## Engineering Mathematics - I - May 2014

### 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.
** 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 marks)
**1 (a) ** Expand $$\log \dfrac {1+x}{1-x} $$ in powers of x using Maclaurin's theorem.(2 marks)
**1 (b)** Define homogeneous functions and composite function and establish the Euler's theorem on homogeneous function.(2 marks)
**1 (c) ** Find the extreme values of the function x^{3} + y^{2} - 3 axy.(3 marks)

### 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 marks)
**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 marks)
**2 (a) ** Define Gamma function and Beta function and also establish the symmetry of Beta function.(2 marks)
**2 (b)** Evaluate the following integral by changing the order of integration: $$ \int^{1}_{0}\int^{c}_{c'}\dfrac {dydx}{\log y}$$(2 marks)
**2 (c) ** Evaluate by definition of definite integral as the limit of a sum $$ \int^{b}_{a}\sin x \ dx $$(3 marks)

### 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 marks)
**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 marks)
**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 marks)
**3 (b)** $$ Solve \ -ydx+xdy= \sqrt{x^2+y^2}dx $$(2 marks)
**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 marks)

### 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 marks)
**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 marks)
**4 (a) ** Determine the rank of the following matrix $$ \begin{bmatrix}4 &2 &3 \\ 8&4 & 6\\ -2&-1 &-1.5 \end{bmatrix} $$(2 marks)
**4 (b)** Solve the system of equation using matrix method. X+3y-2z=0

2x-y+4z=0

x-11y+14z=0(2 marks)
**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 marks)

### 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 marks)
**5 (a)** What do you mean by logical equivalence and prove that the statement (p?q) ? (?p ??q) is a contradiction.(2 marks)
**5 (b)** For a simple graph of n vertices, the number of edge is $$ \dfrac {1}{2} n (n-1) $$(2 marks)
**5 (c) ** Simplify the following circuit

(3 marks)

### 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 marks)
**5 (e) ** Express the following functions into disjunctive normal form f(x,y,z)=x.y'+x.z+x.y(7 marks)