Answer any one question from Q1 & Q2

1 (a)
Expand sin x in powers of (x-π/2). Hence. Find the value of sin 91° correct to 4 decimal places.

7 M

1 (b)
Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.

7 M

2 (a)
if u = xΦ(y/x) + φ(y/x), Prove that \[ x^2\dfrac {\partial^2 u}{\partial x^2}+2xy\dfrac {\partial^2u}{\partial x \partial y}+y^2\dfrac {\partial^2u}{\partial y^2}=0 \]

7 M

2 (b)
Show that the radius of curvature at any point on the cardioid. \[ r=a(1-\cos \theta)\ is \ 2/3 \sqrt{2ar} \]

7 M

Answer any one question from Q3 & Q4

3 (a)
\[ Evaluate \ \lim_{n\rightarrow \infty}\left \{\dfrac {n!}{n^n} \right \}yn \]

7 M

3 (b)
Find the whole area of astroid x

^{u3}+ y^{u3}= a^{u3}
7 M

4 (a)
Find, by triple integration, the volume of the sphere

x

x

^{2}+ y^{2}+ z^{2}= a^{2}
7 M

4 (b)
\[ Prove \ That \ \beta(m,n)= \dfrac {\Gamma(m)\Gamma(n)}{\Gamma(m+n)}\]

7 M

Answer any one question from Q5 & Q6

5 (a)
Solve the differential equation. \[ \dfrac {d^2y}{dx^3}-3\dfrac {d^2y}{dx^2}+4\dfrac {dy}{dx}-2y=e^x+\cos x \]

7 M

5 (b)
Solve the following differential equation by method of variation of parameters

(D

(D

^{2}+ a^{2})y-sec ax.
7 M

6 (a)
Solve the differential equation. \[ x^2\dfrac {d^2y}{dx^2}+2x\dfrac {dy}{dx}-12y=x^3 \log x \]

7 M

6 (b)
Solve \[ \dfrac {dx}{dt}-7x + y=0 \\ \dfrac {dy}{dt}-2x-5y=0 \]

7 M

Answer any one question from Q7 & Q8

7 (a)
Find the normal form of the matrix A and hence find the its rank, where \[ A=\begin{bmatrix}2 &3 &-1 &-1 \\ 1&-1 &-2 &-4 \\ 3&1 &3 &-2 \\ 6&3 &0 &-7 \end{bmatrix} \]

7 M

7 (b)
For the matix \[ A=\begin{bmatrix}1 &1 &2 \\ 1&2 &3 \\ 0&-1 &-1 \end{bmatrix}\] Find non-singular matrices P and Q such that PAQ is in the normal form. Also find rank of A.

7 M

8 (a)
Determine the eigen values and the corresponding eigen vectors of the matrix \[ A=\begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&-4 &3 \end{bmatrix}\]

7 M

8 (b)
Test the consistency of the following system of equation and solve using matrix methods.

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

Answer any one question from Q9 & Q10

9 (a)
Prove that the proposition

P → (q → r) ↔ (p ∧ q) → r is a futology.

P → (q → r) ↔ (p ∧ q) → r is a futology.

7 M

9 (b)
Define the tree and prove that a tree T with n vertices has exactly (n-1) edges.

7 M

10 (a)
Let (B, +, ·, ') be a Boolean algebra and a, b, be any two elements of B. Then prove that

i) (a+b)'=a'·b'

ii) (a·b)'=a'+b'

i) (a+b)'=a'·b'

ii) (a·b)'=a'+b'

7 M

10 (b)
Define the following terms:

i) Support of a fuzzy set.

ii) Complement of a fuzzy set.

iii) Union of two fuzzy set.

iv) Intersection of two fuzzy set.

i) Support of a fuzzy set.

ii) Complement of a fuzzy set.

iii) Union of two fuzzy set.

iv) Intersection of two fuzzy set.

7 M

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