## Engineering Maths 1 - Dec 2015

### First Year Engineering (P Cycle) (Semester 1)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** If y=e^{ax} sin(bx+c) then prove that $ y_n = (a^2 + b^2)^{\frac {n}{2}} \ e^x \sin \left [ (bx + c)+ n \tan^{-1} \left ( \dfrac {b}{a} \right ) \right ] $(6 marks)
**1 (b)** Show that the radius of curvature at any point of the cycloide. x=a(θ+sin θ); y=a (1- cos θ) is 4a cos $ \left ( \dfrac {\theta}{2} \right ) $(7 marks)
**1 (c)** Show that the two curves r=a(1+cos θ) and r=a(1- cos θ) cut each other orthogonally.(7 marks)
**10 (a)** Solve by LU decomposition method,

3x+2y+7z=4

2x+3y+z=5

3x+4y+z=7.(7 marks)
**10 (b)** Reduce the quadratic form 3x^{2}+5y^{2}+3z^{2}-2y^{2}+2zx-2xy the canonical form and specify the matrix of transformation.(6 marks)
**10 (c)** Show that the transformation y_{1}= 2x_{1}+x_{2}+x_{3}, y_{2}=x_{1}+x_{2}+2x_{3}, y_{3}=x_{1}-2x_{3} is regular and also write down the inverse information.(7 marks)
**2 (a)** If x=sin t and y=cos pt then prove that (1-x^{2})y_{n+2} - (2n+1) xy_{n+1} + (p^{2} - n^{2})y_{n} = 0.(7 marks)
**2 (b)** Show that the Pedal equation for the curve r^{m}=a^{m} cos mθ is Pa^{m}=r^{m+1}.(6 marks)
**2 (c)** Derive an expression for radius of curvature in polar form.(7 marks)
**3 (a)** If 'u' is a homogeneous function of degree 'n' in the variable x and y, then prove that $ x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = nu. $(7 marks)
**3 (b)** Using Maclaurin's series prove that, $ \sqrt{1+ \sin 2x} = 1 + x - \dfrac {x^2}{2}- \dfrac {x^3}{3}+ \dfrac {x^4}{24} + \cdots \cdots $(6 marks)
**3 (c)** If z is a function of x and y where x=e^{u} + e^{-v} and y=e^{-u}-e^{v}, then prove that $$ \dfrac {\partial z}{\partial u} - \dfrac {\partial z}{\partial v} = x \dfrac {\partial z}{\partial x} - y \dfrac {\partial z}{\partial y} $$(7 marks)
**4 (a)** If $ u=\sin^{-1} \left [ \dfrac {x^2 + y^2}{x+y} \right ] $ then prove that $ x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}=\tan u. $(7 marks)
**4 (b)** Evaluate $ \lim_{x\to 0} \left [ \dfrac {a^x + b^x + c^x + d^x}{4} \right ]^{\frac {1}{x}} $(6 marks)
**4 (c)** If u=x+y+z, uv=y+z and and uvw=z then show that $ \dfrac {\partial (x \ y \ z)}{\partial (u \ v \ w)}=u^2 v. $(7 marks)
**5 (a)** A particle moves along the curve x=(1-t^{3}), y=(1+t^{2}), z=(2t-5) determine its velocity and acceleration. Also find the components of velocity and acceleration at t=1 in the direction of 2i+j+2k.(7 marks)
**5 (b)** Using differentiation under integral sign evaluate $ \int^1_0 \dfrac {x^n - 1}{\log x}dx, \ a\ge 0 $(6 marks)
**5 (c)** Apply the general rules to trace the curve r=a(1+cos θ).(7 marks)
**6 (a)** Apply the general rule to trace curve y^{2} (a-x) = x^{2}(a+x), a>0.(7 marks)
**6 (b)** Show that $ \overrightarrow {F} = (y^2 - z^2 + 3yz - 2x) \widehat{i} + (3xz + 2xy)\widehat{j}+ (3xy - 2xz + 2z) \widehat{k}$ is both solenoidal and irrotational.(6 marks)
**6 (c)** Show that div (curl A)=0.(7 marks)
**7 (a)** Obtain the reduction formula for $ \int \cos^n xdx $ where 'n' being the positive integer.(7 marks)
**7 (b)** Solve (y cos x+ sin y+y)dx + (sin x+ x cos y+x)dy=0(6 marks)
**7 (c)** Show that the family of curves $ \dfrac {x^2}{a^2+\lambda} + \dfrac {y^2}{b^2 + \lambda}=1 $, where &lambda is a parameter is self orthogonal.(7 marks)
**8 (a)** Evaluate $ \int^{\frac {\pi}{4}}_0 \cos^6 x \sin^6 xdx $(7 marks)
**8 (b)** $$ \text{Solve }e^y \left ( \dfrac {dy}{dx} +1\right )=e^x $$(6 marks)
**8 (c)** A body originally at 80°C cools down to 60°C in 20 minutes. The temperature of air being 40°C. What will be the temperature of the body after 40 minutes from the original?(7 marks)
**9 (a)** Find the Rank of the matrix $ \begin{bmatrix}
1 &2 &3 &4 \\\\5
&6 &7 &8 \\\\8
&7 &0 &5
\end{bmatrix} $(7 marks)
**9 (b)** Find the largest Eigen value and the corresponding Eigen vector of the given matrix 'A' by using the Rayleigh's power method. Take [1 0 0]^{t} as the initial Eigen vector. $$ A=\begin{bmatrix}
2 &0 &1 \\0
&2 &0 \\1
&0 &2
\end{bmatrix} $$(6 marks)
**9 (c)** Solve 2x+y+4z=12, 4x+11y-z=33 and 8x-3y+2z=20 by using Gauss Elimination method.(7 marks)