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## Engineering Mathematics - I - Dec 2012

### First Year Engineering (Set B) (Semester 2)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.

### Answer any one question from Q1 & Q2

**1 (a) ** Expand sin x in powers of (x-?/2). Hence. Find the value of sin 91° correct to 4 decimal places.(7 marks)
**1 (b)** Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.(7 marks)
**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'(7 marks)
**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.(7 marks)
**2 (a) ** if u = x?(y/x) + ?(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 marks)
**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 marks)

### Answer any one question from Q3 & Q4

**3 (a)** $$ Evaluate \ \lim_{n\rightarrow \infty}\left \{\dfrac {n!}{n^n} \right \}yn $$(7 marks)
**3 (b)** Find the whole area of astroid x^{u3} + y^{u3} = a^{u3}(7 marks)
**4 (a) ** Find, by triple integration, the volume of the sphere

x^{2} + y^{2} + z^{2} = a^{2}(7 marks)
**4 (b)** $$ Prove \ That \ \beta(m,n)= \dfrac {\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$(7 marks)

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

(D^{2} + a^{2})y-sec ax.(7 marks)
**6 (a) ** Solve the differential equation. $$ x^2\dfrac {d^2y}{dx^2}+2x\dfrac {dy}{dx}-12y=x^3 \log x $$(7 marks)
**6 (b)** Solve $$ \dfrac {dx}{dt}-7x + y=0 \\ \dfrac {dy}{dt}-2x-5y=0 $$(7 marks)

### 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 marks)
**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 marks)
**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 marks)
**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(7 marks)

### Answer any one question from Q9 & Q10

**9 (a)** Prove that the proposition

P ? (q ? r) ? (p ? q) ? r is a futology.(7 marks)
**9 (b)** Define the tree and prove that a tree T with n vertices has exactly (n-1) edges.(7 marks)