Mathematics 1 - Dec 2013

First Year Engineering (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

1 If the eigen values of the matrix a of order 3×3 are 2, 3 and 1, then find the eigen values of adjoint of A.(2 marks) 10 $$Evaluate \ \int^\pi_0\int^a_0 r \ drd\theta$$ (2 marks)

Answer any one question from Q11 (a) & Q11 (b)

11.(a) (i) Find the eigen values and the eigen vectors of the matrix $$\begin{bmatrix}2 &2 &1 \\1 &3 &1 \\1 &2 &2 \end{bmatrix}$$ (8 marks) 11.(a) (ii) Using Cayley-Hamilton theorem find A^-1 and A⁴, if $$A=\begin{bmatrix} 1&2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix}$$ (8 marks) 11.(b) Reduce the quadratic form 6x²+3y²+3z²-4xy-2yz+4xz into a canonical form by an orthogonal reduction. Hence find its rank and nature.(16 marks)

Answer any one question from Q12 (a) & Q12 (b)

12.(a) (i) Examine the convergence and the divergence of the following series $$1+\dfrac {2}{5}x+\dfrac {6}{9}x^2+\dfrac {14}{17}x^3+.....+\dfrac {2^n-2}{2^n+1}(x^{n-1})+....(x>0).$$ (8 marks) 12.(a) (ii) Discuss the convergence and the divergence of the following series $$\dfrac {1}{2^3}-\dfrac {1}{3^3}(1+2)+\dfrac {1}{4^3}(1+2+3)-\dfrac {1}{5^3}(1+2+3+4)+.....$$ (8 marks) 12.(b) (i) Test the convergence of the series $$\sum^\infty_{n=0}ne^{-n^2}$$ (8 marks) 12.(b) (ii) Test the convergence of the series $$\dfrac {x}{1+x}-\dfrac {x^2}{1+x^2}+\dfrac {x^3}{1+x^3}-\dfrac {x^4}{1+x^4}+\cdots \cdots (0<x<1)$$ (8 marks)

Answer any one question form Q13 (a) & Q13 (b)

13.(a) (i) Find the radius of curvature of the cycloid x=a(θ+sin θ), y=a(1-cos θ)(8 marks) 13.(a) (ii) Find the equation of the evolutes of the parabola y²=4ax.(8 marks) 13.(b) (i) Find the equation of circle of curvature at $$\left ( \dfrac {a}{4},\dfrac {a}{4} \right ) \ on \ \sqrt{x}+\sqrt{y}=\sqrt{a}$$ (8 marks) 13.(b) (ii) Find the envelope of the family of straight lines y=mx-2am-am³, where m is the parameter.(8 marks)

Answer any one question from Q14 (a) & Q14 (b)

14.(a) (i) Expand e^x log (+1+y) in powers of x and y upto the third degree terms using Taylor's theorem.(8 marks) 14.(a) (ii) $$If \ u=\dfrac {yz}{x}, \ v=\dfrac {zx}{y}, \ w=\dfrac {xy}{z}, \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)}$$ (8 marks) 14.(b) (i) Discuss the maxima and minima of f(x,y)=x³y²(1-x-y)(8 marks) 14.(b) (ii) If w=f(y-z, z-x, x-y), then show that $$\dfrac {\partial w}{\partial x}+\dfrac {\partial w}{\partial y}+\dfrac {\partial w}{\partial z}=0$$ (8 marks)

Answer any one question from Q15 (a) & Q15 (b)

15.(a) (i) By changing the order of integration evaluate $$\int^1_0\int^{2-x}_{x^2}xy \ dydx$$ (8 marks) 15.(a) (ii) By changing to polar coordinates, evaluate $$\int^\infty_0\int^{\infty}_0e^{-(x^2+y^2)}dxdy$$ (8 marks) 15.(b) (i) Evaluate ∬ xy dxdy over the positive quadrant of the circle x²+y²=a²(8 marks) 15.(b) (ii) $$Evaluate \ \iiint_v \dfrac {dzdydx}{(x+y+z+1)^3},$$ where V is the region bounded by x=0, y=0, z=0 and x+y+z=1(8 marks) 2 If λ is the eigen value of the matrix A, then prove that λ² is the eigen value of A²(2 marks) 3 Give an example for conditionally convergent series.(2 marks) 4 Test the convergence of the series $$1-\dfrac {1}{2^2}-\dfrac {1}{3^2}+\dfrac {1}{4^2}+\dfrac {1}{5^2}-\dfrac {1}{7^2}-\dfrac {1}{8^2}..... \ to \infty$$ (2 marks) 5 What is the curvature of the circle (x-1)²+(y+2)²=16 at any point on it?(2 marks) 6 Find the envelope of the family of curves $$y=mx+\dfrac {1}{m}$$ where m is the parameter.(2 marks) 7 $$If \ x^y+y^x=1 \ then \ find \ \dfrac{dy}{dx}$$ (2 marks) 8 $$If \ x=r \cos \theta, \ y=r\sin\theta, \ then \ find \ \dfrac {\partial (r, \theta)}{\partial (x,y)}$$ (2 marks) 9 Find the area bounded by the lines x=0, y=1 and y=x, using double integration.(2 marks)