MU Applied Mathematics 1 - December 2011 Exam Question Paper

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})} i s 2 e^{- π / \sqrt{3}} \cos (\frac{π}{3} + \sqrt{3} \cdot \log 2)

$(1+i\sqrt{3})^{(1+l\sqrt{3})} \ is \ 2e^{-\pi/\sqrt{3}} \ \cos \left (\dfrac {\pi}{3}+\sqrt{3}\cdot \log 2 \right )$

5 M

1(d) Test the convergence of

\frac{x}{1.2} + \frac{x^{2}}{3.4} + \frac{x^{3}}{5.6} + \frac{x^{4}}{7.8} + . . . . . . (x > 0, x \neq 1)

$\dfrac{x}{1.2}+\dfrac{x^2}{3.4}+\dfrac{x^3}{5.6}+\dfrac{x^4}{7.8}+......(x > 0, \ x \neq 1 )$

5 M

2(a) If b+ic = (1+a)z and a² + b² + c² = 1 then

P r o v e t h a t = \frac{a + i b}{1 + c} = \frac{1 + i z}{1 - i z}

$Prove \ that = \dfrac{a+ib}{1+c}=\dfrac{1+iz}{1-iz}$

6 M

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 M

2(c) If u=f(e^(y-z),e^(z-x),e^(x-y)) then

P r o v e t h a t = \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0

$Prove \ that = \dfrac{{\partial}u}{{\partial}x}+\dfrac{{\partial}u}{{\partial}y}+\dfrac{{\partial}u}{{\partial}z} =0$

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:

t a n^{- 1} i (\frac{x - a}{x + a}) = \frac{i}{2} l o g (\frac{x}{a})

$tan^{-1}i{\displaystyle{ \Big(\dfrac{x-a}{x+a}}\Big)} = \dfrac{i}{2}log{\Big(\dfrac{x}{a}\Big)}$

8 M

4(a) Prove the following:
x²y_(n+2) + (2n+1)xy_(n+1) + 2n² y_n = 0

i f c o s^{- 1} (\frac{y}{b}) = l o g (\frac{x}{n}) n

$if \ \ \ cos ^{-1} \Big(\dfrac{y}{b} \Big) = log\Big(\dfrac{x}{n} \Big)n$

6 M

4(b) If z = tan(y+ax) + (y-ax)^(3/2)

\frac{\partial^{2} z}{\partial x^{2}} = a^{2} \frac{\partial^{2} z}{\partial y^{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

2 x \frac{\partial z}{\partial y} - y \frac{\partial z}{\partial y} = 4 x

$2x\dfrac{\partial z}{\partial y} -y\dfrac{\partial z}{\partial y}=4x$

8 M

5(a) Separate into real and imaginary parts
cos^-1(3i/4).

6 M

5(b) Prove the following:

x \frac{\partial u}{\partial x} = y \frac{\partial u}{\partial y} = z \frac{\partial u}{\partial z} = 0 i f u = f (\frac{x}{y}, \frac{y}{z}, \frac{z}{x})

$x\dfrac{\partial u}{\partial x} =y\dfrac{\partial u}{\partial y}=z\dfrac{\partial u}{\partial z} = 0 \ \ \ \ if \ \ \ u = f\Big(\dfrac{x}{y},\dfrac{y}{z},\dfrac{z}{x} \Big)$

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 = [\frac{x^{2}}{2} + \frac{x^{4}}{12} + \frac{x^{6}}{45} . . .]

$\log \ \sec \ x = \Big [ \dfrac{x^2}{2}+\dfrac{x^4}{12}+\dfrac{x^6}{45}... \Big]$

6 M

6(c) Evaluate the following:

lim_{x \to 0} \frac{e^{x} s i n x - x - x^{2}}{x^{2} + x l o g (1 - x)}

$\displaystyle \lim_{x\ \to \ 0} \ \dfrac{e^x sinx-x-x^2}{x^2+xlog(1-x)}$

8 M

7(a) If f(x,y) = 0 and ϕ(y,z) = 0 then prove that

\frac{\partial f}{\partial y} . \frac{\partial φ}{\partial z} . \frac{d z}{d x} = \frac{\partial f}{\partial x} . \frac{\partial φ}{\partial y}

$\dfrac{\partial f}{\partial y}.\dfrac{\partial \varphi}{\partial z}.\dfrac{dz}{dx} = \dfrac{\partial f}{\partial x}.\dfrac{\partial \varphi}{\partial y}$

6 M

7(b) Find (1.04)^3.01 by using theory of approximation.

6 M

7(c) Prove the following:

[\bar{b} \times \bar{a} \bar{a} \times \bar{c} \bar{a} \times \bar{b}] = [\bar{a} \bar{b} \bar{c}]^{2}

$\Big [\bar b \times\bar a\ \ \ \bar a \times\bar c \ \ \ \bar a \times \bar b \Big] = \Big[ \bar a \ \ \bar b \ \ \bar c \Big]^2$

8 M

More question papers from Applied Mathematics 1

SPONSORED ADVERTISEMENTS