MU Applied Mathematics 1 - December 2015 Exam Question Paper

MU First Year Engineering (Semester 1)
Applied Mathematics 1
December 2015

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 log tan x=y then prove that sinh(n+1)y+sinh(n-1)y=2 sinh ny⋅cosec2x

3 M

1 (b) If z=log (tan x+tan y) then prove that

\sin 2 x \frac{\partial z}{\partial x} + \sin 2 y \frac{\partial z}{\partial y} = 2

$\sin 2x \dfrac {\partial z}{\partial x} + \sin 2y \dfrac {\partial z}{\partial y} = 2$

3 M

1 (c) If x=r sin θ cos φ, y=r sin θ φ, z=r cos θ, then find

\frac{\partial (r, θ, φ)}{\partial (x, y, z)}

$\dfrac {\partial (r, \theta , \varphi)}{\partial (x,y,z) }$

3 M

1 (d) Prove that

\log \sec x = \frac{x^{2}}{2} + \frac{x^{4}}{12} + \frac{x^{6}}{45} + \dots \dots

$\log \sec x=\dfrac {x^2}{2}+ \dfrac {x^4}{12}+ \dfrac {x^6}{45}+ \cdots \ \cdots$

3 M

1 (e) Find the values of a,b,c and A^-1 when

A = \frac{1}{9} [\begin{matrix} - 8 & 4 & a \\ 1 & 4 & b \\ 4 & 7 & c \end{matrix}]

$A=\dfrac {1}{9} \begin{bmatrix} -8 &4 &a \\ 1 &4 &b \\4 &7 &c \end{bmatrix}$ is or orthogonal.

4 M

1 (f) If y=sinθ+cos θ then prove that

y_{n} = r^{n} \sqrt{1 + (- 1)^{n} \sin 2 θ}

$y_n = r^n \sqrt{1+(-1)^n \sin 2 \theta}$ where θ rx

4 M

2 (a) If z=-1+i√3 then prove that

{(\frac{z}{2})}^{n} + {(\frac{2}{z})}^{n} = {\begin{matrix} 2, & if n is multiple of 3 \\ - 1, & if n is not multiple of 3 \end{matrix}

$\left ( \dfrac {z}{2} \right )^n + \left (\dfrac {2}{z} \right )^n = \left\{\begin{matrix} 2 ,& \text{if n is multiple of 3} \ \ \ \ \ \\-1 , & \text {if n is not multiple of 3} \end{matrix}\right.$

6 M

2 (b)

if A = [\begin{matrix} 1 & 2 & - 2 \\ - 1 & 3 & 0 \\ 0 & - 2 & 1 \end{matrix}]

$\text {if} A=\begin{bmatrix} 1 &2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix}$ then find two non-singular matrices P&Q such that PAQ is in normal form also find ρ(A) and A^-1.

6 M

2 (c) State and prove Euler's theorem for functions of two independent variable hence prove that

(x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}) (x \frac{\partial v}{\partial x} + y \frac{\partial v}{\partial y}) = 0

$\left ( x \dfrac {\partial u}{\partial x}+ y \dfrac {\partial u} {\partial y} \right ) \left( x \dfrac {\partial v} {\partial x} + y\dfrac {\partial v} {\partial y} \right ) = 0$ if x=e^u \tan v, y=e^u, sec v.

8 M

3 (a) Determine the values of a and b such that system

{\begin{matrix} 3 x - 2 y + z = b \\ 5 x - 8 y + 9 z = 3 \\ 2 x + y + a z = - 1 \end{matrix}

$\left\{\begin{matrix} 3x - 2y+z=b \\ 5x-8y+9z=3 \\ 2x+y+az=-1 \end{matrix}\right.$ has i) no solution, ii) a unique solution, iii) infinite number of solutions

6 M

3 (b) Discuss the maximum and minimum of f(x,y)=x³+3xy²-15(x²+y²)+72x

6 M

3 (c) show that

\tan^{- 1} (\frac{x + i y}{x - i y}) = \frac{π}{4} + \frac{i}{2} \log (\frac{x + y}{x - y})

$\tan^{-1} \left ( \dfrac {x+iy}{x-iy} \right ) = \dfrac {\pi}{4} + \dfrac {i}{2}\log \left ( \dfrac {x+y}{x-y} \right )$

8 M

4 (a) If u=xyz, v=x²+y²+x², w=x+y+z then prove that

\frac{\partial x}{\partial u} = \frac{1}{(x - y) (x - z)}

$\dfrac {\partial x}{\partial u} = \dfrac {1}{(x-y)(x-z)}$

6 M

4 (b)

if {\sqrt{i}}^{{\sqrt{i}}^{\sqrt{i} \dots \dots \infty}} = α + i β

$\text {if} \sqrt {i}^{\sqrt{i}^{\sqrt{i}\cdots \cdots \infty}} = \alpha + i\beta$ then prove that

i) α^{2} + β^{2} = e \frac{π β}{2} i i) \tan^{- 1} (\frac{β}{α}) = \frac{π α}{4}

$i) \ \alpha^2 + \beta^2 = e\dfrac {\pi \beta}{2} \\ ii) \ \tan^{-1}\left ( \dfrac {\beta}{\alpha} \right ) = \dfrac {\pi \alpha} {4}$

6 M

4 (c) Apply Crout's method to solve

{\begin{matrix} x - y + 2 z = 2 \\ 3 x + 2 y - 3 z = 2 \\ 4 x - 4 y + 2 z = 2 \end{matrix}

$\left\{\begin{matrix} x-y + 2z=2 \ \ \\ 3x+2y-3z=2 \\ 4x -4y+2z=2 \end{matrix}\right.$

8 M

5 (a) If cos⁶θ+sin⁶θ = α cos 4θ+β then prove that α+β=1.

6 M

5 (b) Find the values of a,b & c such that

lim_{x \to 0} \frac{a e^{x} - b e^{- x} + c x}{x - \sin x} = 4

$\lim_{x\to 0} \dfrac {a e^x - be^{-x}+cx}{x-\sin x}=4$

6 M

5 (c)

if x = \cos [\log (y^{1 / m})]

$\text {if } x =\cos \Big [ \log \big (y^{1/m} \big) \Big ]$ then prove that
(1-x²)y_n+2-(2n+1)xy_n+1-(m²+n²)y_n = 0

8 M

6 (a) Define linear dependence and independence of vectors, Examine for linear dependence of following set of vectors and find the relation between them if dependent

X_{1} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}], X_{2} = [\begin{matrix} 2 \\ 1 \\ 1 \end{matrix}], X_{3} = [\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}]

$X_1 = \begin{bmatrix}1\\-1 \\1\end{bmatrix}, X_2 = \begin{bmatrix}2\\1 \\1\end{bmatrix}, X_3 = \begin{bmatrix}3\\0 \\2\end{bmatrix}$

6 M

6 (b) If z=f(u,v), u=x²-y², v=2xy then prove that

\frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial y^{2}} = 4 \sqrt{u^{2} + v^{2}} (\frac{\partial^{2} z}{\partial u^{2}} + \frac{\partial^{2} z}{\partial v^{2}})

$\dfrac {\partial ^2 z}{\partial x^2} + \dfrac {\partial ^2 z} {\partial y^2} = 4 \sqrt{u^2 + v^2} \left ( \dfrac {\partial ^2 z} {\partial u^2} + \dfrac {\partial ^2 z}{\partial v^2} \right )$

6 M

6 (c) Fit a straight line passing through points (0,1), (1,2), (2,3), (3,4,5), (4,6), (5,7,5).

8 M

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