STUPIDSID
Answer any one question from Q1 & Q2
1 (a)
Expand ea sin-1x in ascending power of x.
7 M
1 (b)
If p= x cos a+y sin a, touches the curve \[ \left ( \dfrac {x}{a} \right )^{\frac {n}{n-1}} + \left ( \dfrac {y}{b} \right )^{\frac {n}{n-1}}=1 \] Prove that: pn=(a cos a)n + (b sin a)n
7 M
2 (a)
Show that the radius of curvature at any point of the cycloid \[ x=a (\theta + \sin \theta ), y=a (1-\cos \theta) \ is \ 4 a \cos \left ( \dfrac {\theta}{2} \right ) \]
7 M
2 (b)
\[ If \ u=\sin^{-1} \left ( \dfrac {x+y}{\sqrt{x}+ \sqrt{y}} \right ), \ prove \ that : \\ i) \ x\dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = \dfrac {1}{2} \tan u \\ ii) \ x^2 \dfrac {\partial^2 u}{\partial x^2}+ 2xy \dfrac {\partial^2 u}{\partial x \partial y} + y^2 \dfrac {\partial^2 u}{\partial y^2} = - \dfrac {\sin u \cos 2 u}{4 \cos^3 u} \]
7 M
Answer any one question from Q3 & Q4
3 (a)
Find the limit as n→∞ of the series: \[ \dfrac {1}{n+1} + \dfrac {1}{n+2}+ \dfrac {1}{n+3} + \cdots \ \cdots + \dfrac {1}{2n} \]
7 M
3 (b)
Find the volume common to the cylinders
x2+y2=a2, x2+z2=a2
x2+y2=a2, x2+z2=a2
7 M
4 (a)
Evaluate: \[ \displaystyle \int^\infty_0 \int^{x}_0 xe^{-x^2/y}dy \ dx \] by changing the order integration.
7 M
4 (b)
Prove that:
\((i)\ \dfrac{\beta (m+1, n)}{m} = \dfrac{\beta (m,n+1)}{n} = \dfrac{\beta (m,n)}{m+n}\\ (ii)\ \Gamma (m) \Gamma \left(m+\dfrac {1}{2} \right) = \dfrac {\sqrt{\pi}}{2^{2m-1}} \Gamma(2m)\)
\((i)\ \dfrac{\beta (m+1, n)}{m} = \dfrac{\beta (m,n+1)}{n} = \dfrac{\beta (m,n)}{m+n}\\ (ii)\ \Gamma (m) \Gamma \left(m+\dfrac {1}{2} \right) = \dfrac {\sqrt{\pi}}{2^{2m-1}} \Gamma(2m)\)
7 M
Answer any one question from Q5 & Q6
5 (a)
Solve the equation: \[ (y-x) \dfrac{dy}{dx} = a^2 \]
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5 (b)
Solve the equation: \[ \dfrac {d^2 y}{dx^2} + 4y = \sec 2x \\ \] by the method of variation of parameters.
7 M
6 (a)
Solve the equation: \[ x^2 \dfrac {d^2 y}{dx^2} -2x \dfrac {dy}{dx} - 4y = x^2 + \log x \]
7 M
6 (b)
Solve the simultaneous equations: \[ \dfrac {dx}{dt} + y = \sin t \\ \dfrac {dy}{dt}+x \cos t \]
7 M
Answer any one question from Q7 & Q8
7 (a)
Reduce the matrix: \[ A= \begin{bmatrix} 2 &3 &4 &5 \\3 &4 &5 &6 \\4 &5 &6 &7 \\9 &10 &11 &12 \end{bmatrix} \] to normal form and find its range.
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7 (b)
Find the eigen values and eigen vectors of the matrix: \[ A=\begin{bmatrix}2 &1 &1 \\1 &2 &1 \\0 &0 &1 \end{bmatrix} \]
7 M
8 (a)
Text for consistency and solve:
5x+3y+7z=4
3x+26y+2z=9
7x+2y+10z=5
5x+3y+7z=4
3x+26y+2z=9
7x+2y+10z=5
7 M
8 (b)
Verify Cayley-Hamilton theorem for the matrix: \[ A=\begin{bmatrix} 1 &2 &1 \\0 &1 &-1 \\3 &-1 &1 \end{bmatrix} \] and find its inverse.
7 M
Answer any one question from Q9 & Q10
9 (a)
Define the following terms with examples:
i) Simple graph
ii) Degree of a vertex
iii) Isomorphic graphs
iv) Spanning tree
i) Simple graph
ii) Degree of a vertex
iii) Isomorphic graphs
iv) Spanning tree
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9 (b)
Express the following function into disjunctive normal form:
f(x,y,z)=(x+y+z)(x·y+x'·z)'
f(x,y,z)=(x+y+z)(x·y+x'·z)'
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10 (a)
Let X={a, b, c, d} be a universe of discourse and A, B be the fuzzy sets on X defined by: \[ A= \left \{ \dfrac {0.3}{a}, \dfrac {0.5}{b}, \dfrac {0.6}{c}, \dfrac {0.4}{d} \right \} \\ B= \left \{ \dfrac {0.2}{a}, \dfrac {0.6}{b}, \dfrac {0.3}{c}, \dfrac {0.7}{d} \right \} \]
Find:
i) Height of A ∪ B
ii) α-cut of A ∩ B for \alpha;=0.4
(A ∪ B)'
iv) A' ∩ B'
i) Height of A ∪ B
ii) α-cut of A ∩ B for \alpha;=0.4
(A ∪ B)'
iv) A' ∩ B'
7 M
10 (b)
Prove that the number of vertices of odd degree in a graph in always even.
7 M
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