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 n^th 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 α³+β³(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 x²u_xx+2xy u_xy + y²u_yy(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^{i?eta} 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 = e^x cos2x cosx find y_n(4 marks) 5(c)(ii) If y = e^tan-1x prove that (1+x²)y_n+2+[2 (n+1) x-1]y_n+1+n (n+1)y_n = 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 :