Algebraic & Transcendental Equations
We will cover following topics
PYQs
Regula-Falsi Method
1) Develop an algorithm for Regula-Falsi method to find a root of f(x)=0 starting with two initial iterates x0 and x1 to the root such that sign (f(x0))≠sign(f(x1)). Take n as the maximum number of iterations allowed and epsilon be the prescribed error.
[2009, 30M]
2) Find the smallest positive root of equation xex−cosx=0 using Regula-Falsi method. Do three iterations.
[2008, 12M]
3) Use the method of false position to find a real root of x3−5x−7=0 lying between 2 and 3 and correct to 3 places of decimals.
[2007, 12M]
4) Find a real root of the equation f(x)=x3−2x−5=0 by the method of false position.
[2002, 12M]
Newton-Raphson Method
1) Apply Newton-Raphson method, to find a real root of transcendental equation xlog10x=1.2, correct to three decimal places.
[2019, 10M]
2) Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.
[2017, 15M]
3) Apply Newton-Raphson method to determine a root of the equation cosx−xex=0 correct up to four decimal places.
[2014, 10M]
4) Develop an algorithm for Newton-Raphson method to solve f(x)=0 starting with initial iterate x0, n be the number of iterations allowed, epsilon be the prescribed relative error and delta be the prescribed lower bound for f′(x).
[2013, 20M]
5) Use Newton-Raphson method to find the real root of the equation 3x=cosx+1 correct to four decimal places.
[2012, 12M]
6) Find the positive root of the equation 10xe−x2−1=0 correct up to 6 decimal places by using Newton-Raphson method. Carry out computations only for three iterations.
[2010, 12M]
7) Draw a flow chart for solving equation F(x)=0 correct to five decimal places by Newton-Raphson method.
[2008, 30M]
8) How many positive and negative roots of the equation ex−5sinx=0 exist? Find the smallest positive root correct to 3 decimals, using Newton-Raphson method.
[2004, 10M]
9) Find the positive root of the equation 2e−x=1x+2+1x+1 using Newton-Raphson method correct to four decimal places. Also show that the following scheme has error of second order:
xn+1=12xn(1+ax2n).
[2003, 30M]