STUPIDSID
1(a)
Find the percentage error in the area of an ellipse if 1% error is made in measuring the major and minor axes.
2 M
1(b)
Discuss the maxima and minima of the function x3+y3-3axy.
2 M
1(c)
Find the radius of curvature at any point of the curve x=acost, y=bsint
3 M
Solve any one question from Q.1(d) &Q.1(e)
1(d)
Use Taylor's theorem to prove that
\[\tan^{-1}(x+h)=\tan^{-1}x+(h\sin\theta).\dfrac{\sin\theta}{1}-(h\sin\theta)^2\frac{\sin 2\theta}{2}\\ +(h\sin\theta)^3\dfrac{\sin 3\theta}{3}-\cdots+(-1)^{n-1}(h\sin\theta)^n\dfrac{\sin n\theta}{n}+\cdots\]
Where θ=cot-1x.
\[\tan^{-1}(x+h)=\tan^{-1}x+(h\sin\theta).\dfrac{\sin\theta}{1}-(h\sin\theta)^2\frac{\sin 2\theta}{2}\\ +(h\sin\theta)^3\dfrac{\sin 3\theta}{3}-\cdots+(-1)^{n-1}(h\sin\theta)^n\dfrac{\sin n\theta}{n}+\cdots\]
Where θ=cot-1x.
7 M
1(e)
Expand ercosx by Maclaurin's theorem.
7 M
2(a)
Find the limit when n → ∞ of the series \[\sum ^n_{r=1}\dfrac{n^2}{(n^2+r^2)^{3/2}}\]
2 M
2(b)
Evaluate \[\int ^1_0\left ( \log \dfrac{1}{y} \right )^{n-1}dy\].
2 M
2(c)
Change the order of integration in \[\int^q_0 \int ^a_y \dfrac{xdxdy}{x^2+y^2}\] and hence evaluate the same.
3 M
Solve any one question from Q.2(d) &Q.2(e)
2(d)
Prove that \[\displaystyle \beta (m,n)=\dfrac{\Gamma m \Gamma n}{\Gamma (m+n)}, m>0,n>0\] .
7 M
2(e)
Find the volume bounded by the paraboloid x2+4y2+z=4 and the xy-plane. Also sketch the curve.
7 M
3(a)
Solve (ycosx+siny+y)dx+(sinx+xcosy+x)dy=0
2 M
3(b)
Solve p2+2pycotx-y2=0
2 M
3(c)
Solve (D2+2D+1)y=xcosx.
3 M
Solve any one question from Q.3(d) &Q.3(e)
3(d)
Solve \[x^2\dfrac{d^2y}{dx^2}+5x\dfrac{dy}{dx}+4y=x \log x\]
7 M
3(e)
Solve by the method of variation of parameters \[\dfrac{d^2y}{dx^2}+y=cosec x\] .
7 M
4(a)
Find the rank of the matrix A, where \[A=\begin{pmatrix}
2 & 1 & -1\\
0 & 3 & -2\\
2 & 4 & -3
\end{pmatrix}
\]. If λ be an eigen value of a non singular matrix A.
2 M
4(b)
Show that λ-1 is an eigen value of A-1.
2 M
4(c)
Solve the equations:
x1+3x2+2x3=0, 2x1-x2+3x3=0, 3x1-5x2+4x3=0, x1+172+4x3=0.
x1+3x2+2x3=0, 2x1-x2+3x3=0, 3x1-5x2+4x3=0, x1+172+4x3=0.
3 M
Solve any one question from Q.4(d) &Q.4(e)
4(d)
Find the eigen values and eigen vectors of the matrix
\[A=\begin{pmatrix} -2 & 2 & -3\\ 2 & 1 & -6\\ -1 & -2 & 0 \end{pmatrix} \]
\[A=\begin{pmatrix} -2 & 2 & -3\\ 2 & 1 & -6\\ -1 & -2 & 0 \end{pmatrix} \]
7 M
4(e)
Verify Cayley - Hamilton theorem for the matrix.
\[A=\begin{pmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{pmatrix} \] . Hence compute A-1.
\[A=\begin{pmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2 \end{pmatrix} \] . Hence compute A-1.
7 M
5(a)
Define the following with examples:
i) Union of two fuzzy set
ii) Intersection of two fuzzy set
i) Union of two fuzzy set
ii) Intersection of two fuzzy set
2 M
5(b)
Define the following :
i) Simple graph
ii) Multigraph
iii) Degree of vertex
iv) Isolated vertex.
i) Simple graph
ii) Multigraph
iii) Degree of vertex
iv) Isolated vertex.
2 M
5(c)
Prove that the number of vertices of odd degree in a graph is always even.
3 M
Solve any one question from Q.5(d) &Q.5(e)
5(d)
If (B, +,•, ') be a Boolean algebra and a,b be any two elements of B. Then show that (a+b)'=a'.b' ∀ a,b ∈ B.
7 M
5(e)
Write the function f(x, y, z) = x.y'+x.z+x.y into conjunctive normal form in three variables.
7 M
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