AU Mathematics 1 - May 2013 Exam Question Paper

AU First Year Engineering (Semester 1)
Mathematics 1
May 2013

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 A^-1 where

A = [\begin{matrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \end{matrix}]

$A=\begin{bmatrix}3 &1 &4 \\0 &2 &6 \\0 &0 &5 \end{bmatrix}$

2 M

2 Write down the matrix of the quadratic form 2x²+8z²+4xy+10xz-2yz

2 M

3 Find the equation of the sphere on the line joining the points (2, -3, 1) and (1, -2, -1) as diameter

2 M

4 Define right circular cone.

2 M

5 Find the readius of curvature of the curve y=e^x at x=0

2 M

6 Find the envelope of the lines x/t+yt=2c, 't' being a parameter.

2 M

F i n d \frac{\partial u}{\partial x} a n d \frac{\partial u}{\partial y} i f u = y^{x}

$Find \ \dfrac {\partial u}{\partial x} \ and \ \dfrac {\partial u}{\partial y} \ if \ u=y^x$

2 M

i f x = r \cos θ, y = r \sin θ f i n d \frac{\partial (r, θ)}{\partial (x, y)}

$if \ x=r \cos \theta, \ y=r\sin \theta \ find \ \dfrac {\partial (r, \theta)}{\partial (x,y)}$

2 M

E v a l u a t e \int_{1}^{b} \int_{1}^{a} \frac{d x d y}{x y}

$Evaluate \ \int^b_1 \int^a_1 \dfrac {dxdy}{xy}$

2 M

10 Change the order of Integration in

\int_{0}^{a} \int_{0}^{y} f (x, y) d x d y

$\int^a_0 \int^y_0 f(x, y)dxdy$

2 M

Answer any one question form Q11 (a) or Q11 (b)

11.(a) (i) Find the eigen values and eigen vectors of the matrix

A = [\begin{matrix} 2 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 2 \end{matrix}]

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

8 M

11.(a)(ii) Show that the matrix

A = [\begin{matrix} 2 & - 1 & 2 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{matrix}]

$A=\begin {bmatrix}2&-1&2\\-1&2&-1\\1&-1&2 \end{bmatrix}$ satisfies its own characteristics equation. Find also its inverse.

8 M

11.(b) Reduce the quadratic form 3x²+5y²+3z²-2xy-2yz+2zx canonical form

16 M

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

12.(a) (i) Find the equation of the sphere passing through the points (4, -1, 2), (0, -2, 3), (1, 5, -1), (2, 0, 1)

8 M

12.(a) (ii) Find the equation of the right circular cylinder whose axis is

\frac{x - 1}{2} = \frac{y}{3} = \frac{z - 3}{1}

$\dfrac {x-1}{2}=\dfrac {y}{3}=\dfrac {z-3}{1}$ and radius '2'.

8 M

12.(b) (i) Find the two tangent planes to the sphere x²+y²+z²-4x+2y-6z+5=0 which are parallel to the plane 2x+2y=z. Find their points of contacts

8 M

12.(b) (ii) Find the equation of the cone formed by rotating the line 2x+3y=5, z=0 about the y-axis.

8 M

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

13.(a) (i) Find the evolute of the parabola x²=4ay

8 M

13.(a) (ii) Find the radius of curvature of the curve x³+xy²-6y²=0 at (3,3).

8 M

13.(b) (i) Find the centre of curvature of the curve y=x³-6x²+3z+1 at the point (1, -1).

8 M

13.(b) (ii) Find the readius of curvature of the curve x=a(cost+ t sin t); y=a(sin t - t cos t) at 't'.

8 M

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

14.(a) (i)

I f u = x y + y z + z x w h e r e x = \frac{1}{t}, y = e^{t} a n d z = e^{- t} f i n d \frac{d u}{d t}

$If \ u=xy+yz+zx \ where \ x= \dfrac{1}{t}, \ y=e^t \ and \ z=e^{-t} \ find \ \dfrac {du}{dt}$

8 M

14.(a) (ii) Test for maxima and minima of the function f(x,y)=x³+y³-12x-3y+20

8 M

14.(b) (i) Expand e^x sin y in power of x and y as far as the terms of the 3^rd degree using Taylor's expansion.

8 M

14.(b) (ii) Find the dimensions of the rectangular box, open at the top, of maximum capacity whose surface area is 432 square meter.

8 M

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

15.(a) (i) Change the order of integration in

\int_{0}^{a} \int_{y}^{a} \frac{x}{x^{2} + y^{2}} d x d y

$\int^a_0\int^a_y \dfrac {x}{x^2+y^2}dx \ dy$ and hence evaluate it.

8 M

15.(a) (ii) Using double integral find the area of the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

$\dfrac {x^2}{a^2}+\dfrac{y^2}{b^2}=1$

8 M

15.(b) (i)

E v a l u a t e \int_{0}^{\log 2} \int_{0}^{x} \int_{0}^{x + \log y} e^{x + y + z} d z d y d x

$Evaluate \ \int^{\log 2}_0 \int^{x}_0\int^{x+\log y}_0 e^{x+y+z}dzdydx$

8 M

15.(b) ii) Using double integral find the area bounded by the parabolas y²=4ax and x²=4ay.

8 M

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