1 (a)
Expand \[\log \dfrac {1+x}{1-x} \] in powers of x using Maclaurin's theorem.

2 M

1 (b)
Define homogeneous functions and composite function and establish the Euler's theorem on homogeneous function.

2 M

1 (c)
Find the extreme values of the function x

^{3}+ y^{2}- 3 axy.
3 M

Answer any one question from Q1. (d) & Q1. (e)

1 (d)
If the sides and angles of a triangle ABC vary in such a way that its circum radius remains constant, prove that \[ \dfrac{da}{\cos A}+ \dfrac {db}{\cos B}+ \dfrac {dc}{\cos C}=0\]

7 M

1 (e)
Prove that the radius of curvature for the catenary Y=c cosh (x/c) is equal to the portion of the normal intercepted between the curve and the x-axis and that it varies as the square of the ordinate.

7 M

2 (a)
Define Gamma function and Beta function and also establish the symmetry of Beta function.

2 M

2 (b)
Evaluate the following integral by changing the order of integration: \[ \int^{1}_{0}\int^{c}_{c'}\dfrac {dydx}{\log y}\]

2 M

2 (c)
Evaluate by definition of definite integral as the limit of a sum \[ \int^{b}_{a}\sin x \ dx \]

3 M

Answer any one question from Q2. (d) & Q2. (e)

2 (d)
Find the volume bounded by the cylinder x

^{2}+ y^{2}= 4 and the plans y + z = 4 and z=0.
7 M

2 (e)
Prove that: \[ \lim_{n\rightarrow \infty} \left [\left(1+ \dfrac{1^2}{n^2 }\right) \left(1+ \dfrac{2^2}{n^2}\right)\left(1+\dfrac{3^2}{n^2}\right)...\left(1+\dfrac{n^2}{n^2}\right)\right ]^{\frac{1}{4}}\\[2ex]=2e^{\frac {x-4}{2}} \]

7 M

3 (a)
Define the order and degree of a differential equation with one example also explain that the elimination of n arbitary constants from an equation leads us to which order derivative and hence a differential equation of which order.

2 M

3 (b)
\[ Solve \ -ydx+xdy= \sqrt{x^2+y^2}dx \]

2 M

3 (c)
A bacteria population is known to have a rate of growth to itself. If between noon and 2 pm the population triples, at what time, no controls being exerted should becomes 100 times what it was at soon.

3 M

Answer any one question from Q3. (d) & Q3. (e)

3 (d)
\[ Solve \ x^3\dfrac {d^3y}{dx^3}+3x^2\dfrac {d^2y}{dx^2}+x\dfrac {dy}{dx}+y=x+\log x. \]

7 M

3 (e)
Solve the following differential equation by using the method of variation of parameters. \[ \dfrac {d^2y}{dx^2}-2\dfrac {dy}{dx}+2y=e^x \tan x \]

7 M

4 (e)
Find the eigen values of A and using Cayley-Hamilton theorem. Find A

^{n}(n is a positive integer); given that \[ \begin{bmatrix}1&2 \\ 4&3 \end{bmatrix} \]
7 M

4 (a)
Determine the rank of the following matrix \[ \begin{bmatrix}4 &2 &3 \\ 8&4 & 6\\ -2&-1 &-1.5 \end{bmatrix} \]

2 M

4 (b)
Solve the system of equation using matrix method. X+3y-2z=0

2x-y+4z=0

x-11y+14z=0

2x-y+4z=0

x-11y+14z=0

2 M

4 (c)
If A is a non-singular matrix, prove that the eigen values of A

^{-1}are the reciprocal of the eigen values of A.
3 M

Answer any one question from Q4. (d) & Q4. (e)

4 (d)
Find the eigen values eigen vectors of the matrix \[ \begin{bmatrix}-2&2 &-3 \\ 2&1 &- 6\\ -1&-2 &0\end{bmatrix} \]

7 M

5 (a)
What do you mean by logical equivalence and prove that the statement (p?q) ? (?p ??q) is a contradiction.

2 M

5 (b)
For a simple graph of n vertices, the number of edge is \[ \dfrac {1}{2} n (n-1) \]

2 M

5 (c)
Simplify the following circuit

3 M

Answer any one question from Q5. (d) & Q5. (e)

5 (d)
A simple graph with n vertices and k compoents can have at most \[ \dfrac {(n-k)(n-k+1)}{2}\] edges.

7 M

5 (e)
Express the following functions into disjunctive normal form f(x,y,z)=x.y'+x.z+x.y

7 M

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