Numerical Methods - Engineering Mathematics - Aerospace Engineering Previous Year Questions

1

2007 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2007

The Euler iteration formula for numerically integrating a first order nonlinear differential equation of the form \( \dot{x} = f(x) \), with a constant step size of \( \Delta t \) is

A

\(x_{k+1} = x_k - \Delta t \times f(x_k) \)

B

\(x_{k+1} = x_k + (\Delta t^2 / 2) \times f(x_k) \)

C

\(x_{k+1} = x_k - (1/\Delta t) \times f(x_k) \)

D

\(x_{k+1} = x_k + \Delta t \times f(x_k) \)

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2

2007 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2007

The Newton-Raphson iteration formula to find a cube root of a positive number \( c \) is

A

\(x_{k+1} = \frac{2x_k^3 + \sqrt[3]{c}}{3x_k^2} \)

B

\(x_{k+1} = \frac{2x_k^3 - \sqrt[3]{c}}{3x_k^2} \)

C

\(x_{k+1} = \frac{2x_k^3 + c}{3x_k^2} \)

D

\(x_{k+1} = \frac{x_k^3 + c}{3x_k^2} \)

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3

2009 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2009

The value of the integral \(\int_0^{\pi} \frac{dx}{1+\sin x}\) evaluated using the trapezoidal rule with two equal intervals is approximately

A

1.27

B

1.81

C

1.41

D

0.71

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4

2010 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2010

In finding a root of the equation: \(x^2 - 6x + 5 = 0\) the Newton-Raphson method achieves an order of convergence equal to:

A

1.0

B

1.67

C

2.0

D

2.5

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5

2011 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2011

Consider the function \( f(x) = x - \sin(x) \). The Newton-Raphson iteration formula to find the root of the function starting from an initial guess \( x^{(0)} \) at iteration \( k \) is

A

\(x^{(k+1)} = \frac{\sin x^{(k)} - x^{(k)} \cos x^{(k)}}{1 - \cos x^{(k)}} \)

B

\(x^{(k+1)} = \frac{\sin x^{(k)} - x^{(k)} \cos x^{(k)}}{1 + \cos x^{(k)}} \)

C

\(x^{(k+1)} = \frac{\sin x^{(k)} + x^{(k)} \cos x^{(k)}}{1 - \cos x^{(k)}} \)

D

\(x^{(k+1)} = \frac{\sin x^{(k)} + x^{(k)} \cos x^{(k)}}{1 + \cos x^{(k)}} \)

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6

2012 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2012

The integration \(\int_0^1 x^3 dx\) computed using trapezoidal rule with \(n = 4\) intervals is ____.

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7

2014 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2014

The value of \( I = \int_0^1 1000 x^4 dx \), obtained by using Simpson’s rule with 2 equally spaced intervals is,

A

200

B

400

C

180

D

208

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8

2016 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2016

Use Newton-Raphson method to solve the equation: \(x e^x = 1\). Begin with the initial guess \(x_0 = 0.5\). The solution after one step is x = ______.

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9

2017 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2017

3-point Gaussian integration formula is given by: \(\int_{-1}^1 f(x)dx \approx \sum_{j=1}^3 A_j f(x_j)\) with \(x_1 = 0, x_2 = -x_3 = -\sqrt{\frac{3}{5}}, A_1 = \frac{8}{9}, A_2 = A_3 = \frac{5}{9}\). This formula exactly integrates

A

\(f(x) = 5 - x^7\)

B

\(f(x) = 2 + 3x + 6x^4\)

C

\(f(x) = 13 + 6x^3 + x^6\)

D

\(f(x) = e^{-x^2}\)

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10

2019 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2019

For the system of springs and masses shown below, k = 1250 N/m and m = 10 kg. The highest natural frequency, α of the system is ___ radians/s (round off to the nearest integer).

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11

2020 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2020

If \(\int_{0}^{1} (x^3 - 2x + 1) dx\) is evaluated numerically using trapezoidal rule with four intervals, the difference between the numerically evaluated value and the analytical value of the integral is equal to ____ (round off to three decimal places).

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12

2021 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2021

A two degree of freedom spring-mass system undergoing free vibration with generalized coordinates \( x_1 \) and \( x_2 \) has natural frequencies \( \omega_1 = 233.9 \) rad/s and \( \omega_2 = 324.5 \) rad/s, respectively. The corresponding mode shapes are \( \phi_1 = \begin{bmatrix} 1 \\ -3.16 \end{bmatrix} \) and \( \phi_2 = \begin{bmatrix} 1 \\ 3.16 \end{bmatrix} \). If the system is disturbed with certain deflections and zero initial velocities, then which of the following statement(s) is/are true?

A

An initial deflection of \(x_1(0) = 6.32 \) cm and \(x_2(0) = -3.16 \) cm would make the system oscillate with only the second natural frequency.

B

An initial deflection of \(x_1(0) = 2 \) cm and \(x_2(0) = -6.32 \) cm would make the system oscillate with only the first natural frequency.

C

An initial deflection of \(x_1(0) = 2 \) cm and \(x_2(0) = -2 \) cm would make the system oscillate with a linear combination of first and second natural frequencies.

D

An initial deflection of \(x_1(0) = 1 \) cm and \(x_2(0) = -6.32 \) cm would make the system oscillate with only the first natural frequency.

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13

2021 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2021

The definite integral \( \int_1^5 x^3 dx \) is evaluated using four equal intervals by two methods – first by the trapezoidal rule and then by the Simpson’s one-third rule. The absolute value of the difference between the two calculations is _____ (round off to two decimal places).

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14

2024 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2024

Using Trapezoidal rule with one interval, the approximate value of the definite integral: \[ \int_{1}^{2} \frac{dx}{1+x^2} = \_\_\_\_ \] (rounded off to 2 decimal places).

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15

2025 · Aerospace Engineering · Engineering Mathematics · Numerical Methods

Aerospace Engineering (AE) 2025

An approximate solution of the equation \(x^3 - 17 = 0\) is to be obtained using the Newton-Raphson method. If the initial guess is \(x_0 = 2\), the value at the end of the first iteration is \(x_1 = \) ____ (rounded off to two decimal places).

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