Set No. Code No:RR/NR 210101 II B.Tech. I Semester Regular Examination November, 2003
1
MATHEMATICS - II (Common to all Branches)
2.a)
or ld
1.a) b)
Max. Marks:80 Answer any Five questions All question carry equal marks --Define inverse of a matrix. Prove that inverse of a matrix is unique. Find the inverse of the matrix 1 3 1 3 3 1 2 4 4
.in
Time: 3 hours
b)
Find the characteristic roots of the matrix 6 -2 2 -2 3 -1 2 -1 3 For the matrix, 1 2 3 A 0 3 2 0 0 2
c)
find the Eigen values of 3 A + 5 A - 6A + 2I If 1, 2,........, n are the eigen values of A, find the eigen values of the matrix (A-I)2
Define the following with an example i) Hermitian matrix ii) Skew-Hermitain matrix iii) Unitary matrix iv) Orthogonal matrix. Show that the eigen value of an unitary matrix is of unit modulus.
nt
3.a)
2
uW
3
b)
Define a periodic function. Find the Fourier expansion for the function f x x x 2 ,1 x 1
Aj
4.a) b) 5.a)
b)
Prove that the function f x x, 0 x π can be expanded in a series of sines as sin x sin 2 x sin 3x x 2 ....... 2 3 1 Find the half range Fourier sine series of f x x, 0 x 2 (Contd…2)
Code No:RR/NR210101
:: 2 ::
Set No.1
Form the partial differential equations by eliminating the arbitrary constants from a) (x-a)2+ (y-b)2 + z2 = r2 and b) solve x(y-z)p +y(z-x)q = z( x-y).
7.a) b)
y3 zx + x2 zy = 0 uxt = e–t cos x with u( x,0) = 0 and u (0,t) = 0 Solve by separation of variables.
8.
Find the Fourier transform of 1 – x2 if |x| < 1 f(x) = 0 if |x| > 1
or ld
x cos x – sin x
Hence evaluate ∫ ---------------------0 x3
cos x/2 dx
nt
uW
-*-*-*-
Aj
.in
6.
Set No. Code No.RR/NR 210101
2
II B.Tech. I Semester Regular Examination November, 2003 MATHEMATICS - II (Common to all Branches)
Max. Marks:80 Answer any Five questions All question carry equal marks ---
Find the inverse of the matrix, where 3 3 4 A = 2 3 4 0 1 1 Verify that A3 = A-1.
2.a) b)
-1
If A and B are n rowed square matrices and if A is invertible show that A B and -1 BA have same eigen values. Show that eigen values of a triangular matrix are just the diagonal elements of the matrix. Define : i) Spectral Matrix ii) Quadratic Form iii) Canonical form. Reduce the quadratic form 3x²+ 5y²+3z²-2yz+2zx –2xy to the canonical form. Specify the matrix of transformation
nt
3.a)
uW
b)
1 2 2 1 If A = 2 1 2 3 2 2 1 Prove that A-1 = A1.
or ld
1.a)
.in
Time: 3 hours
b)
Write the Dirichlet’s conditions for the existence of Fourier series of a function f(x) in the interval α, α 2π
Aj
4.a)
b)
Prove that x 2
π2 n Cos nx 4 1 3 n2 n 1
(Contd…2)
Code No:RR/NR210101 5.a) b)
:: 2 ::
Set No.2
Find a Fourier sine series for f x ax b, in 0 x 1 Find the half range sine series for f x x π - x , in 0 x π . Deduce that 1 1 1 1 π3 ........ 32 13 33 5 3 7 3
b)
Form the partial differential equations by eliminating the arbitrary function from xyz = f( x2 + y2 + z2) Solve the partial differential equations z2 (p2 + q2) + x2 + y2 Solve by the method of separation of variables a) ux = 2 ut + u where u(x,0) = 6 e-3x b)4ux + uy = 3u given u = 3 e-y –e-5y when x=0.
8.
Find the Fourier transform of 1 for |x| < a f(x) = 0 for |x| > a > 0 and hence evaluate
∫
or ld
7.
sin x ----------------- dx x
uW
0 sin as . cos xs and ∫ -------------------- ds - s
nt
-*-*-*-
Aj
.in
6.a)
Set No. Code No.RR/NR 210101 II B.Tech. I Semester Regular Examination November, 2003
3
MATHEMATICS - II (Common to all Branches) Max. Marks:80 Answer any Five questions All question carry equal marks ---
b)
If
Prove that AA-1 = I Find the Inverse of the matrix. a ib c id if A = c id a ib a2 + b2 + c2 + d2 = 1
b) 3.a)
If is an eigen value of a non singular matrix A, show that i) |A| is an eigen value of the matrix adj A. ii) I is an eigen value of A-1 If is an eigen value of an orthogonal matrix show that l/ is also an eigen value.
uW
2.a)
or ld
1.a)
1 2 3 A = 3 2 1 4 2 1
.in
Time: 3 hours
Write the matrix of the quadratic form x1² + 2x2² -7x3²-4x1x2+8x1x3+5x2x3
Write down the quadratic form corresponding to the matrix, 1 2 5 A 2 0 3 5 3 4
Aj
nt
b)
c)
Reduce the quadratic form 3x1²+3x2²+3x3²+2x1 x2+2x1x3-2x2x3 into sum of squares by an orthogonal transformation and give the matrix of transformation. (Contd…2)
Code No:RR/NR210101
Given that f x x x 2 for π x π find the Fourier expansion of f x . Deduce that
5.a)
b)
Set No.3
π2 1 1 1 1 2 2 2 ........ 6 2 3 4
Find the half range cosine series f x x 2 - x , in 0 x 2 and hence find the 1 1 1 1 sum of series 2 2 2 2 ........ 1 2 3 4 Find the half range cosine series for the function f x x 2 , in 0 x π and hence 1 1 1 1 find the sum of the series 2 2 2 2 ........ 1 2 3 4
.in
4.
:: 2 ::
Form the partial differential equations i) (x-a)2 + (y-b)2 +z2 =c2 ii) z = ax+ by +a2+b2 iii) z = f(x - it) + g( x - it)
7.
Find the temperature in a thin metal rod of length L, with both the ends insulated x and with initial temperature in the rod is sin . L
8.a) b)
State and prove Parseval’s identity. Using Paseval’s identity prove that
uW
or ld
6.
∫
0
sin t
---------------------t
2
dt = /2
Aj
nt
-*-*-*-
Set No. Code No. RR/NR210101 II B.Tech. I Semester Regular Examination November, 2003
4
MATHEMATICS - II (Common to all Branches) Max. Marks:80 Answer any Five questions All question carry equal marks ---
2.a) b)
Show that if 1, 2 …… . n are latent roots of a matrix A, then A3 has the latent roots 13, 23, …….. n3. If is eigen value of a matrix A, show that i) K is an eigen vaule of AK ii) K is an eigen value of KA. Is the matrix
2
-3 1
4
3 1
-3
1
nt
3.a)
or ld
b)
Find the non singular matrices P and Q such that PAQ is in the normal form of the matrix and find the rank of matrix 1 2 3 2 A = 2 2 1 3 3 0 4 1 Determine the rank of the matrix. 2 1 3 4 0 3 4 1 A= 2 3 7 5 2 5 11 6
uW
1.a)
.in
Time: 3 hours
9
Aj
is orthogonal.
b)
If A and B are orthogonal matrices, prove that AB is also orthogonal.
(Contd…2)
Code No: RR/NR210101
5.a) b) c)
Find the Fourier series representing f x x, 0 x 2π . Sketch the graph of f x from – 4 to 4. Prove that the function f x x, 0 x π can be expanded in a series of sines as sin x sin 2 x sin 3 x x 2 ....... 2 3 1 2x 1 π , π x 0 Is f x even ? 1 2x , 0 x π π If, so find the Fourier series for the function 1 1 1 1 π3 Deduce that 3 3 3 3 ........ 8 1 3 5 7
.in
b)
Set No.4
or ld
4.a)
:: 2 ::
Form the partial differential equations i) Z =f (x) + ey g (x) ii) Z= y2 + 2f(1/x +log y) iii) F (xy +z2, x + y + z)=0
7.
An insulated rod of length L has its ends A and B maintained at 0°C and 100°C respectively until study state conditions prevail. If B is suddenly reduced to 0°C and maintained to 0°C. find the temperature at a distance x from A at time t.
8.a)
Find Fourier cosine transform of cos x, 0
1 Find Fourier sine transform of -----x
Aj
nt
b)
uW
6.
-*-*-*-
