2 +7y 2 +3z 2 –12xy–8yz+4xz to the canonical form [6+10] 7. (a) Find the half range cosine series for the function f (x) = (x − 1)2 in the interval 0 < x < 1 ∞ P 2 1 = π8 Hence show that (2n−1)2 n=1

(b) State and prove Fourier integral theorem.

[10+6]

8. (a) Form the partial differential equation by eliminating the arbitrary constants a, b from 2z = (x + a)1/2 + (y − a)1/2 + b. (b) Solve the partial differential equation.,z 4 p2 + z 4 q 2 = x2 y 2 . £ ¤ z (c) Find Z −1 z2 +11z+24 . [5+5+6] ?????

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Code No: R5100204

I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICAL METHODS (Common to Electrical & Electronics Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Find a root of ex sinx=1 using Newton Raphson’s method. (b) Use Lagrange’s formula to calculate f(3) from the following table x : 0 1 2 4 5 6 f(x) : 1 14 15 5 6 19

[8+8]

2. Fit a parabola of the form y=a+bx+cx2 x 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y 1.1 1.3 1.6 2.0 2.7 3.4 4.1

[16]

dy 3. Use Euler’s modified method to find y(1.1), y(1.2) and y(1.3) correct to three decimal places given dx = xy 1/3 , y(1) =1 [16]

4. (a) Find the rank of the matrix by reducing it to the echelon form   1 0 −5 6 2   3 −2 1  5 −2 −9 14  4 −2 −4 8 (b) For what values of η the equations x + y + z = 1; x + 2y + 4z =η; x + 4y + 10z = η 2 . are consistent and solve them completely in each case.

[8+8]

5. Verify theorem and hence evaluate A−1 , if  Cayley Hamilton  1 3 7 A= 1 2 3  1 2 1

[16]

6. (a) Define the following: i. Hermitian matrix ii. Skew-Hermitain matrix iii. Unitary matrix iv. Orthogonal matrix. (b) Show that the eigen values of an unitary matrix is of unit modulus. ½ t, 0 < t ≤ π2 7. (a) Represent the following function by a Fourier sin series. f (t) = π π 2, 2 < t ≤ π ∞ 2 2) R λ sin λx dλ (b) Using Fourier integral theorem prove that e−ax – e−bx = 2(b −−a π (λ2 +a2 ) (λ2 +b2 ) 0

8. (a) Form the partial differential equations by eliminating the arbitrary functions Z = y 2 + 2f (1/x + logy) (b) Solve the partial differential equation (x2 − y 2 − z 2 )p + 2xyq = 2xz. (c) State and Prove damping rule. ?????

[8+8]

[8+8]

[5+6+6]

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Code No: R5100204

I B.Tech (R05) Supplementary Examinations, June 2009 MATHEMATICAL METHODS (Common to Electrical & Electronics Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Find a real root of x3 –x2 –2=0 using Regula falsi method. (b) Find log 337.5 using Gauss forward formula from the table x logx 2. Evaluate

310 2.49136

R4

320 2.50515

330 2.51851

340 2.53148

350 2.54407

360 2.5563

ex dx taking h = 1 using cubic spline method.

[8+8] [16]

0

3. Use Runga kutta method to solve 10 4. (a) Find  1  1 1

dy dx

= x2 + y 2 y(0) = 1 for the interval 0 < x ≤ .4 with h=.1 [16]

the rank ofthe matrix by reducing it to the normal form 3 6 −1 4 5 1  5 4 3

(b) Solve the following method.  2 5. For the matrix A =  0 1

tridiagonal system x1 –2x2 = 3, x1 –x2 -4x3 = –6 2x2 +5x3 =8 by a simple [8+8]  1 1 1 0  find A4 using cayley Hamilton theorem [16] 1 2   −1 2 −2 6. (a) Prove that the matrix 13  −2 1 2  is orthogonal. 2 2 1 (b) Find the eigen values  and the corresponding eigenvectors of the matrix  2−i 0 i  0 1+i 0  i 0 2−i

[8+8]

7. (a) Expand ( L2 - x) in −L < x < L. (b) Expand f(x) = cos x; 0 < x < π in half range sine series. ½ x if 0 < x < π/2 (c) Find the finite Fourier cosine transform of f(x) = π − x if π/2 < x < π [5+5+6] a 8. (a) Form the partial differential equation by eliminating the arbitrary function f from xy + yz + zx = f(z / (x+y)). (b) Solve the partial differential equation (2z – y) p + (x + z) q + (2 x + y) = 0. (c) Solve the difference equation, using Z - transforms yn+2 − 4yn+1 + 3yn = 0 given that y0 = 2 and y1 = 4. [5+5+6] ?????