Question Paper: Engineering Mathematics -I : Question Paper Dec 2014 - First Year Engineering (Set A) (Semester 1) | Rajiv Gandhi Proudyogiki Vishwavidyalaya (RGPV)
0

## Engineering Mathematics -I - Dec 2014

### First Year Engineering (Set A) (Semester 1)

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)