This set of Numerical Analysis Interview Questions and Answers for freshers focuses on “Bisection Method – 2”.

1. Using Bisection method find the root of 3x^{2} = 5x+2 in the interval [0,3].

a) 0.617

b) 0.527

c) 0.517

d) 0.717

View Answer

Explanation: Function f(x) = 3x

^{2}– 5x – 2 = 0. The Iteration table is given as follows

No. | a | b | c | f(a)*f(c) |

1 | 0 | 3 | 2.25 | Positive |

2 | 1.5 | 3 | 2.25 | Negative |

3 | 1.5 | 2.25 | 1.875 | Positive |

4 | 1.875 | 2.25 | 2.0625 | Negative |

5 | 1.875 | 2.0625 | 1.96875 | Positive |

6 | 1.96875 | 2.0625 | 2.015625 | Negative |

7 | 1.96875 | 2.015625 | 1.9921875 | Positive |

8 | 1.9921875 | 2.015625 | 2.00390625 | Negative |

9 | 1.9921875 | 2.00390625 | 1.99806875 | Positive |

10 | 1.998046875 | 2.00390625 | 2.000976563 | Negative |

The difference between final values is less than 0.01 hence we stop the iterations. So one of the roots of f(x) = 3x^{2} – 5x-2 = 0 is approximately 2.000976563.

2. Find the root of xe^{-x} -0.3 = 0 using Bisection Method in the interval [1,5].

a) 1.68

b) 1.86

c) 1.88

d) 1.66

View Answer

Explanation: Iteration table is given as follows:

No. | a | b | c | f(a)*f(c) |

1 | 1.0 | 5.0 | 3.0 | (-) |

2 | 1.0 | 3.0 | 2.0 | (-) |

3 | 1.0 | 2.0 | 1.5 | (+) |

4 | 1.5 | 2.0 | 1.75 | (+) |

5 | 1.75 | 2.0 | 1.875 | (-) |

6 | 1.75 | 1.875 | 1.8125 | (-) |

7 | 1.75 | 1.8125 | 1.78125 | (+) |

8 | 1.78125 | 1.8125 | 1.796875 | (-) |

9 | 1.78125 | 1.796875 | 1.7890625 | (-) |

After 9 iterations the difference between the last 2 values is less than 0.01. Thus the approximated value of x is 1.7890625.

3. If a function is defined at 2 points 3 and 7 as f(3)=8 and f(7)=12, it is sufficient to find the roots through Bisection Method.

a) True

b) False

View Answer

Explanation: If a function is real and continuous in the region from a to b and f(a) and f(b) and if both have different signs then there is at least one real root between a and b. This is because the function has to cross the x-axis at least once. The given values are both positive so it is insufficient to find out the roots through Bisection method.

4. The function on which we use the bisection method is not necessary to be continuous on the interval we choose.

a) True

b) False

View Answer

Explanation: The Bisection Method states that function under Bisection Method must be continuous on the interval on which the Bisection Method is applied. Moreover the function has to be a real valued to be able to find the roots by Bisection Method.

5. If it is provided that f(3) = 4 is one of the initial points. What can be the choice of second point for solving by Bisection Method?

a) -5 such that f(-5) = -26

b) 0 such that f(0) = 5

c) -3 such that f(-3) = -2

d) 13 such that f(13) = 2

View Answer

Explanation: The Bisection Method involves the procedure of solving a numerical through a bracketing method. This involves defining 2 points where a bracket is small enough and such that f(x) crosses the x-axis at least once in that interval. Hence option -3 such that f(-3) = -2 is more convenient.

6. What is the percentage decrease in an interval containing root after iteration is applied by Bisection Method?

a) 20%

b) 30%

c) 40%

d) 50%

View Answer

Explanation: The Bisection Method employs the reduction of any interval by 50% after each iteration. Hence it is also called as Binary Reduction method.

7. Find the approximated value of x till 6 iterations for x^{3}-4x+9=0 using Bisection Method. Take a = -3 and b = -2.

a) -2.703125

b) -1.903125

c) -3.903125

d) -0.703125

View Answer

Explanation: Iteration table is given as follows:

No. | a | b | c | f(a)*f(c) |

1 | -3.0 | -2.0 | -2.5 | (-) |

2 | -3.0 | -2.5 | -2.75 | (-) |

3 | -2.75 | -2.5 | -2.625 | (-) |

4 | -2.75 | -2.625 | -2.6875 | (-) |

5 | -2.75 | -2.6875 | -2.71875 | (+) |

6 | -2.71875 | -2.6875 | -2.703125 | (-) |

Hence we stop the iterations after 6. Therefore the approximated value of x is -2.703125.

8. The algorithm provided to find the roots of the function using Bisection Method is given by ____________

a) Bolzano’s theorem

b) Mean Value theorem

c) Bisection theorem

d) Secant theorem

View Answer

Explanation: The Bolzano’s theorem states that if a function is real and continuous in the region from a to b and f(a) and f(b) and if both have different signs then there is at least one real root between a and b. This principle or algorithm is used to find the roots of function using Bisection Method.

9. A function is defined as f(x) = x^{3} – x – 11 = 0. Between the interval [2,3] find the root of the function by Bisection Method upto 8 iteration?

a) 1.7334

b) 1.7364

c) 1.7354

d) 1.7344

View Answer

Explanation: The Iteration Table is given as follows:

No. | a | b | c | f(a)*f(c) |

1 | 2.0 | 3.0 | 2.5 | (-) |

2 | 2.0 | 2.5 | 2.25 | (+) |

3 | 2.25 | 2.5 | 2.375 | (+) |

4 | 2.375 | 2.5 | 2.4375 | (-) |

5 | 2.375 | 2.4375 | 2.40625 | (-) |

6 | 2.375 | 2.40625 | 2.390625 | (-) |

7 | 2.375 | 2.390625 | 2.3828125 | (-) |

8 | 2.375 | 2.3828125 | 2.37890625 | (-) |

Thus, with the 9^{th} iteration, we note that the final value. Hence 2.37890625 is our approximation of the root.

10. The Bisection method has which of the following convergences?

a) Linear

b) Quadratic

c) Cubic

d) Quaternary

View Answer

Explanation: Bisection Method has a linear convergence. It is generally ½ for most functions.

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