Numerical Methods - Engineering Mathematics - Civil Engineering Previous Year Questions

1

2007 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2007

The following equation needs to be numerically solved using the Newton-Raphson method. \(x^3 + 4x - 9 = 0\) The iterative equation for this purpose is (k indicates the iteration level)

A

\(x_{k+1} = \frac{2x_k^3 + 9}{3x_k^2 + 4}\)

B

\(x_{k+1} = \frac{3x_k^2 + 4}{2x_k^3 + 9}\)

C

\(x_{k+1} = x_k - 3x_k^2 + 4\)

D

\(x_{k+1} = \frac{4x_k^2 + 3}{9x_k^2 + 2}\)

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2

2016 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2016 [Session 1]

Newton-Raphson method is to be used to find root of equation \(3x - e^x + \sin x = 0\). If the initial trial value for the root is taken as 0.333, the next approximation for the root would be ______ (note: answer up to three decimal)

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3

2018 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2018 [Session 2]

The quadratic equation \(2x^2 - 3x + 3 = 0\) is to be solved numerically starting with an initial guess as \(x_0 = 2\). The new estimate of \(x\) after the first iteration using Newton-Raphson method is ______

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4

2019 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2019 [Session 2]

The value of the function \(f(x)\) is given at \(n\) distinct values of \(x\) and its value is to be interpolated at the point \(x^*\), using all the \(n\) points. The estimate is obtained first by the Lagrange polynomial, denoted by \(I_L\), and then by the Newton polynomial, denoted by \(I_N\). Which one of the following statements is correct?

A

\(I_L\) is always greater than \(I_N\)

B

\(I_L\) and \(I_N\) are always equal

C

\(I_L\) is always less than \(I_N\)

D

No definite relation exists between \(I_L\) and \(I_N\)

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5

2020 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2020 [Session 2]

The integral \( \int_0^1 (5x^3 + 4x^2 + 3x + 2)dx \) is estimated numerically using three alternative methods namely the rectangular, trapezoidal and Simpson's rules with a common step size. In this context, which one of the following statements is TRUE?

A

Simpson's rule as well as rectangular rule of estimation will give NON-zero error.

B

Simpson's rule, rectangular rule as well as trapezoidal rule of estimation will give NON-zero error.

C

Only the rectangular rule of estimation will give zero error.

D

Only Simpson's rule of estimation will give zero error.

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6

2021 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2021 [Session 2]

Numerically integrate, \(f(x) = 10x - 20x^2\) from lower limit \(a = 0\) to upper limit \(b = 0.5\). Use Trapezoidal rule with five equal subdivisions. The value (in units, round off to two decimal places) obtained is ________

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7

2024 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2024 [Session 1]

The smallest positive root of the equation \[ x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0 \] lies in the range

A

0 < x ≤ 2

B

2 < x ≤ 4

C

6 ≤ x ≤ 8

D

10 ≤ x ≤ 100

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8

2024 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2024 [Session 1]

Consider the data of \( f(x) \) given in the table.

\( i \)	0	1	2
\( x_i \)	1	2	3
\( f(x_i) \)	0	0.3010	0.4771

The value of \( f(1.5) \) estimated using second-order Newton's interpolation formula is __________ (rounded off to 2 decimal places).

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9

2024 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2024 [Session 2]

The second derivative of a function \(f\) is computed using the fourth-order Central Divided Difference method with a step length \(h\).

The CORRECT expression for the second derivative is

A

\(\frac{1}{12h^2} \left[ -f_{i+2} + 16 f_{i+1} - 30 f_i + 16 f_{i-1} - f_{i-2} \right]\)

B

\(\frac{1}{12h^2} \left[ f_{i+2} + 16 f_{i+1} - 30 f_i + 16 f_{i-1} - f_{i-2} \right]\)

C

\(\frac{1}{12h^2} \left[ -f_{i+2} + 16 f_{i+1} - 30 f_i + 16 f_{i-1} + f_{i-2} \right]\)

D

\(\frac{1}{12h^2} \left[ -f_{i+2} - 16 f_{i+1} + 30 f_i - 16 f_{i-1} - f_{i-2} \right]\)

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10

2025 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2025 [Session 1]

Consider the differential equation given below. Using the Euler method with the step size (h) of 0.5, the value of y at x = 1.0 is equal to ______ (rounded off to 1 decimal place). \[ \frac{dy}{dx} = y + 2x - x^2 ; \quad y(0) = 1 \quad (0 \le x < \infty) \]

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11

2010 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2010

The table below gives values of a function F(x) obtained for values of x at intervals of 0.25.

x	0	0.25	0.5	0.75	1.0
F(x)	1	0.9412	0.8	0.64	0.50

The value of the integral of the function between the limits 0 to 1 using Simpson's rule is

A

0.7854

B

2.3562

C

3.1416

D

7.5000

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12

2011 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2011

The square root of a number \(N\) is to be obtained by applying the Newton Raphson iterations to the equation \(x^2 - N = 0\). If \(i\) denotes the iteration index, the correct iterative scheme will be

A

\(x_{i+1} = \frac{1}{2} \left( x_i + \frac{N}{x_i} \right)\)

B

\(x_{i+1} = \frac{1}{2} \left( x_i^2 + \frac{N}{x_i^2} \right)\)

C

\(x_{i+1} = \frac{1}{2} \left( x_i + \frac{N^2}{x_i} \right)\)

D

\(x_{i+1} = \frac{1}{2} \left( x_i - \frac{N}{x_i} \right)\)

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13

2012 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2012

The estimate of \(\int_{0.2}^{1.2} \frac{dx}{x}\) obtained using Simpson's rule with three-point function evaluation exceeds the exact value by

A

0.235

B

0.068

C

0.024

D

0.012

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14

2012 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2012

The error in \( \frac{d}{d x} f(x) \big|_{x=x_0} \) for a continuous function estimated with \( h = 0.03 \) using the central difference formula \( \frac{d}{d x} f(x) \big|_{x=x_0} \approx \frac{f(x_0 + h) - f(x_0 - h)}{2h} \), is \( 2 \times 10^{-3} \). The values of \( x_0 \) and \( f(x_0) \) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for \( h = 0.02 \) is approximately

A

\(1.3 \times 10^{-4} \)

B

\(3.0 \times 10^{-4} \)

C

\(4.5 \times 10^{-4} \)

D

\(9.0 \times 10^{-4} \)

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15

2013 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2013

Find the magnitude of the error (correct to two decimal places) in the estimation of following integral using Simpson’s \(\frac{1}{3}\) Rule. Take the step length as 1. \[\int_{0}^{4} (x^4 + 10) dx\]

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16

2026 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2026

Starting with the first approximation as \( x = 0.5 \), the second approximation for the root of the following function by the Newton-Raphson method is _____ (rounded off to two decimal places). \( f(x) = e^{-x} - x \)

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17

2026 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2026

Values of \( y \) for different values of \( x \) are tabulated below.

\( x \)	−2	1	2
\( y \)	28	4	16

If a second-degree interpolating polynomial \( P_2(x) \) is used to represent \( y \), the value of \( P_2(0) \) is _____ (rounded off to the nearest integer).

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18

2026 · Civil Engineering · Engineering Mathematics · Numerical Methods

Civil Engineering (CE) 2026

A fifth-degree polynomial in x is defined for x > 0. All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at a point by using the first-order forward as well as the first-order backward difference methods. Identical step lengths are used for both the methods.
Following statements are made.
(I) Forward difference method underestimates the true derivative.
(II) Backward difference method overestimates the true derivative.
Which one of the following options is CORRECT?

A

Both statements (I) and (II) are FALSE.

B

Both statements (I) and (II) are TRUE.

C

Statement (I) is TRUE and statement (II) is FALSE.

D

Statement (I) is FALSE and statement (II) is TRUE.

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