STUPIDSID
Answer any one question from Q1 & Q2
1 (a)
\[ (\sin^{-1}x)^2=\dfrac {2}{2!}x^2+\dfrac {2.2^2}{4!}x^4+\dfrac {2.2^2.4^2}{6!}x^6+.... \\ and \ hence \ deduce \\\theta^2=2\dfrac {\sin^2\theta}{2!}+2^2\dfrac {2\sin^4\theta}{4!}+2^2.4^2\dfrac {2\sin^6\theta}{6!}+.... \]
7 M
1 (b)
if u(x,y,z)=log(tan x + tan y + tan z), prove that \[ \sin 2x\dfrac {\partial u}{\partial x}+\sin 2y\dfrac {\partial u}{\partial y}+\sin 2z\dfrac {\partial u}{\partial z}=2 \]
7 M
2 (a)
Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.
7 M
2 (b)
Determine the curvature of the parabola y2=2 px at
(i) an arbitary point (x,y).
(ii) the point (P/2, P) and
(iii) the point (0,0)
(i) an arbitary point (x,y).
(ii) the point (P/2, P) and
(iii) the point (0,0)
7 M
Answer any one question from Q3 & Q4
3 (a)
Evaluate by expressing the limit of a sum in the form of a definite integral: \[ \lim_{x\rightarrow \infty} \left [ \left ( 1+\dfrac {1}{n^2} \right )\left (1+ \dfrac {2^2}{n^2} \right ) \left (1+ \dfrac {3^2}{n^2} \right ).... \left (1+ \dfrac {n^2}{n^2} \right )\right ]^{1/n} \]
7 M
3 (b)
Define B(m,n). Prove that
B(m,n)=B(m+1,n)+B(m,n+1)m,n>0.
B(m,n)=B(m+1,n)+B(m,n+1)m,n>0.
7 M
4 (a)
Evaluate the following integral by changing the order of integration : \[ \int^1_0\int^{\sqrt{2-x^2}}_x \dfrac {xdydx}{\sqrt{x^2+y^2}} \]
7 M
4 (b)
Find the volume cut from the sphere x2 + y2 + z2=a2 by the cylinder x2 + y2=ax.
7 M
Answer any one question from Q5 & Q6
5 (a)
Solve (3x2y2 + 2xy)dx + (2x3y3 - x2)dy=0
7 M
5 (b)
\[ Solve \ y-x=x\dfrac {dy}{dx}+\left (\dfrac {dy}{dx} \right )^2 \]
7 M
6 (a)
\[ Solve \ \dfrac {d^2y}{dx^2}-6 \dfrac {dy}{dx}+13y=8e^{3x}\sin 4x+2^x \]
7 M
6 (b)
\[ Solve \ \dfrac {dx}{dt}+4x+3y=t \\ \dfrac {dy}{dt}+2x+5y=e^t \]
7 M
Answer any one question from Q6 & Q7
7 (a)
Define rank of a matrix. Find the rank of matrix A, where \[ A=\begin{bmatrix}1^2 &2^2 &3^2 &4^2 \\ 2^2&3^2 &4^2 &5^2 \\ 3^2&4^2 &5^2 &6^2 \\ 4^2 &5^2 &6^2 &7^2 \end{bmatrix} \]
7 M
7 (b)
Solve completely the system of equation 2w+3x-y-z=0, 4w-6x-2y+2z=0, -6w+12x+3y-4z=0
7 M
8 (a)
Determine the eigen values and eigen vectors of the matrix \[ A=\begin{bmatrix} -2&2 &-3 \\ 2&1 &-6 \\ -1&-2 &0 \end{bmatrix} \]
7 M
8 (b)
Show that Caley-Hamilton theorem is satisfied by the matrix A. \[ where \ A=\begin{bmatrix}0 &0 &1 \\ 3& 1&0 \\ -2&1 &4 \end{bmatrix}\] Hence find A-1.
7 M
9 (a)
Write the following function into disjunctive normal form of 3 variable x,y,z:
(i) x' + y'
(ii) xy' + x'y
(i) x' + y'
(ii) xy' + x'y
7 M
9 (b)
In a Boolean algebra B. Prove that the identity elements 0,1 ? B are unique and prove 0'=1,1'=0
7 M
10 (a)
Define the following terms giving example:
(i) Support of fuzzy set.
(ii) Complement of a fuzzy set.
(iii) Union of two fuzzy sets.
(iv) Intersection of two fuzzy sets.
(i) Support of fuzzy set.
(ii) Complement of a fuzzy set.
(iii) Union of two fuzzy sets.
(iv) Intersection of two fuzzy sets.
7 M
10 (b)
Prove that the number of vertices of odd degree in a graph is always even.
7 M
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