Question Paper: Applied Mathematics 1 : Question Paper May 2015 - First Year Engineering (Semester 1) | Mumbai University (MU)
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## Applied Mathematics 1 - May 2015

### 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) $$If \ \tan \dfrac {x}{2} = \tan h \ \dfrac {u}{2} \ then \ S.T. \\ u=\log \tan \left ( \dfrac {\pi}{4} + \dfrac {x}{2} \right )$$(3 marks) 1 (b) $$If \ u = x^y \ find \ \dfrac {\partial^3 u}{\partial x \partial y \partial x}$$(3 marks) 1 (c) If ux=yz,vy=zx, wz=xy find $$j \left [ \dfrac {u,v,w}{x,y,z} \right ]$$(3 marks) 1 (d) $$If y = (x-1)^n \ then \ P.T. \ y+ \dfrac {y_1}{1!} + \dfrac{y_2}{2!}+ \dfrac {y_3}{3!}+ \cdots \ \cdots \dfrac {y_n}{n!}= x^n$$(3 marks) 1 (e) $$P.T.\ sinhx = X + \dfrac {x^3}{3!} + \dfrac {x^5}{5!} + \dfrac {x^7}{7!}+$$(4 marks) 1 (f) Express the matrix A as sum of Hermition and skew Hermition matrix where $$\begin{bmatrix} 3i &-1+i &3-2i \\1+i &-i &1+2i \\-3-2i &-1+2i &0 \end{bmatrix}$$(4 marks) 2 (a) Solve x7+x4+i(x3+1)=0(6 marks) 2 (b) Reduce the matrix A to normal form and hence find its rank where $$A=\begin{bmatrix}0 &1 &-3 &-1 \\1 &0 &4 &3 \\3 &1 &0 &2 \\1 &1 &-2 &0 \end{bmatrix}$$(6 marks) 2 (c) State and prove Euler's theorem for three variable and hence find $$x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}+ z \dfrac {\partial u}{\partial z} \ where \\ u= \dfrac {x^3 y^3 z^3}{x^3+ y^3 +z^3}$$(8 marks) 3 (a) Solve the following system of equations
2x-2y-5z=0
4x-y+z=0
3z-2y+3z=0
x-3y+7z=0
(6 marks)
3 (b) Find the maximum and minimum values of
x3+3xy2-3x2-3y2+4
(6 marks)
3 (c) Separate into real and imaginary parts of tanh-1 (x+iy).(8 marks) 4 (a) If u=2xy, v=x2-y2 and x=rcos?, y=rsin? then find $$\dfrac {\partial (u_1v)}{\partial (\partial_1 \theta)}$$(6 marks) 4 (b) If iii... ? =A+i B, prove that $$\tan \left ( \dfrac {\pi A}{2} \right )= \dfrac {B}{A} \ and \ A^2 + B^2 = e^{-\pi B}$$(6 marks) 4 (c) Solve by crouts methods the system of equations
3x+2y+7z=4
2x+3y+z=5
3x+4y+z=7.
(8 marks)
5 (a) By using De Moivre's theorem Express $$\dfrac {\sin 7\theta }{\sin \theta}$$ in powers of sinθ only.(6 marks) 5 (b) By using Taylor's series expand tan-1 x in positive powers of (x-1) upto first four non-zero terms.(6 marks) 5 (c) if y=sin [log (x2+2x+1)] prove that (x+1)2 yn+2 + (2n+1 (x+1) )yn+1+ (n2+4)yn=0(8 marks) 6 (a) Determine linear dependance or independance of vectors
x1=[1,3,4,2] x2==[3,-5,2,6]
x=[2,-1,3,4] and if dependent find the relation between them.
(6 marks)
6 (b) If u =x2-y2, v=2xy and z=f(u,v) prove that $$\left ( \dfrac {\partial z}{\partial x} \right )^2 + \left ( \dfrac {\partial z}{\partial y} \right )^2 = 4\sqrt{u^2+v^2} \left [ \left ( \dfrac{\partial z}{\partial u} \right )^2 + \left ( \dfrac {\partial z}{\partial v} \right )^2 \right ]$$(6 marks) 6 (c) Evaluate $$i) \ \lim_{x\to 0} \dfrac {\sin x. \sin^{-1} x-x^2}{x^6}$$ ii) Fit straight line to the following data
(x,y)= (-1, -5), (1,1), (2,4), (3,7), (4, 10)
Estimate y when x=7.
(4 marks)