Numerical Methods - Engineering Mathematics - Petroleum Engineering Previous Year Questions

1

2016 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2016

For a function f(x), the values of the function in the interval [0, 1] are given in the table below.

x	f(x)
0.0	1.0
0.2	1.24
0.4	1.56
0.6	1.96
0.8	2.44
1.0	3.0

The value of the integral \(\int_{0}^{1} f(x) dx\) according to the trapezoidal rule is __________.

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2

2017 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2017

If \(\frac{d^2 y}{dx^2} + f(x, y) = 0\) is to be solved using the conditions \(y(0) = a\) and \(y(1) = b\), which of the following numerical method(s) can be used?

A

Euler with shooting method

B

Euler without shooting method

C

4th order Runge-Kutta with shooting method

D

Both (A) and (C)

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3

2017 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2017

The numerical method used to find the root of a non-linear algebraic equation, that converges quadratically, is:

A

Bisection method

B

Regula-falsi method (Method of False Position)

C

Newton-Raphson method

D

None of above

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4

2017 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2017

Single step integration (step size = 0.5) of \(I = \int_0^1 x^2 e^x dx\), evaluated numerically using the Simpson's 1/3 rule, is __________. (write answer with three decimal places)

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5

2017 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2017

Solve \(\frac{dy}{dx} = -y\) numerically from \(x = 0\) to 1 using explicit, forward, first order Euler method with initial condition of \(y(0) = 1\) and step size (\(h\)) of 0.2. The absolute value of error in \(y(1)\) calculated using analytical and numerical solution is ____% (calculate the error using analytical solution as the basis and use three decimal places).

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6

2018 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2018

Solve the given differential equation using the 2^nd order Runge-Kutta (RK2) method:
\[ \frac{dy}{dt} = t - \sqrt{y} \,; \, \text{Initial condition:} \, y(t=0) = 4 \]
Use the following form of RK2 method with an integration step-size, \( h = 0.5 \):
\[ k_1 = f(t_i, y_i) \, ; \, k_2 = f(t_i + 0.5h, y_i + 0.5k_1 h) \]
\[ y_{i+1} = y_i + k_2 h \]
The value of \( y(t = 0.5) \) = _____________. (rounded-off to two decimal places)

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7

2019 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2019

A company has purchased an asset by investing Rs.30,000. The useful life of the asset is 5 years and it has no salvage value at the end of its useful life. The depreciation cost (in Rs.) for the 2nd year using sum-of-years-digit (SYD) method is

A

10,000

B

8,000

C

6,000

D

4,000

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8

2019 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2019

The man-hours required (\(T_n\)) to manufacture the \(n^{th}\) unit in a plant is given by \(T_n = T_1 n^b\), where \(b = -0.322\) at the 80% learning rate. If the manufacturing time for the first unit (\(T_1\)) is 80 man-hours, the total time (in man-hours) required to manufacture the first 4 units, at 80% learning rate, is

A

322.11

B

251.35

C

103.76

D

51.19

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9

2019 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2019

The numerical value of the definite integral \(\int_{0}^{1} e^{-x} dx\) using trapezoidal rule with function evaluations at points \(x = 0, 0.5\) and \(1\) is _____ (round off to 3 decimal places)

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10

2019 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2019

The sales data of a product for 5 years are

Year	2014	2015	2016	2017	2018
Sales (units)	280	268	259	270	287

Assume the forecast for the year 2014 as 260 units. Using an exponential smoothing method with smoothing constant \( \alpha = 0.5 \), the sales forecast (units) for the year 2019, is ______

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11

2020 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2020

Match the following P. Gauss-Seidel method I. Interpolation Q. Forward Newton-Gauss method II. Non-linear differential equation R. Runge-Kutta method III. Linear algebraic equation

A

P-I, Q-II, R-III

B

P-II, Q-I, R-III

C

P-I, Q-III, R-II

D

P-III, Q-I, R-II

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12

2020 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2020

The equation \( x^3 - 3x - 5 = 0 \) is to be solved using the Newton-Raphson method. Starting with an initial guess of 2, the value of \( x \) after three iterations (rounded off to three decimal places) is ________.

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13

2021 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2021

The estimate for the root of the function \( f(x) = e^{2x} + 2x \) after one iteration with an initial guess of \( x_0 = 0 \), using the Newton-Raphson method is __________ (correct up to two decimal places).

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14

2022 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2022

Simpson’s one-third rule will give the exact value of the integral, \( I = \int_{a}^{b} [b_{0} + b_{1}x + b_{2}x^{2} + \cdots + b_{n}x^{n}] dx \) (where \( a, b, b_{0}, b_{1}, b_{2}, \cdots, b_{n} \) are numeric constants), if the values of \( n \) are

A

1

B

2

C

3

D

4

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15

2022 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2022

Consider the following ordinary differential equation
\[ \frac{dy}{dx} = x^2 y \]
The initial value is \(y(0) = 1\) and the step-size is 0.1. Solving this differential equation by Euler’s first-order method, the value of \(y(0.2)\) is _____ (rounded off to three decimal places).

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16

2024 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2024

Given \(f(x) = 2 + 20x + 30x^5\).
The value of \(\int_0^2 f(x)dx\) using Simpson's 1/3^rd rule with only one interior point is __________.

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17

2025 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2025

Let \( f(x) = \ln x \). The first derivative \( f'(x) \) is to be calculated at \( x = 1 \) using numerical differentiation. \( f'(1) \) is calculated using first order forward difference (\( f'_{FD} \)), first order backward difference (\( f'_{BD} \)), and second order central difference (\( f'_{CD} \)), using interval width \( h = 0.1 \).
The CORRECT order of the values of \( f'_{FD} \), \( f'_{BD} \), and \( f'_{CD} \) is

A

\(f'_{FD} > f'_{CD} > f'_{BD} \)

B

\(f'_{CD} > f'_{BD} > f'_{FD} \)

C

\(f'_{BD} > f'_{FD} > f'_{CD} \)

D

\(f'_{BD} > f'_{CD} > f'_{FD} \)

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18

2023 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2023

In the \( 4^{th} \) order Runge-Kutta method for solving ordinary differential equations with step size \( h < 1 \), the ratio of the order of local error to the order of global error is

A

\(h \)

B

\(h^2 \)

C

\(\frac{1}{h} \)

D

\(\frac{1}{h^2} \)

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19

2023 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2023

Using Simpson's one-third rule (with step size h = 0.25), the area under the curve y = e^{-x^2}, from x = 0 to x = 1 is ______ (rounded to two decimal places).

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20

2026 · Petroleum Engineering · Engineering Mathematics · Numerical Methods

Petroleum Engineering (PE) 2026

The values of a function f(X) at three different X are given as:
f(0.90) = 7.75
f(1.00) = 9.00
f(1.10) = 10.35
Using the central difference approximation method, which of the following is the CORRECT estimate of first differential, f'(1.00) and second differential, f''(1.00), respectively?

A

10.00 and 13.00

B

13.00 and 10.00

C

10.00 and 10.00

D

13.00 and 13.00

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