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)