Engineering Mathematics - I - May 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.

Solve the any two question from Q1(a) Q1(b) and Q1(c) Q1(d)

1 (a) Prove that : [ an^{-1}(x+h)= an^{-1}x+h sin z cdot dfrac {sin z}{1}- dfrac {(hsin z)^2}{2} cdot sin 2z+ cdots cdots \ where z=cot^{-1}x. ](7 marks) 1 (b) [ If u=xphi(y/x)+ psi(y/x), then prove that: \ x^2 dfrac {partial^2 u}{partial x^2} + 2xy dfrac {partial^2 u}{partial x partial y}+ y^2 dfrac {partial^2u}{partial y^2}=0 ](7 marks) 1 (c) What error in the common logarithm of a number will be produed by an error of 1% in the number?(7 marks) 1 (d) Find the maxima and minima of the following function: [ sin x + sin y + sin (x+y)in left [ 0 le x le dfrac {pi}{2}, 0 le y le dfrac {pi}{2} ight ] ](7 marks)

Solve the any two question from Q2(a) Q2(b) and Q2(c) Q2(d)

2 (a) Find ab-initio the value of the integral: [ displaystyle int^{pi/2}_0 sin x dx ](7 marks) 2 (b) Evaluate: [ displaystyle int^{infty}_0 dfrac {x^8 (1-x^6)}{(1+x)^{24}}dx ](7 marks) 2 (c) Evaluate: [ lim_{n o infty} left {dfrac {?n}{n^n} ight }^{1/n} ](7 marks) 2 (d) Change the order of integration: [ displaystyle int^4_0 int^{2sqrt{x}}_{x^2/4}dx dy ] Hence evaluate it.(7 marks)

Solve the any two question from Q3(a) Q3(b) and Q3(c) Q3(d)

3 (a) Solve: y(xy+2x²y²) dx + x (xy-x²y²) dy=0(7 marks) 3 (b) Solve: [ dfrac {d^2y}{dx^2}+ 4 y = e^x + sin 2 x ](7 marks) 3 (c) Solve: p(p y)= x (x+y) [ where p=dfrac {dy}{dx} ](7 marks) 3 (d) Solve: [ dfrac {dx}{dt}+ y = sin t \ dfrac {dy}{dt}+ x - cos t ] Give that x=2 and y=2, when t=0(7 marks)

Solve the any two question from Q4(a) Q4(b) and Q4(c) Q4(d)

4 (a) Find the rank of the matrix: [A=?egin{bmatrix} 1 &4 &3 &6 &1 \0 &2 &3 &1 &4 \0 &0 &1 &3 &7 \0 &0 &0 &-1 &3 \0 &0 &0 &0 &0 end{bmatrix}_{5 imes 5} ] by defining it in Echelon form.(4 marks) 4 (b) Find the eigen values and eigen vectors of the matrix: [ A=?egin{bmatrix} 3 &-4 &4 \1 &-2 &4 \1 &-1 &3 end{bmatrix} ](10 marks) 4 (c) Find the values of k such that the system of equations:
x+ky+3z=0
4x+3y+kz=0
2x+y+2z=0
has non-trivial solution.(7 marks) 4 (d) Verify the Cayley-Hamilton theorem for the matrix: [ A=?egin{bmatrix} 2 &-1 &1 -1 &2 &1 \1 &-1 &2 end{bmatrix} ](7 marks)

Solve the any two question from Q5(a) Q5(b) and Q5(c) Q5(d)

5 (a) Define the following terms for a graph:
(i) Subgraph
(ii) Degree of vertex
(iii) Composition and De-composition
(iv) Rooted tree(7 marks) 5 (b) Define fuzzy logic and its applications in science and engineering.(7 marks) 5 (c) Prepare a truth table to get the negative of the statement ?Sita is dull and careless.?(7 marks) 5 (d) Prove that :
a?b+ b?c+ c?a= (a+b)?(b+c)?(c+a) ? a,b,c, ? B(7 marks)