Question Paper: Applied Mathematics 1 : Question Paper May 2016 - First Year Engineering (Semester 1) | Mumbai University (MU)
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Applied Mathematics 1 - May 2016

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) Solve the equation 7coshx+8sinhx = 1 for real values of x.(3 marks) 1(b) If ( z(x+y)=(x-y) ext{find}left ( dfrac{partial z}{partial x} -dfrac{partial z}{partial y} ight )^2 )(3 marks) 1(c) If ( u=r^2cos 2 heta, v=r^2sin 2 heta ext {find}dfrac{partial (u,v)}{partial (r, heta)} )(3 marks) 1(d) Prove that ( sec^2x=1+x^2+dfrac{2x^4}{3}+cdots )(3 marks) 1(e) find the rank of the Matrix by reducing it to normal form. [egin{bmatrix} 1 & 1 & 1\ 1 & -1 & -1\ 3 & 1 & 1 end{bmatrix}](3 marks) 1(f) Find nth derivatives of (dfrac{x}{(x-1)(x-2)(x-3)} )(3 marks) 2(a) If α, β are the roots of the equation ( x^2-2sqrt{3}.x+4=0 )
find the value of α33
(6 marks)
2(b) Examine whether the vectors
X1 = [3 1 1], X2 = [2 0 -1], X3 = [4 2 1] are linearly independent.
(6 marks)
2(c)(i) State an prove Euler's theorem for a homogeneous function in two variables.(4 marks) 2(c)(ii) If y=xcosu
Find the value of x2uxx+2xy uxy + y2uyy
(4 marks)
3(a) Is the following system has trivial or non trivial solution? Obtain the non trival solution if exist.
3x1 + 4x2 - x3 - 9x4 = 0
2x1 + 3x2 + 2x3 - 3x4 = 0
2x1 + x2 - 14x3 - 12x4 = 0
x1 + 3x2 + 13x3 + 3x4 = 0
(6 marks)
3(b) Discuss the stationary points for Maxima and Minima of
x3 + xy2 - 12x2 - 2y2 + 21x + 10
(6 marks)
3(c)(i) If tan (x+iy) = a+ib prove that ( anh 2y=dfrac{2b}{1+a^2+b^2} )(4 marks) 3(c)(ii) Separate into real and imaginary parts of Log (3+4i)(4 marks) 4(a) if x = u cosv, y = u sinv
( ext {Prove that}dfrac{partial (u,v)}{partial (x,y)},dfrac{partial (x,y)}{partial (u,v)}=1 )
(6 marks)
4(b) Show that ( logleft [ e^{ialpha}+e^{ieta} ight ]=logleft [ 2cosleft ( dfrac{alpha -eta }{2} ight ) ight ]+ileft ( dfrac{alpha +eta }{2} ight ) )(6 marks) 4(c)(i) Solve the system of equation by Gauss Jordan Method
x + 2y + 6z = 22, 3x + 4y + z = 26, 6x - y - z = 19
(4 marks)
4(c)(ii) Solve the system of equation by Gauss Siedel Method.
2x - 4y + 49z = 49
43x + 2y + 25z = 23
3x + 53y + 3z = 91
(4 marks)
5(a) Prove that ( cos^6 heta+sin^6 heta=dfrac{1}{8}[3cos 4 heta+5] )(6 marks) 5(b) Find the value of a and b [ ext{if}lim_{x ightarrow 0}dfrac{x(1+acos x)-bsin x}{x^3}=1](4 marks) 5(c)(i) if y = ex cos2x cosx find yn(4 marks) 5(c)(ii) If y = etan-1x prove that (1+x2)yn+2+[2 (n+1) x-1]yn+1+n (n+1)yn = 0(4 marks) 6(a) Find non-Singular Matrices P & Q such that, ( A=egin{bmatrix} 1 & 2 & 3 & 4\ 2 & 1 & 4 & 3\ 3 & 0 & 5 & -10 end{bmatrix} ) is reduced to normal form. Also find rank.(6 marks) 6(b) ( u=f(e^{y-z},e^{z-x},e^{x-y}) ext{find} dfrac{partial u}{partial x}+dfrac{partial u}{partial y}+dfrac{partial u}{partial z} )(6 marks) 6(c)(i) Fit a straight line to the following data :

Year x: 1951 1961 1971 1981 1991
Production y: 10 12 8 10 15
(4 marks) 6(c)(ii) Fit a second degree parabolic curve to the following data :
x : 1 2 3 4 5 6 7 8 9
y : 2 6 7 8 10 11 11 10 9
(4 marks)

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