Question
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Find the real root of the following polynomial equation using any method. Express the answer accurate to three decimal places. 4x4-12x3 +13x2-12x + 9 = 0 (15 marks) Question 2 The paraboloid the sphere xy +21, and the plane z 2x+y intersect near the point (2,0,3). Perform two iterations of the three-dimensional Newton method to obtain a better estimate of the point of intersection. Two iterations means starting with (2,0,3) as a first approximation, find a second approximation, then use the second approximation to find a third approximation. Solve matrix systems using any method and work to 3 decimal places. (35 marks) Question 3 The function f(x) = xsin(2x) is zero at the origin and is zero again at multiples ofx2 Use first the Trapezoidal rule, then Simpson's 1/3 rule, with a 0.3 step size, to approximate the integral under the first 'hump', i.e. up to x 1.5. Repeat the Trapezoidal rule to get the integral from x = 0 to x-3 using a 0.5 step size. (20 marks) Question4 Obtain a least squares exponential fit for the following data. Temperature, T (F) Solubility, S (%) 100 185 239 285 2.4 3.4 7.0 11.1 19.6 (15 marks Question 5 The following system of differential equations describes the fuel concentration and temperature in the ignition stroke of a combustion engine: Using the initial conditions shown, solve the equations using the Euler method for 0SIS1 Ar0.2
Explanation / Answer
Question 1)
Answer : We will solve this by factoring method.
We have to find the real roots of given given polynomial for that set given polynomial equal to 0.
4x4 -12x3 +13x2 - 12x + 9 = 0
This can be rewritten as : 4x4 + 4x2 - 12x3 - 12x + 9x2 + 9 = 0
Now groupping first, middle and last two terms we get and factoring common factor we get :
4x2(x2 + 1) - 12x(x2 + 1) + 9(x2 + 1) = 0
Taking common (x2 + 1) we get :
(4x2 -12x + 9)(x2 + 1) = 0
Here (4x2 -12x + 9) = (2x - 3)2
So then we have : (2x - 3)2(x2 + 1) = 0
This gives you (2x - 3)2 = 0 and (x2 + 1) = 0
2x - 3 = 0 and x2 = -1
x = 3/2 and x = i, -i, (x = i and -i are the complex roots)
Real root of given polynomial is x = 3/2.
