## Applied Mathematics 1 - May 2013

### First Year Engineering (Semester 1)

TOTAL MARKS: 80

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

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

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1(a)** If cosh x = sec θ, prove that x = log (sec θ + tan θ)(3 marks)
**1(b)** If u = log (x^{2}+y^{2}), prove that

$$\dfrac{\partial^2 u}{\partial x \ \partial y} = \dfrac{\partial^2 u}{\partial y \ \partial x}$$(3 marks)
**1(c)** If x = r cosθ, y = r sinθ;

$$ \dfrac{\partial(x,y)}{\partial(r, \theta)} $$(3 marks)
**1(d)** Expand log(1 + x + x^{2} + x^{3}) in powers of x upto x^{8}.(3 marks)
**1(e)** Show that every square matrix can be uniquely expressed as sum of a symmetric and Skew-symmetric matrix.(4 marks)
**1(f)** If y = cos x.cos 2x.cos 3x then find its n^{th} order derivative

(4 marks)
**2(a)** Solve the equation x^{6}-i=0.(6 marks)
**2(b)** Reduce matrix A to normal form and find its rank where

$$A={ \left[ \begin{array}{ccc}
1 & 2 & 3 &2 \\
2 & 3 & 5 & 1 \\
1 & 3 & 4 &5 \end{array} \right]}$$(6 marks)
**2(c)** State and prove Euler's theorem for a homogeneous function in two variable. And hence find

$$x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} \ \ \ where \ \ u = \dfrac{\sqrt{x}+\sqrt{y}}{x+y}$$(8 marks)
**3(a)** Determine the values of λ so that the equations

x+y+z=1,

x+2y+4z= λ ,

x+4y+10z=λ^{2}

have a solution and solve them completely in each case.(6 marks)
**3(b)** Find the stationary values of

x^{3} + y^{3} - 3axy, a > 0(6 marks)
**3(c)** Separate into real and imaginary parts

tan^{-1}(e^{iθ})(8 marks)
**4(a)** If x = u cos v and y = u sin v

$$\dfrac{\partial(x,y)}{\partial(u,v)}.\dfrac{\partial(u,v)}{\partial(x,y)} =1$$(6 marks)
**4(b)** If tan[log(x + iy)] = a+ib,

$$prove \ that \ tan[log(x^2+y^2)]=\dfrac{2a}{1-a^2-b^2}$$

where a^{2} + b^{2} ≠ 1.(6 marks)
**4(c)** Using Gauss- Seidel iteration method solve,

10x_{1} + x_{2} + x_{3} = 12,

2x_{1} + 10x_{2} + x_{3}=13,

2x_{1} + 2x_{2} + 10x_{3} = 14

Upto three iterations.(8 marks)
**5(a)** In a series of sines of multiple of θ, expand sin^{7} θ(6 marks)
**5(b)** Evaluate the following:

$$\lim_{x\rightarrow 1}\dfrac{x^x-x}{x-1-logx}$$(6 marks)
**5(c)** Prove the following if y^{1/m} + y^{-1/m} = 2x;

(x^{2} -1) y_{(n+2)} + (2n+1)xy_{(n+1)} + (n^{2}-m^{2})y_{n} = 0 (8 marks)
**6(a)** Examine the following vectors for linear dependence/independence

X_{1} = (a,b,c), X_{2} = (b,c,a), X_{3} = (c,a,b)

where a+b+c ≠ to zero. (6 marks)
**6(b)** If z = f(x,y) , x=(e^{u} + e^{-v}), y=(e^{-u} - e^{v})

$$ \dfrac{\partial z}{\partial u}-\dfrac{\partial z}{\partial v} = x\dfrac{\partial z}{\partial x}-y\dfrac{\partial z}{\partial y}$$(6 marks)
**6(c)** Fit a straight line to following data and also estimate the production in 1957.

Year | 1951 | 1961 | 1971 | 1981 | 1991 |

Production in Thousand Tones | 10 | 12 | 8 | 10 | 13 |