Numerical Methods | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) PDF Download

Q1: Consider the data of f(x) given in the table.  
The value of f(1.5) estimated using second-order Newton's interpolation formula is (rounded off to 2 decimal places). (2024 SET-1)
Ans: 0.16 to 0.18
Sol: It is based on  2^nd order interpolation

Here,

f(x) = 0.3010(x − 1) + ((-0.1249)/2) (x − 1)(x − 2)
f(x) = 0.3010(1.5 − 1) + ((-0.1249)/2) (1.5 − 1)(1.5 − 2)
f(1.5) = 0.1505 + 0.0156 = 0.1661

Q1: A function f(x), that is smooth and convex-shaped between interval (x₁, x_u) is shown in the figure. This function is observed at odd number of regularly spaced points. If the area under the function is computed numerically, then ____ (2023 SET-1)
(a) the numerical value of the area obtained using the trapezoidal rule will be less than the actual.
(b) the numerical value of the area obtained using the trapezoidal rule will be more than the actual.
(c) the numerical value of the area obtained using the trapezoidal rule will be exactly equal to the actual.
(d) with the given details, the numerical value of area cannot be obtained using trapezoidal rule.
Ans: (a)
Sol: Approximated function has under estimation so numerical value of the area obtained using trapezoidal rule will be less than the actual.

Q1: Consider the following recursive iteration scheme for different values of variable P with the initial guess x₁ = 1 :

For P = 2, x₅  is obtained to be 1.414, rounded-off to three decimal places. For P = 3, x₅ is obtained to be 1.732, rounded-off to three decimal places.
If P = 10, the numerical value of x₅ is _________ . (round off to three decimal places) (2022 SET-1)
Ans: 3.1 to 3.2
Sol:
Converges when x_{n + 1} = x_n = α


When P = 2, x₅ = √2 = 1.4124
When P = 3, x₅ = √3 = 1.732
When P = 10, x₅ = √10 = 3.162

Q1: Numerically integrate, f(x) = 10x − 20x² 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 __________ (2021 SET-2)
Ans: 0.38 to 0.42
Sol: y = 10x - 20x²
a = 0, b = 0.5, n = 5
So, h = (b-a)/n = 0.1
And

= 0.40

Q2: The value of  using the trapezoidal rule with four equal subintervals is (2021 SET-1)
(a) 1.718
(b) 1.727
(c) 2.192
(d) 2.718
Ans: (b)
Sol: Let
Then

The formula of trapezoidal rule to the given data is given by


Q1: The integral  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 statement is TRUE? (2020 SET-2)
(a) Simpsons 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 given zero error.
(d) Only Simpson's rule of estimation will give zero error.
Ans: (d)
Sol: Because integral is a polynomial of 3rd degree so Simpson's rule will give error free answer.

Q1: 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? [2019 : 1 Mark, Set-ll]
(a) I_L is always greater than I_N 
(b) No definite relation exists between I_L and I_N
(c) I_L and I_N are always equal
(d) I_L is always less than I_N        
Ans: (c)

Q1: The quadratic equation 2x² - 3x + 3 = 0 is to be solved numerically starting with an initial guess as x₀ = 2. The new estimate of x after the first iteration using Newton-Raphson method is ______.    [2018 : 1 Mark, Set-II]
Ans: 1 to 1
Sol: Given 
f(x) = 2x² - 3x + 3, x₀ = 2
f'(x) = 4x - 3
By Newton-Rapshon


Q1: Consider the equation du/dt = 3t² + 1 with u = 0 at t = 0. This is numerically solved by using the forward Euler method with a step size. Δt = 2. The absolute error in the solution in the end of the first time step is _____________.    [2017 : 2 Marks, Set-I]
Ans: 7.95 to 8.05
Sol:  du/dt = 3t² + 1
f(u , f) = 3t² + 1
u0 = 0
t0 = 0
Δt = 2
By Euler’s method

After first iteration u = 2 when t = 2

Absolute error = Exact value - approx value
= 10 -2
= 8

Q1: Newton-Raphson method is to be used to find foot of equation 3x - e^x + sinx = 0. If the initial trail value of the roots is taken as 0.333, the next approximation for the root would be __________.    [2016 : 1 Mark, Set-I]
Ans: 0.355 to 0.365
Sol: According to Newton-Raphson Method:

or, f' (x) = 3 - e^x + cosx
⇒

∴ X₁ = 0.36

Q1: For step-size, Δx = 0.4, the value of following integral using Simpson’s 1/3 rule is _________.  [2015 : 2 Marks, Set-II]

Ans: 1.36 to 1.37
Sol: a = 0,b = 0.8, Δx = 0.4

f{x) = 0.2 + 25x - 200x² + 675x³ - 900x⁴ + 400x⁵

By Simpson’s 1/3 Rule

0 = y(0) = 0.2

y₁ = y (0.4) = 2.456

y₂ = y(0.8) = 0.232

= 1.367

Q2: In Newton-Raphson iterative method, the initial guess value (x_ini) is considered as zero while finding the roots of the equation:
f(x) = -2 + 6x - 4X² + 0.5.x³.
The correction, Δx, to be added to x_ini in the first iteration is ___________.  [2015 : 1 Mark, Set-II]
Ans: 2.32 to 2.34
Sol: f (x) = - 2 + 6x - 4x² + 0.5x³
f '(x) = 6 - 8x + 1 .5x²
x_ini = 0
By Newton Raphson Method,

⇒

∴

Q3: The quadratic equation x² - 4x + 4 = 0 is to be solved numerically, starting with the initial guess x₀ = 3. The Newton-Raphson method is applied once to get a new estimate and then the Secant method is applied once using the initial guess and this new estimate. The estimated value of the root after the application of the Secant method is  _____.    [2015 : 2 Marks, Set-I]
Ans: 2.32 to 2.34
Sol: f(x) = x² - 4x + 4
f'(x) = 2x - 4
x₀ = 3
f (3) = 1, f'(3) = 2
By Newton Rapshon method,

f(5/2) = 25/4 -10 + 4 = 1/4
By Secant method,

Q4: The integral  is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If I is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?  [2015 : 1 Mark, Set-I]
(a) J > I 
(b) J < I 
(c) J = I 
(d) insufficient data to determine the relationship    
Ans: (a)
Sol:

Exact value is computed by integration which follows thee exact shape of graph while computing the area.
Whereas, in Trapezoidal rule, the lines joining each points are considered straight lines which is not the exact variation of graph all the time like as shown in figure.
∴ J > I
OR

Here, f(x) = x²
or,  f'(x) = 2x
or, f''(x) = 2 > 0
Since f''(x) is positive, the error is negative.
Since error = exact - approximate.
= I - J
and since error is negative in this case J > I is true.

Q1: The magnitude as the error (correct to two decimal places) in the estimation of following integral using simpson 1/3 rule. Take the step length as 1.    [2013 : 2 Marks]

Sol: Using Simpson’s Rule,



= 245.33
The value of integral,


∴Magnitude of error = 245.33 - 244.8 = 0.53

Q1: The error in  for a continuous function estimated with h = 0.03 using the central difference formula   The values of x₀ and f(x₀) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately.  [2011 : 2 Marks]
(a) 1.3 x 10^-4 
(b) 3.0 x 10^-4 
(c) 4.5 x 10^-4 
(d) 9.0 x 10^-4     
Ans: (d)
Sol: Error in central difference formula is 0(h²)
This means,
error ∝ h²
If error for h = 0.03 is 2 x 10^-3 then,
Error for h = 0.02 is approximately

Q2: The estimate of  Obtained using Simpson's rule with three-point function evaluation exceeds the exact value by    [2011 : 2 Marks]
(a) 0.235 
(b) 0.068 
(c) 0.024 
(d) 0.012
Ans: (d)
Sol: 

Approximate value by Simpson’s rule with 3 point is,

(n_pt is the number of pts and n_i is the number of intervals)
Hero
The tabic is

So the estimate exceeds the exact value by,
Approximate value - Exact value
= 0.012499
≈ 0.012

Q3: The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x² - N = 0. If i denotes the iteration index, the correct iterative scheme will be  [2011 : 1 Mark]

Ans: (a)
Sol: 



Q1: The table below gives values of a function F(x) obtained for values of x at intervals of 0.25.

The value of the integral of the function between the limits 0 to 1 using Simpson’s rule is   [2010 : 2 Marks]
(a) 0.7854 
(b) 2.3562 
(c) 3.1416 
(d) 7.5000
Ans: (a)
Sol: