## 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 3x^{2}+5y^{2}+3z^{2}-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 x^{2}+y^{2}+z^{2}-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 x^{2}=4ay(8 marks)
**13.(a) (ii)** Find the radius of curvature of the curve x^{3}+xy^{2}-6y^{2}=0 at (3,3).(8 marks)
**13.(b) (i)** Find the centre of curvature of the curve y=x^{3}-6x^{2}+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)=x^{3}+y^{3}-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 y^{2}=4ax and x^{2}=4ay.(8 marks)
**2** Write down the matrix of the quadratic form 2x^{2}+8z^{2}+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)