Numerical Methods and Calculus

Question 21

• I only
• II only
• Both I and II
• Neither I nor II

Question 22

A non-zero polynomial f(x) of degree 3 has roots at x = 1, x = 2 and x = 3. Which one of the following must be TRUE?

• f(0)f(4) < 0
• f(0)f(4) > 0
• f(0) + f(4) < 0
• f(0) + f(4) > 0

Question 23

In the Newton-Raphson method, an initial guess of x0 = 2 is made and the sequence x0, x1, x2 … is obtained for the function

0.75x3 – 2x2 – 2x + 4 = 0

Consider the statements
(I) x3 = 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 24

If [Tex]\\int_{0}^{2\\Pi} |x sinx|dx = k \\Pi,[/Tex] then the value of k is equal to __________.

• 2
• 3
• 4
• 5

Question 25

• I Only
• II Only
• Both I and II
• Neither I or II

Question 26

The value of integral given below is: [Tex]\\int_{0}^{\\pi} x^2 cosx dx[/Tex] (A)[Tex]-2 \\pi[/Tex] (B)[Tex]\\pi[/Tex] (C)[Tex]-\\pi[/Tex] (D)[Tex]2\\pi[/Tex]

• A
• B
• C
• D

Question 27

Let G(x) = 1/(1 - x)² = , where | x | < 1. What is g(i) ?

• i
• i+1
• 2i
• 2ⁱ

Question 28

Find the value of [Tex]\\lim_{x\\rightarrow 0}\\frac{tanx-x}{x^3}[/Tex]

• 1/3
• -1/6
• 1/2
• None of these

Question 29

A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale). If we use the Newton-Raphson 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 30

The trapezoidal rule for integration give exact result when the integrand is a polynomial of degree:

• 0 but not 1
• 1 but not 0
• 0 or 1
• 2