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MU First Year Engineering (Semester 1)
Applied Mathematics 1
December 2011
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(a) If arg(z+1) = π/6 and arg(z-1) = 2π/3 find z, a complex number.
5 M
1(b) Prove that tanh-1 (sinθ) = cosh-1 (secθ)
5 M
1(c) Prove that real part of $(1+i\sqrt{3})^{(1+l\sqrt{3})} \ is \ 2e^{-\pi/\sqrt{3}} \ \cos \left (\dfrac {\pi}{3}+\sqrt{3}\cdot \log 2 \right )$
5 M
1(d) Test the convergence of
$\dfrac{x}{1.2}+\dfrac{x^2}{3.4}+\dfrac{x^3}{5.6}+\dfrac{x^4}{7.8}+......(x > 0, \ x \neq 1 )$
5 M

2(a) If b+ic = (1+a)z and a2 + b2 + c2 = 1 then
$Prove \ that = \dfrac{a+ib}{1+c}=\dfrac{1+iz}{1-iz}$
6 M
2(b) Find the roots α, α2, α34 of the equation x5 - 1 = 0 and show that
(1-α)(1-α2)(1-α3)(1-α4) = 5
6 M
2(c) If u=f(e(y-z),e(z-x),e(x-y)) then
$Prove \ that = \dfrac{{\partial}u}{{\partial}x}+\dfrac{{\partial}u}{{\partial}y}+\dfrac{{\partial}u}{{\partial}z} =0$
8 M

3(a) Prove that arg z1 - arg z2 = π/2 if
|z1 + z2 |= |z1 - z2 | where z2 , z1 are complex numbers.
6 M
3(b) Prove that αn + βn = 2cos n θ cosecn θ
if α and β roots of the equation
z2sin2 θ - z sin2θ + 1 = 0 .
6 M
3(c) Show the following:
$tan^{-1}i{\displaystyle{ \Big(\dfrac{x-a}{x+a}}\Big)} = \dfrac{i}{2}log{\Big(\dfrac{x}{a}\Big)}$
8 M

4(a) Prove the following:
x2y(n+2) + (2n+1)xy(n+1) + 2n2 yn = 0
$if \ \ \ cos ^{-1} \Big(\dfrac{y}{b} \Big) = log\Big(\dfrac{x}{n} \Big)n$
6 M
4(b) If z = tan(y+ax) + (y-ax)(3/2)
$\dfrac{\partial^2z}{\partial x^2} =a^2 \dfrac{\partial^2z}{\partial y^2}$
6 M
4(c) If the given function, f(xy2,z - 2x) = 0, thenprove that
$2x\dfrac{\partial z}{\partial y} -y\dfrac{\partial z}{\partial y}=4x$
8 M

5(a) Separate into real and imaginary parts
cos-1(3i/4).
6 M
5(b) Prove the following:
$x\dfrac{\partial u}{\partial x} =y\dfrac{\partial u}{\partial y}=z\dfrac{\partial u}{\partial z} = 0 \ \ \ \ if \ \ \ u = f\Big(\dfrac{x}{y},\dfrac{y}{z},\dfrac{z}{x} \Big)$
6 M
5(c) Examine the function
f(x,y) = y2 + 4xy + 3x2 + x3 for extreme values.
8 M

6(a) Find x, if a = xi + 12j - k; b = 2i + 2j + k; c = i + k are coplanar. Also find unit vector in the direction of a
6 M
6(b) Prove the following:
$\log \ \sec \ x = \Big [ \dfrac{x^2}{2}+\dfrac{x^4}{12}+\dfrac{x^6}{45}... \Big]$
6 M
6(c) Evaluate the following:
$\displaystyle \lim_{x\ \to \ 0} \ \dfrac{e^x sinx-x-x^2}{x^2+xlog(1-x)}$
8 M

7(a) If f(x,y) = 0 and ϕ(y,z) = 0 then prove that
$\dfrac{\partial f}{\partial y}.\dfrac{\partial \varphi}{\partial z}.\dfrac{dz}{dx} = \dfrac{\partial f}{\partial x}.\dfrac{\partial \varphi}{\partial y}$
6 M
7(b) Find (1.04)3.01 by using theory of approximation.
6 M
7(c) Prove the following:
$\Big [\bar b \times\bar a\ \ \ \bar a \times\bar c \ \ \ \bar a \times \bar b \Big] = \Big[ \bar a \ \ \bar b \ \ \bar c \Big]^2$
8 M

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