MCQ ON THE SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

MCQ ON THE SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS:

MCQ on the solution of algebraic and transcendental equations are discussed here with answers underlined. You can test your knowledge. 

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Question 1: The Iterative formula for Newton Raphson method is

A) `x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}`

B) `x_{n+1} = \frac{x_n - f(x_n)}{f’(x_n)}`

C) `x_{n+1} = \frac{x_n + f(x_n)}{f’(x_n)}`

D) None

Explanation: `x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}`.

Question 2: In Newton Raphson method if the `f(x)` is constant then `\ldots`

A) `f''(x)=0`

B) `f(x)=0`

C) `f'(x)=0`

D) `f'(x)=C`

Explanation: `f'(x)=0`

Question 3: The Newton-Raphson method fails if `\ldots`

A) `f'(x_0)=0`

B) `f''(x_0)=0`

C) `f(x_0)=0`

D) `f'''(x_0)=0`

Explanation: `f'(x_0)=0`.

Question 4: The Newton-Raphson method is also called as `\ldots`

A) Tangent method

B) Secant method

C) Chord Method

D) None

Explanation: tangent method

Question 5: The formula of the Bisection method is

A) `X= \frac{(a+b)}{2}`

B) `X= a+\frac{b}{2}`

C) `X=\frac{ab}{2}`

D) None

Explanation: `X= \frac{(a+b)}{2}`

Question 6: The Newton-Raphson method applies to the solution of

A) Both algebraic and transcendental equations

B) Both algebraic and transcendental and also when the roots are complex

C) Algebraic equations only

D) Transcendental equations only

Explanation: Both algebraic and transcendental equations

Question 7: `\ldots` lies in the category of iterative method.

A) Bisection method

B) Regula Falsi method

C) Secant Method

D) All of these

Explanation: All of these.

Question 8: Find the interval of the equation x- cosx=0 using the bisection method

A) `(0.5, 1)`

B) `(0, 0.5)`

C) `(1,2)`

D) None

Explanation: `(0.5, 1)`

Question 9: Which method is faster than the Bisection method.

A) Gauss-Elimination

B) Gauss- Jordan

C) Runge-Kutta method

D) Regula- Falsi method

Explanation: Regula- Falsi method

Question 10: In which of the following methods proper choice of initial value is very important?

A) Newton Raphson Method

B) Bisection Method

C) Regula -Falsi

D) Iterative method

Explanation: Newton Raphson Method

Question 11: For the solution of non-linear equations which method uses the midpoint procedure?

A) Bisection Method

B) Regula -Falsi

C) Newton Raphson Method

D) None

Explanation: Bisection Method.

Question 12: Which is the faster method?

A) Newton Raphson Method

B) Bisection Method

C) Regula -Falsi

D) None

Explanation: Newton Raphson Method

Question 13: False Position method is also known as

A) Regula Falsi method

B) Newton Raphson method

C) Bisection method

D) None

Explanation: Regula Falsi Method

Question 14: The root of a non-linear lies in the interval `(a,b)` when

A) `f(a)` and `f(b)` have same sign

B) `f(a)` and `f(b)` have opposite sign

C) None

D) Both A) and B)

Explanation: `f(a)` and `f(b)` have opposite sign

Question 15: The order of convergence of the Regula-Falsi method is

A) 1.618

B) 1.5

C) 1.321

D) 2.371

Explanation: 1.618.

Question 16: Which of the following statements applies to the bisection method used for finding roots of functions?

A) Guaranteed to work for all continuous functions

B) converges within a few iterations

C) Is faster than the Newton-Raphson method

D) None

Explanation: Guaranteed to work for all continuous functions

Question 17: Iterative formula to find the `\frac{1}{N}` is

A) `x_{n+1}=x_n(2-Nx_n)`

B) `x_{n+1}=\frac{1}{2} (x_n +\frac{N}{x_n})`

C) `x_{n+1}=\frac{1}{2} (x_n +\frac{1}{Nx_n})`

D) `x_{n+1}=\frac{1}{k} ((k-1)x_n +\frac{N}{x_{n}^{k-1}})`

Explanation: `x_{n+1}=x_n(2-Nx_n)`

Question 18: Iterative formula to find the `\sqrt{ N}` is

A) `x_{n+1}=x_n(2-Nx_n)`

B) `x_{n+1}=\frac{1}{2} (x_n +\frac{N}{x_n})`

C) `x_{n+1}=\frac{1}{2} (x_n +\frac{N}{x_n})`

D) `x_{n+1}=\frac{1}{2} (x_n +\frac{1}{Nx_n})`

Explanation: `x_{n+1}=\frac{1}{2} (x_n +\frac{N}{x_n})`

Question 19: Iterative formula to find the `\frac{1}{\sqrt{ N}}` is

A) `x_{n+1}=\frac{1}{2} (x_n +\frac{1}{Nx_n})`

B) `x_{n+1}=\frac{1}{k} ((k-1)x_n +\frac{N}{x_{n}^{k-1}})`

C) `x_{n+1}=\frac{1}{2} (x_n +\frac{N}{x_n})`

D) `x_{n+1}=x_n(2-Nx_n)`

Explanation: `x_{n+1}=\frac{1}{2} (x_n +\frac{1}{Nx_n})`.

Question 20: Iterative formula to find the `{[k]\sqrt{ N}}` is

A) `x_{n+1}=\frac{1}{k} ((k-1)x_n +\frac{N}{x_{n}^{k-1}})`

B) `x_{n+1}=x_n(2-Nx_n)`

C) `x_{n+1}=\frac{1}{2} (x_n +\frac{N}{x_n})`

D) `x_{n+1}=\frac{1}{2} (x_n +\frac{1}{Nx_n})`

Explanation: `x_{n+1}=\frac{1}{k} ((k-1)x_n +\frac{N}{x_{n}^{k-1}})`

Question 21: Equation `x^2-30 =0` real root lies between

A) `(5, 6)`

B) `(6, 7)`

C) `(1, 2)`

D) None

Explanation: `(5, 6)`

Question 22: Newton Raphson method is used to find

A) real and complex roots

B) only real roots

C) Only complex roots`f'(x)=0`

D) `f'(x)=C`

Explanation: real and complex roots

Question 23: The cube root of 41, using Newton-Raphson method is

A) `3.4482`

B) `4.456`

C) `1.3456`

D) None

Explanation: `3.4482`.

Question 24: The negative root of the equation `x^3 -21x+3500=0`, using Newton-Raphson method is

A) `-16.56`

B) `-13.56`

C) 17.67

D) None

Explanation: -16.56

Question 25: Newton Raphson Formula converges

A) Initial approximation is chosen sufficiently close to the root

B) Final approximation is chosen sufficiently close to the root

C) Initial approximation is chosen sufficiently away to the root

D) None

Explanation: Initial approximation is chosen sufficiently close to the root

Question 26: In the following equations which one is the best example of transcendental equation

A) `x^3-3x^2-12=0`

B) `x^2-30=0`

C) `e^x-cosx=0`

D) None

Explanation: `e^x-cosx=0`

Question 27: The root of equation `x^3-2x-1=0`

A) ` 1 and 2`

B) `0 and 1`

C) `2 and 3`

D) None

Explanation: `1 and 2`.

Question 28: The Bisection method is used to solve which type of equations

A) Algebraic equations

B) Transcendental equation

C) Both Algebraic and Transcendental equations

D) None

Explanation: Both Algebraic and Transcendental equations

Question 29: The equation `x logx=1.2` can be solved by

A) Bisection method

B) Regula-Falsi method

C) Newton Raphson method

D) All of these

Explanation: All of these

Question 30: If a function is real and continuous in the region from a to b and f (a) and f (b) have opposite signs then there is no real root between a and b

A) false

B) can't say

C) true

Explanation: true

Question 31: The Bisection method has which of the following convergences

A) Linear

B) Quadratic

C) Cubic

D) None

Explanation: Linear

Question 32: The formula used for solving the equation using the Regula-falsi method is `x_2 = \frac{x_1f(x_0)-x_0f(x_1)}{f(x_0)-f(x_1)}`

A) True

B) False

C) Can't say

Explanation: true

Question 33: The rate of convergence of Newton-raphson method is

A) Quadratic

B) Linear

C) Cubic

D) None

Explanation: Quadratic

Question 34: The Newton-Raphson method of finding roots of nonlinear equations falls under the category of which of the following methods?

A) graphical

B) random

C) open

D) None

Explanation: Open

Question 35: The real root of the equation `x logx=1.2` lies in

A) `(2, 3)`

B) `(3, 4)`

C) `(0, 2) `

D) None

Explanation: `(2, 3)`.

Question 36: How many times do we need to iterate using bisection method to find the root of the equation `xlogx 1.2 — O` to four decimal places if root lies in 2.74 and 2.75

A) 7

B) 4

C) The Method can't be used since derivative is zero

D) None

Explanation: 7

Question 37: Use the Bisection method to find the second approximation to 3 decimal places of the equation `x—cosx=0` if root lies in 0.73 and 0.74

A) `0.7375`

B) `0.734`

C) `0.73875`

D) None

Explanation: `0.7375`

Question 38: How many times do we need to iterate using Regula Falsi method to find the root of the equation `xe^x — cosx = 0` to four decimal places if root lies in 0.51 and 0.52

A) The Method can't be used since derivative is zero

B) `4`

C) `2`

D) None

Explanation: 2

Question 39: Newton Raphson method will always converge to a solution for `f (x) =0` on the interval `[a, b]` if certain conditions met. Which of the following is true

A) f is continuous on the interval [a, b]

B) f (a) and f (b) have same signs

C) f" (x) does change sign on the interval [a, b]

D) f' (x) = O on the interval [a, b]

Explanation: f is continuous on the interval [a, b]

Question 40: In Bisection method if roots lies between a and b the `f(a) f(b)`

A) `\le 0`

B) ` \ge 0`

C) `=0`

D) None

Explanation: ` le 0`

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