Numerical Solutions of Nonlinear Equations - Numerical Analysis - Mathematics Previous Year Questions

1

2009 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2009

Let \(\phi : [0, 1] \to \mathbb{R}\) be three times continuously differentiable. Suppose that the iterates defined by \(x_{n+1} = \phi(x_n), n \geq 0\) converge to the fixed point \(\xi\) of \(\phi\). If the order of convergence is three then

A

\(\phi'(\xi) = 0, \phi''(\xi) = 0\)

B

\(\phi'(\xi) \neq 0, \phi''(\xi) = 0\)

C

\(\phi'(\xi) = 0, \phi''(\xi) \neq 0\)

D

\(\phi'(\xi) \neq 0, \phi''(\xi) \neq 0\)

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2

2011 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2011

While solving the equation \(x^2 - 3x + 1 = 0\) using the Newton-Raphson method with the initial guess of a root as 1, the value of the root after one iteration is

A

1.5

B

1

C

0.5

D

0

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3

2012 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2012

The approximate eigenvalue of the matrix \(A=\begin{bmatrix} -15 & 4 & 3 \\ 10 & -12 & 6 \\ 20 & -4 & 2 \end{bmatrix}\) obtained after two iterations of Power method, with the initial vector \([1\ 1\ 1]^T\), is

A

7.768

B

9.468

C

10.548

D

19.468

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4

2012 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2012

The root of the equation \(xe^x=1\) between 0 and 1, obtained by using two iterations of bisection method, is

A

0.25

B

0.50

C

0.75

D

0.65

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5

2012 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2012

Consider the system of equations
\( \begin{pmatrix} 5 & -1 & 1 \\ 2 & 4 & 0 \\ 1 & 1 & 5 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 10 \\ 12 \\ -1 \end{pmatrix} \)
Using Jacobi's method with the initial guess \( \begin{bmatrix} x^{(0)} & y^{(0)} & z^{(0)} \end{bmatrix}^{T} = \begin{bmatrix} 2.0 & 3.0 & 0.0 \end{bmatrix}^{T} \), the approximate solution \( \begin{bmatrix} x^{(2)} & y^{(2)} & z^{(2)} \end{bmatrix}^{T} \) after two iterations, is

A

\(\begin{bmatrix} 2.64 & -1.70 & -1.12 \end{bmatrix}^{T} \)

B

\(\begin{bmatrix} 2.64 & -1.70 & 1.12 \end{bmatrix}^{T} \)

C

\(\begin{bmatrix} 2.64 & 1.70 & -1.12 \end{bmatrix}^{T} \)

D

\(\begin{bmatrix} 2.64 & 1.70 & 1.12 \end{bmatrix}^{T} \)

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6

2014 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2014

Using the Newton-Raphson method with the initial guess \(x^{(0)} = 6\), the approximate value of the real root of \(x \log_{10} x = 4.77\), after the second iteration, is ______________

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7

2014 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2014

Using the Gauss-Seidel iteration method with the initial guess \(\left\{x_{1}^{(0)}=3.5, x_{2}^{(0)}=2.25, x_{3}^{(0)}=1.625\right\}\), the second approximation \(\left\{x_{1}^{(2)}, x_{2}^{(2)}, x_{3}^{(2)}\right\}\) for the solution to the system of equations
\(2 x_{1}-x_{2}=7\)
\(-x_{1}+2 x_{2}-x_{3}=1\)
\(-x_{2}+2 x_{3}=1\),
is

A

\(x_{1}^{(2)}=5.3125, x_{2}^{(2)}=4.4491, x_{3}^{(2)}=2.1563\)

B

\(x_{1}^{(2)}=5.3125, x_{2}^{(2)}=4.3125, x_{3}^{(2)}=2.6563\)

C

\(x_{1}^{(2)}=5.3125, x_{2}^{(2)}=4.4491, x_{3}^{(2)}=2.6563\)

D

\(x_{1}^{(2)}=5.4991, x_{2}^{(2)}=4.4491, x_{3}^{(2)}=2.1563\)

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8

2016 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2016

The number of roots of the equation \( x^2 - \cos(x) = 0 \) in the interval \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \) is equal to ______________

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9

2016 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2016

For the fixed point iteration \( x_{k+1} = g(x_k), \; k=0,1,2,....., \) consider the following statements P and Q: (P) : If \( g(x) = 1 + \frac{2}{x} \) then the fixed point iteration converges to 2 for all \( x_0 \in [1,100] \). (Q) : If \( g(x) = \sqrt{2+x} \) then the fixed point iteration converges to 2 for all \( x_0 \in [0,100] \). Which of the above statements hold TRUE?

A

both P and Q

B

only P

C

only Q

D

Neither P nor Q

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10

2018 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2018

Let \(f : \mathbb{R} \to \mathbb{R}\) be a twice continuously differentiable function. The order of convergence of the secant method for finding root of the equation \(f(x) = 0\) is

A

\(\frac{1+\sqrt{5}}{2}\)

B

\(\frac{2}{1+\sqrt{5}}\)

C

\(\frac{1+\sqrt{5}}{3}\)

D

\(\frac{3}{1+\sqrt{5}}\)

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11

2019 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2019

Let \( f:[a,b] \to \mathbb{R} \) (the set of all real numbers) be any function which is twice differentiable in \( (a,b) \) with only one root \( \alpha \) in \( (a,b) \). Let \( f'(x) \) and \( f''(x) \) denote the first and second order derivatives of \( f(x) \) with respect to \( x \). If \( \alpha \) is a simple root and is computed by the Newton-Raphson method, then the method converges if

A

\(|f'(x) f''(x)| < |f'(x)|^2 \), for all \(x \in (a,b) \)

B

\(|f(x) f''(x)| < |f'(x)|^2 \), for all \(x \in (a,b) \)

C

\(|f'(x) f''(x)| > |f'(x)|^2 \), for all \(x \in (a,b) \)

D

\(|f(x) f''(x)| > |f'(x)|^2 \), for all \(x \in (a,b) \)

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12

2021 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2021

Consider the fixed-point iteration
\[x_{n+1} = \varphi(x_n), \ n \geq 0,\]
with \[\varphi(x) = 3 + (x - 3)^3, \ x \in (2.5, 3.5),\]
and the initial approximation \(x_0 = 3.25\).
Then, the order of convergence of the fixed-point iteration method is

A

1

B

2

C

3

D

4

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13

2021 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2021

Let \( f(x) = x^4 + 2x^3 - 11x^2 - 12x + 36 \) for \( x \in \mathbb{R} \). The order of convergence of the Newton-Raphson method \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}, \quad n \geq 0, \) with \( x_0 = 2.1 \), for finding the root \( \alpha = 2 \) of the equation \( f(x) = 0 \) is ________.

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14

2023 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2023

Let \(f\) be a twice continuously differentiable function on \([a, b]\) such that \(f'(x) < 0\) and \(f''(x) < 0\) for all \(x \in (a, b)\). Let \(f(\zeta) = 0\) for some \(\zeta \in (a, b)\). The Newton-Raphson method to compute \(\zeta\) is given by
\(x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}\), \(k = 0, 1, 2, ...\)
for an initial guess \(x_0\).
If \(x_k \in (\zeta, b)\) for some \(k \geq 0\), then which of the following statements is/are correct?

A

\(x_{k+1} > \zeta\)

B

\(x_{k+1} < \zeta\)

C

\(x_{k+1} < x_k\)

D

For every \(\eta \in (\zeta, x_k)\), \(\frac{f'(\eta)}{f'(x_k)} > 1\)

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15

2024 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2024

Let \(p_1 < p_2\) be the two fixed points of the function \(g(x) = e^x - 2\), where \(x \in \mathbb{R}\). For \(x_0 \in \mathbb{R}\), let the sequence \((x_n)_{n \geq 1}\) be generated by the fixed point iteration \(x_n = g(x_{n-1}), \quad n \geq 1.\) Which one of the following is/are correct?

A

\((x_n)_{n \geq 0}\) converges to \(p_1\) for any \(x_0 \in (p_1, p_2)\)

B

\((x_n)_{n \geq 0}\) converges to \(p_2\) for any \(x_0 \in (p_1, p_2)\)

C

\((x_n)_{n \geq 0}\) converges to \(p_2\) for any \(x_0 > p_2\)

D

\((x_n)_{n \geq 0}\) converges to \(p_1\) for any \(x_0 < p_1\)

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16

2025 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2025

To find a real root of the equation \( x^3 + 4x^2 - 10 = 0 \) in the interval \( \left(1, \frac{3}{2}\right) \) by using the fixed-point iteration scheme, consider the following two statements:
S1: The iteration scheme \( x_{k+1} = \sqrt{\frac{10}{4+x_k}} \), \( k = 0, 1, 2, ... \), converges for any initial guess \( x_0 \in \left(1, \frac{3}{2}\right) \).
S2: The iteration scheme \( x_{k+1} = \frac{1}{2} \sqrt{10 - x_k^3} \), \( k = 0, 1, 2, ... \), diverges for some initial guess \( x_0 \in \left(1, \frac{3}{2}\right) \).
Then, which one of the following is correct?

A

S1 is TRUE and S2 is FALSE

B

S2 is TRUE and S1 is FALSE

C

both S1 and S2 are TRUE

D

neither S1 nor S2 is TRUE

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17

2007 · Mathematics · Numerical Analysis · Numerical Solutions of Nonlinear Equations

Mathematics (MA) 2007

Suppose that \(x_0\) is sufficiently close to 3. Which of the following iterations \(x_{n+1} = g(x_n)\) will converge to the fixed point \(x = 3\)?

A

\(x_{n+1} = -16 + 6x_n + \frac{3}{x_n}\)

B

\(x_{n+1} = \sqrt{3+2x_n}\)

C

\(x_{n+1} = \frac{3}{x_n-2}\)

D

\(x_{n+1} = \frac{x_n^2-3}{2}\)

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