Solve any one question from Q.1(a,b,c) & Q.2(a,b,c)
1(a)
Show that system of Linear equations is consistent. Find solution:
x+2y+3z=6
2x+3y=11
4x+y-5z= -3.
x+2y+3z=6
2x+3y=11
4x+y-5z= -3.
4 M
1(b)
Find eign values and eign vectors of the matrix: A=[5412]
4 M
1(c)
Prove that:(coshx−sinx)n=coshnx−sinh nx.
4 M
2(a)
Are the following vectors are linearly dependent? If so find relation: X1=(3,2,7),X2=(2,4,1),X3=(1,−2,6).
4 M
2(b)
If α, β roots of equation x2−2x+4=0,/ prove that:α3+βn=2n+1cosnπ3.
4 M
2(c)
If (a+ib)p=mx+iy,/ prove that: yx=2tan−1balog(a2+b2).
4 M
Solve any one question from Q.3(a, b, c) Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)
3(a)
Test the covergence of the series i)∑xna+√nii)1√5−12√6+13√7
4 M
3(b)
Show that: log[1+e2xex]=log2+x22−x412+x645......
4 M
3(c)
Find the nth derivative of: \[y=e^{x}\cos x.\cos 2x.
4 M
Solve any one question from Q.4(a, b, c)
4(a)
i)xlim→0[π4x−π2x(eπx+1)]ii)xlim→0(2x+5x+7x3)1/x.
4 M
4(b)
Using Taylor's theorem expand x3−2x2+3x+1/ in powers of (x-1).
4 M
4(c)
If y=easin−1x,/ prove that:(1−x2)yn+2−(2n+1)x yn+1−(n2+a2)yn=0
4 M
Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)
5(a)
Find the value of n for which z=tne−r2/4t/ satisfies the partial differential equation:1r2[∂∂r(r2∂z∂r)]=∂z∂t.
6 M
5(b)
If T=sin(xyx2+y2)+√x2+y2+x2yx+y,/ find the value of x∂t∂x+y∂t∂y.
7 M
5(c)
If z = f(x, y), where x=ucosα−vsinα,y=usinα−vcosα,/ where α is constant, show that: (∂z∂x)2+(∂z∂y)2=(∂z∂u)2+(∂z∂v)2.
6 M
Solve any two question from Q.6 (a, b, c)
6(a)
If x2=a√u+b√v andy2=a√u−b√v/ when a and b are constants, prove that: (∂u∂x)y(∂x∂u)v=12(∂v∂y)x(∂y∂v)u.
6 M
6(b)
If u=tan−1(√x3+y3√x+√y),then show that:x2∂2u∂x2+2xy∂2u∂x∂y+y2∂2u∂y2=−2sin3ucosu./
7 M
6(c)
If u=x2−y2,v=2xy andz=f(u,v),then show that:x∂z∂x−y∂z∂y=2√u2+v2∂z∂u./
6 M
Solve any one question from Q.7(a, b, c) &Q.8(a, b, c)
7(a)
If u+v=x2+y2,u−v=x+2yFind ∂u∂x treating y constant./
4 M
7(b)
Examine for functional dependence:u=x−yx+z,v=x+zy+z.
4 M
7(c)
Find stationary points of : f(x,y)=x3y2(1−x−y)/ and find fmax where it exists.
5 M
8(a)
If x=v2+w2,y=w2+u2,z=u2+v2, prove that JJ'=1./
4 M
8(b)
Find the percentage error in computing the parallel resistance r of three resistances r1, r2, r3 from the formula: 1r=1r1+1r2+1r3if r1,r2,r3 are in error by 2% each
4 M
8(c)
Find the stationary points of:T(x,y,z)=8x2+4yz−16z+600/ if the condition 4x2+y2+4z2=16/ is satisfied.
5 M
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