Question: Engineering Mathematics -I : May 2014 - First Year Engineering (Set A) (Semester 1) | Rajiv Gandhi Proudyogiki Vishwavidyalaya (RGPV)
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Engineering Mathematics -I - May 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.
4 (e) Find the eigen values of A and using Cayley-Hamilton theorem. Find An (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 x3 + y2 - 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)

2 (d) Find the volume bounded by the cylinder x2 + y2 = 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)

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