## 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 A^{4}, if $$ A=\begin{bmatrix} 1&2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix} $$(8 marks)
**11.(b)** Reduce the quadratic form 6x^{2}+3y^{2}+3z^{2}-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 y^{2}=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-am^{3}, 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)=x^{3}y^{2}(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 x^{2}+y^{2}=a^{2}(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 λ^{2} is the eigen value of A^{2}(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)