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I B.Tech Semester Supplimentary Examinations, June 2009 MATHEMATICAL METHODS ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Telematics, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Find a real root of the equation x^4 -x-10=0 by bisection method. (b) From the following table of half yearly premium for policies at quinquennial ages estimate the premium for policies at the age of 63. [8+8] Age: x: 45 50 55 60 65 Premium: y: 114.84 96.16 83.32 74.48 68.
- (a) Fit acurve of the form y = aebx^ from the following data. x 1 2 3 4 5 6 y 1.6 4.5 13.8 40.2 125 300
(b) Evaluate
∫^1
0
e−x 2 taking h = 0.2 using
i. Simpson’s 13 rd ii. Trapenzoidal rule. [8+8]
- Find the solution of dy dx = x − y at x=.4 subject to the condition y=1, at x=0 and h=.1 using Milne’s method. Use Euler’s modified method to evaluate y(.1), y(2) and y(.3). [16]
- (a) Define the rank of the matrix and find the rank of the following matrix.
(b) Determine whether the following equations will have a non-trivial solution if so solve them. 4x + 2y + z + 3w = 0, 6x + 3y + 4z + 7w = 0, 2x + y + w = 0. [8+8]
- (a) Find the eigen values and the corresponding eigen vectors of the matrix.
(b) If λ 1 , λ 2 ,........, λn are the eigen values of A, then prove that the eigen values of (A -kI) are λ 1 − k, λ 2 − k, λ 3 − k, ..........., λn − k. [10+6]
- (a) Prove that every Hermitian matrix can be written as A+iB where A is real and Symmetric and B is real and Skew-Symmetric. (b) Reduce the quadratic form x^21 + 3x^22 + 3x^23 − 2 x 2 x 3 to a canonical form. [8+8]
- (a) Obtain a half range cosine series for f (x) =
kx, 0 ≤ x ≤ L 2 k (L − x) , L 2 ≤ x ≤ L Deduce the sum of the series 112 + 312 + 512 + 712 ........
(b) Show that Fourier transform of e
−x^2 (^2) is reciprocal [8+8]
- (a) Form the partial differential equation by eliminating the arbitrary constants from (x − a)^2 + (y − b)^2 + z^2 = r^2 (b) Solve the partial differential equation z^2 (p^2 + q^2 ) = x^2 + y^2 (c) Find the Z – transform of sinαk, k ≥ 0 [5+6+5]
- (a) Obtain the Fourier series for the function f (x) =
πx, 0 ≤ x ≤ 1 π (2 − x) , 1 ≤ x ≤ 2 (b) Find the Fourier cosine transform of 5−^2 x^ + 2e−^5 x. [10+6]
- (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 − − 4 yn+1 + 3yn = 0 given that y 0 = 2 and y 1 = 4. [5+5+6]
I B.Tech Semester Supplimentary Examinations, June 2009 MATHEMATICAL METHODS ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Telematics, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Find a real root of x^3 –2x^2 –4=0 using bisection method. (b) Find the polynomial from the following data x 0 1 2 3 4 y 5 12 37 86 165
Using Newton’s forward formula. [8+8]
- (a) Fit a straight line from the following table x 0 1 2 3 4 y 1 5 10 22 38
(b) Using trapezoidal rule,approximately calculate the value of
1 dx/
(1 + x) with i. Four intervals and ii. Eight intervals. [8+8]
- Use Runga Kutta fourth order method to evaluate y(.1) and y(.2), given that dy dx =^ x^ +^ y, y(0) = 1^ [16]
- (a) Find the rank of the matrix
A =
by reducing it to the normal form.
(b) Find whether the following equations are consistent, if so solve them.
x 1 + 2x 2 + 3x 3 = 16
x 1 + x 2 − 3 x 3 = − 9 x 1 − 2 x 2 + 2x 3 = 8. [8+8]
I B.Tech Semester Supplimentary Examinations, June 2009 MATHEMATICAL METHODS ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Computer Science & Engineering, Electronics & Instrumentation Engineering, Bio-Medical Engineering, Information Technology, Electronics & Control Engineering, Computer Science & Systems Engineering, Electronics & Telematics, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆ ⋆ ⋆ ⋆ ⋆
- (a) Prove that Newton’s method has a quadratic convergence. (b) Find f(1.3), given x = -2, -1, 0, 1, 2, 3; f(x)= 6, 5, 2, -2 ,3, 6. Using Gauss forward difference method. [8+8]
- Fit a non-linear weighted least squares parabola of the form y=a+bx+cx^2 from the following data x 0 1 2 3 4 y 1 1.8 1.3 2.5 6. w .5 1 2 2 3
[16]
- Find y(.2) and y(.4) using Taylor’s series method given that dy dx = xy^2 + 1 and y(0)=1 [16]
- (a) Determine whether the following equations will have a non-trivial solution. If so, solve them.
x + y − 2 z + 3w = 0, x − 2 y + z − w = 0
4 x + y − 5 z + 8w = 0, 5 x − 7 y + 2z − w = 0
(b) Find the value of k such that the rank of
2 k 7 3 6 10
(^) is 2 [8+8]
- Verify Cayley Hamilton theorem and hence evaluate A−^1 , if
A =
[16]
- (a) Using Lagrange’s reduction transform x^21 + 2x^22 − 7 x^23 − 4 x 1 x 2 + 8x 1 x 3 to canonical form. (b) Prove that inverse of a unitary matrix is unitary. (c) Prove that two eigen vectors of a symmetrix are orthogonal. [6+5+5]
- (a) Represent the following function by a Fourier sin series. f (t) =
t, 0 < t ≤ π 2 π 2 ,^
π 2 <^ t^ ≤^ π
(b) Using Fourier integral theorem prove that e−ax– e−bx^ = 2(b
(^2) −−a (^2) ) π
∫^ ∞
0
λ sin λx dλ (λ^2 +a^2 ) (λ^2 +b^2 ) [8+8]
- (a) Form the partial differential equation by eliminating the arbitrary constants z=f(x–y). (b) Solve the partial differential equation (y^2 + z^2 − −x^2 )p − 2 xyq + 2xz = 0. (c) Solve the difference equation using z-transforms un+2 − 5 un+1 +6un = 4n^ given that u 0 = 0 u^1 = 1. [5+5+6]
