Mathematics 1 - May 2013

First Year Engineering (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

1 Find the eigen values of A^-1 where $$ A=\begin{bmatrix}3 &1 &4 \\0 &2 &6 \\0 &0 &5 \end{bmatrix}$$(2 marks) 10 Change the order of Integration in $$ \int^a_0 \int^y_0 f(x, y)dxdy$$(2 marks)

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

11.(a) (i) Find the eigen values and eigen vectors of the matrix $$ A=\begin{bmatrix} 2&0 &1 \\0 &2 &0 \\1 &0 &2 \end{bmatrix} $$(8 marks) 11.(a)(ii) Show that the 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 marks) 11.(b) Reduce the quadratic form 3x²+5y²+3z²-2xy-2yz+2zx canonical form(16 marks)

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 marks) 12.(a) (ii) Find the equation of the right circular cylinder whose axis is $$ \dfrac {x-1}{2}=\dfrac {y}{3}=\dfrac {z-3}{1} $$ and radius '2'.(8 marks) 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 marks) 12.(b) (ii) Find the equation of the cone formed by rotating the line 2x+3y=5, z=0 about the y-axis.(8 marks)

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

13.(a) (i) Find the evolute of the parabola x²=4ay(8 marks) 13.(a) (ii) Find the radius of curvature of the curve x³+xy²-6y²=0 at (3,3).(8 marks) 13.(b) (i) Find the centre of curvature of the curve y=x³-6x²+3z+1 at the point (1, -1).(8 marks) 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 marks)

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

14.(a) (i) $$ If \ u=xy+yz+zx \ where \ x= \dfrac{1}{t}, \ y=e^t \ and \ z=e^{-t} \ find \ \dfrac {du}{dt} $$(8 marks) 14.(a) (ii) Test for maxima and minima of the function f(x,y)=x³+y³-12x-3y+20(8 marks) 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 marks) 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 marks)

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

15.(a) (i) Change the order of integration in $$ \int^a_0\int^a_y \dfrac {x}{x^2+y^2}dx \ dy $$ and hence evaluate it.(8 marks) 15.(a) (ii) Using double integral find the area of the ellipse $$ \dfrac {x^2}{a^2}+\dfrac{y^2}{b^2}=1 $$(8 marks) 15.(b) (i) $$ Evaluate \ \int^{\log 2}_0 \int^{x}_0\int^{x+\log y}_0 e^{x+y+z}dzdydx $$(8 marks) 15.(b) ii) Using double integral find the area bounded by the parabolas y²=4ax and x²=4ay.(8 marks) 2 Write down the matrix of the quadratic form 2x²+8z²+4xy+10xz-2yz(2 marks) 3 Find the equation of the sphere on the line joining the points (2, -3, 1) and (1, -2, -1) as diameter(2 marks) 4 Define right circular cone.(2 marks) 5 Find the readius of curvature of the curve y=e^x at x=0(2 marks) 6 Find the envelope of the lines x/t+yt=2c, 't' being a parameter.(2 marks) 7 $$ Find \ \dfrac {\partial u}{\partial x} \ and \ \dfrac {\partial u}{\partial y} \ if \ u=y^x $$(2 marks) 8 $$ if \ x=r \cos \theta, \ y=r\sin \theta \ find \ \dfrac {\partial (r, \theta)}{\partial (x,y)} $$(2 marks) 9 $$ Evaluate \ \int^b_1 \int^a_1 \dfrac {dxdy}{xy} $$(2 marks)