STUPIDSID
Solve the any two question from Q1(a) Q1(b) and Q1(c) Q1(d)
1 (a)
Prove that : [ an^{-1}(x+h)= an^{-1}x+h sin z cdot dfrac {sin z}{1}- dfrac {(hsin z)^2}{2} cdot sin 2z+ cdots cdots \ where z=cot^{-1}x. ]
7 M
1 (b)
[ If u=xphi(y/x)+ psi(y/x), then prove that: \ x^2 dfrac {partial^2 u}{partial x^2} + 2xy dfrac {partial^2 u}{partial x partial y}+ y^2 dfrac {partial^2u}{partial y^2}=0 ]
7 M
1 (c)
What error in the common logarithm of a number will be produed by an error of 1% in the number?
7 M
1 (d)
Find the maxima and minima of the following function: [ sin x + sin y + sin (x+y)in left [ 0 le x le dfrac {pi}{2}, 0 le y le dfrac {pi}{2}
ight ] ]
7 M
Solve the any two question from Q2(a) Q2(b) and Q2(c) Q2(d)
2 (a)
Find ab-initio the value of the integral: [ displaystyle int^{pi/2}_0 sin x dx ]
7 M
2 (b)
Evaluate: [ displaystyle int^{infty}_0 dfrac {x^8 (1-x^6)}{(1+x)^{24}}dx ]
7 M
2 (c)
Evaluate: [ lim_{n o infty} left {dfrac {?n}{n^n}
ight }^{1/n} ]
7 M
2 (d)
Change the order of integration: [ displaystyle int^4_0 int^{2sqrt{x}}_{x^2/4}dx dy ] Hence evaluate it.
7 M
Solve the any two question from Q3(a) Q3(b) and Q3(c) Q3(d)
3 (a)
Solve: y(xy+2x2y2) dx + x (xy-x2y2) dy=0
7 M
3 (b)
Solve: [ dfrac {d^2y}{dx^2}+ 4 y = e^x + sin 2 x ]
7 M
3 (c)
Solve: p(p y)= x (x+y) [ where p=dfrac {dy}{dx} ]
7 M
3 (d)
Solve: [ dfrac {dx}{dt}+ y = sin t \ dfrac {dy}{dt}+ x - cos t ] Give that x=2 and y=2, when t=0
7 M
Solve the any two question from Q4(a) Q4(b) and Q4(c) Q4(d)
4 (a)
Find the rank of the matrix: [A=egin{bmatrix}
1 &4 &3 &6 &1 \0
&2 &3 &1 &4 \0
&0 &1 &3 &7 \0
&0 &0 &-1 &3 \0
&0 &0 &0 &0
end{bmatrix}_{5 imes 5} ] by defining it in Echelon form.
4 M
4 (b)
Find the eigen values and eigen vectors of the matrix: [ A=egin{bmatrix}
3 &-4 &4 \1
&-2 &4 \1
&-1 &3
end{bmatrix} ]
10 M
4 (c)
Find the values of k such that the system of equations:
x+ky+3z=0
4x+3y+kz=0
2x+y+2z=0
has non-trivial solution.
x+ky+3z=0
4x+3y+kz=0
2x+y+2z=0
has non-trivial solution.
7 M
4 (d)
Verify the Cayley-Hamilton theorem for the matrix: [ A=egin{bmatrix}
2 &-1 &1 \-1
&2 &1 \1
&-1 &2
end{bmatrix} ]
7 M
Solve the any two question from Q5(a) Q5(b) and Q5(c) Q5(d)
5 (a)
Define the following terms for a graph:
(i) Subgraph
(ii) Degree of vertex
(iii) Composition and De-composition
(iv) Rooted tree
(i) Subgraph
(ii) Degree of vertex
(iii) Composition and De-composition
(iv) Rooted tree
7 M
5 (b)
Define fuzzy logic and its applications in science and engineering.
7 M
5 (c)
Prepare a truth table to get the negative of the statement ?Sita is dull and careless.?
7 M
5 (d)
Prove that :
a⋅b+ b⋅c+ c⋅a= (a+b)⋅(b+c)⋅(c+a) ∀ a,b,c, ∈ B
a⋅b+ b⋅c+ c⋅a= (a+b)⋅(b+c)⋅(c+a) ∀ a,b,c, ∈ B
7 M
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