NewtonRaphsonMethod
Question 1 
In the NewtonRaphson method, an initial guess of x0 = 2 is made and the sequence x0, x1, x2 … is obtained for the function
0.75x^{3} – 2x^{2} – 2x + 4 = 0 Consider the statements (I) x_{3} = 0. (II) The method converges to a solution in a finite number of iterations.Which of the following is TRUE?
Only I  
Only II  
Both I and II  
Neither I nor II 
Question 1 Explanation:
Note: Numerical methods are not in GATE CS syllabus.
Question 2 
NewtonRaphson method is used to compute a root of the equation x^{2}  13 = 0 with 3.5 as the initial value. The approximation after one iteration is
3.575  
3.676  
3.667  
3.607

Question 2 Explanation:
Note: Out of syllabus.
Question 3 
square of R
 
reciprocal of R  
square root of R
 
logarithm of R

Question 3 Explanation:
Note: Out of syllabus.
Question 4 
Consider the function f(x) = x^{2}  2x  1. Suppose an execution of the Newton
Raphson method to find a zero of f(x) starts with an approximation x_{0} = 2 0f x.
What is the value of x_{2}, 'the approximation of x' that the algorithm produces after two iterations, rounded to three decimal places?
2.417  
2.419  
2.423  
2.425 
Question 4 Explanation:
Note: Out of syllabus.
Question 5 
1.5
 
√2
 
1.6  
1.4 
Question 5 Explanation:
Given series is
x_{n+1}=x_{n}/2+9/8x_{n} ⟶ (I); x_{0}=0.5
Equation based on NewtonRapson is
x_{n+1}=x_{n}f(x_{n})/f'(x_{n})⟶ (II)
Equate I and II
x_{n}f(x_{n})/f'(x_{n})=x_{n}/2+9/8x_{n}
x_{n}f(x_{n})/f'(x_{n})=x_{n}x_{n}/2+9/8x_{n}
x_{n}f(x_{n})/f'(x_{n})=x_{n}(4x^{n2}9)/8x_{n}
So, f(x)=4x^{n2}9
4x^{2}9=0
4x^{2}=9
x^{2}=9/4
x=±3/2
x=±1.5
Equation based on NewtonRapson is
x_{n+1}=x_{n}f(x_{n})/f'(x_{n})⟶ (II)
Equate I and II
x_{n}f(x_{n})/f'(x_{n})=x_{n}/2+9/8x_{n}
x_{n}f(x_{n})/f'(x_{n})=x_{n}x_{n}/2+9/8x_{n}
x_{n}f(x_{n})/f'(x_{n})=x_{n}(4x^{n2}9)/8x_{n}
So, f(x)=4x^{n2}9
4x^{2}9=0
4x^{2}=9
x^{2}=9/4
x=±3/2
x=±1.5
Question 6 
A piecewise linear function f(x) is plotted using thick solid lines in the figure below
If we use the NewtonRaphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be
1.3, 0.6, and 0.6 respectively
 
0.6, 0.6, and 1.3 respectively
 
1.3, 1.3, and 0.6 respectively
 
1.3, 0.6, and 1.3 respectively 
Question 6 Explanation:
Note: Out of syllabus.
Question 7 
X^{2} = 3  
X^{3} = 3  
X^{2} = 2  
X^{3} = 2 
Question 7 Explanation:
Newton Rampson formula,
Question 8 
The NewtonRaphson method is to be used to find the root of the equation f(x)=0 where x_{o} is the initial approximation and f' is the derivative of f. The method converges
always  
only if f is a polynomial  
only if f(x_{0}) < 0  
None of the above 
Question 8 Explanation:
Note: Out of syllabus.
Question 9 
The NewtonRaphson method is used to find the root of the equation x^{2} − 2 = 0. If the iterations are started form –1, the iterations will
converge to 1  
converge to √2  
converge to  √2  
not converge 
Question 9 Explanation:
Question 10 
Question 10 Explanation:
Note: Out of syllabus.
Question 11 
The iteration formula to find the square root of a positive real number b using the Newton Raphson method is
None of the above 
Question 11 Explanation:
Note: Out of syllabus.
There are 11 questions to complete.