AU Mathematics 1 - May 2014 Exam Question Paper

AU First Year Engineering (Semester 1)
Mathematics 1
May 2014

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 Find the Eigen values of the inverse of the matrix

$A=\begin{bmatrix} 2&1 &0 \\0 &3 &4 \\0 &0 &4 \end{bmatrix}$

2 M

2 If 2, -1, -3 are the Eigen values of the matrix A, then find the Eigen values of the matrix A²-2I

2 M

3 Find the nature of the series 1+2+3+....+n+.

2 M

4 Define Cauchy's integral test.

2 M

5 Find the centre of curvature of y=x² at the origin.

2 M

6 Define Involutes and Evolutes.

2 M

$Evaluate: \ \lim_ {x\to \infty,y\to 2} \dfrac {xy+5}{x^2+2y^2}$

2 M

$If \ x^y+y^x=c, \ then \ find \ \dfrac {dy}{dx}$

2 M

$Evaluate \ \int^{5}_0\int^2_0x^2+y^2 \ dxdy$

2 M

$Evaluate \ \int^{\frac {\pi}{2}}_0\int^{\sin \theta}_0 rd\theta dr$

2 M

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

11.(a) (i) Verify Cayley Hamilton theorem for the matrix

$\begin{bmatrix} 1&0 &3 \\2 &1 &-1 \\1 &-1 &1 \end{bmatrix}$ hence find it?s A^-1

8 M

11.(a) (ii) Find the Eigen values and Eigen vectors of

$\begin{bmatrix} -2&2 &-3 \\2 &1 &-6 \\-1 &-2 &0 \end{bmatrix}$

8 M

11.(b) (i) Reduce the quadratic form

$x^2_1+5x^2_2+x^2_3+2x_1x_2+2x_2x_3+6x_3x_1$ to the canonical form through orthogonal transformation and find its nature.

10 M

11.(b) (ii) Prove that the Eigen values of a real symmetric matrix are real.

6 M

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

12.(a) (i) Prove that the harmonic series is divergent

8 M

12.(a) (ii) Test the convergence of the series

$\dfrac {1}{4.7.10}+\dfrac {4}{7.10.13}+\dfrac {9}{10.13.16}+.....$

8 M

12.(b) (i) Find the nature of the series

$\sum^\infty_{n=2}\dfrac {1}{n(\log n)^p}$ by Cauchy's integral test.

8 M

12.(b) (ii) Test the convergence of the series

$1+\dfrac {2^p}{2!}+\dfrac {3^p}{3!}+\dfrac {4^p}{4!}+.....$ by D'Alembert's ratio test.

8 M

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

13.(a) (i) Find the envelope of

$\dfrac {x}{a}+\dfrac {y}{b}=1$ subject to aⁿ+bⁿ=cⁿ, where c is constant.

8 M

13.(a) (ii) Find the Evolute of

$\sqrt{x}+\sqrt{y}=\sqrt{a}$

8 M

13.(b) (i) Find the equation of the circle of curvature of

$\dfrac {x^2}{4}+\dfrac {y^2}{9}=2 \ at \ (2,3)$

8 M

13.(b) (ii) Find the radius of curvature at any point on x=e^t cos t, y=e^t sin t.

8 M

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

14.(a) (i) Find the extreme value of x²+y²+z² subject to the condition x+y+z=3a

8 M

14.(a) (ii)

$If \ u=(x-y)f\left (\dfrac {y}{x} \right ),\ then \ find \ x^2\dfrac {\partial^2u}{\partial x^2}+2xy\dfrac {\partial^2 u}{\partial x \partial y}+ y^2\dfrac {\partial^2 u}{\partial y^2}$

8 M

14.(b) (i)

$If \ u=\dfrac {yz}{x}, \ v=\dfrac {zx}{y}, \ w=\dfrac {xy}{z} \ the \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)}$

8 M

14.(b) (ii) Expand e^x cos y at

$\left ( 0, \dfrac {\pi}{2} \right )$ upto the third terms using Taylor's series.

8 M

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

15.(a) (i) Find the volume of region bounded by the paraboloid z=x²+y² and the plane z=4

8 M

15.(a) (i) Find the surface area of the section of the cylinder x²+y²=a² made by the plane x+y+z=a

8 M

15.(b) (i) Change the order of Integration

$\int^a_0\int^{2a-x}_{\frac {x^2}{a}} xy \ dxdy$ and hence evaluate it.

10 M

15.(b) (ii) Find the area of the cardioid r=a(1+cos ?)

6 M

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