Applied Mathematics 1 - Dec 2011

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 arg(z+1) = π/6 and arg(z-1) = 2π/3 find z, a complex number.(5 marks) 1(b) Prove that tanh^-1 (sinθ) = cosh^-1 (secθ)(5 marks) 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 marks) 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 marks) 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 marks) 2(b) Find the roots α, α², α³,α⁴ of the equation x⁵ - 1 = 0 and show that
(1-α)(1-α²)(1-α³)(1-α⁴) = 5(6 marks) 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 marks) 3(a) Prove that arg z₁ - arg z₂ = π/2 if
|z₁ + z₂ |= |z₁ - z₂ | where z₂ , z₁ are complex numbers.(6 marks) 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 marks) 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 marks) 4(a) Prove the following:
x²y_(n+2) + (2n+1)xy_(n+1) + 2n² y_n = 0
$$if \ \ \ cos ^{-1} \Big(\dfrac{y}{b} \Big) = log\Big(\dfrac{x}{n} \Big)n$$(6 marks) 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 marks) 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 marks) 5(a) Separate into real and imaginary parts
cos^-1(3i/4).(6 marks) 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 marks) 5(c) Examine the function
f(x,y) = y² + 4xy + 3x² + x³ for extreme values.(8 marks) 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 marks) 6(b) Prove the following:
$$\log \ \sec \ x = \Big [ \dfrac{x^2}{2}+\dfrac{x^4}{12}+\dfrac{x^6}{45}... \Big]$$(6 marks) 6(c) Evaluate the following:
$$ \displaystyle \lim_{x\ \to \ 0} \ \dfrac{e^x sinx-x-x^2}{x^2+xlog(1-x)}$$(8 marks) 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 marks) 7(b) Find (1.04)^3.01 by using theory of approximation.(6 marks) 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 marks)