Q: The smallest positive root of the equation x⁵ - 5x⁴ - 10x³ + 50x² + 9x - 45 = 0 lies in the range

0 < x ≤ 2. The correct option is right because evaluating the polynomial at various points shows that it changes sign between 0 and 2, indicating a root exists in that interval. Specifically, the function is negative at 0 and positive at 2, confirming the presence of a root in the specified range.

Q: For a flowing fluid, a dimensionless combination of velocity (V), length scale (l), and acceleration due to gravity (g) would be

V²/(gl). The correct option, V²/(gl), is dimensionless because it combines the units of velocity squared (L²/T²) with the units of gravity (L/T²) and length (L), resulting in a ratio that cancels out all dimensions, leaving a pure number.

Q: 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

1/(12h²) [−f_(i+2) + 16 f_(i+1) − 30 f_i + 16 f_(i−1) − f_(i−2)]. The correct expression accurately represents the second derivative using the fourth-order Central Divided Difference method by appropriately weighing the function values at points around the target point, ensuring that the contributions from points further away are balanced to achieve higher accuracy.

Q: In the IEEE floating point representation the hexadecimal value 0x00000000 corresponds to

the special value +0. The hexadecimal value 0x00000000 in IEEE floating point representation indicates a special case where the exponent and the fraction are both zero, which specifically denotes the value +0.

Q: Let r denote number system radix. The only value(s) of r that satisfy the equation √121_r = 11_r, is/are

any value > 2. In base r, 121 = r^2+2r+1 = (r+1)^2, so sqrt(121_r) = r+1 = 11_r for every base. The only constraint is that digit '2' must be valid, requiring r>2. Hence any value > 2 works.

Question 1

The smallest positive root of the equation x⁵ - 5x⁴ - 10x³ + 50x² + 9x - 45 = 0 lies in the range

Accepted Answer

0 < x ≤ 2. The correct option is right because evaluating the polynomial at various points shows that it changes sign between 0 and 2, indicating a root exists in that interval. Specifically, the function is negative at 0 and positive at 2, confirming the presence of a root in the specified range.

Question 2

For a flowing fluid, a dimensionless combination of velocity (V), length scale (l), and acceleration due to gravity (g) would be

Accepted Answer

V²/(gl). The correct option, V²/(gl), is dimensionless because it combines the units of velocity squared (L²/T²) with the units of gravity (L/T²) and length (L), resulting in a ratio that cancels out all dimensions, leaving a pure number.

Question 3

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

Accepted Answer

1/(12h²) [−f_(i+2) + 16 f_(i+1) − 30 f_i + 16 f_(i−1) − f_(i−2)]. The correct expression accurately represents the second derivative using the fourth-order Central Divided Difference method by appropriately weighing the function values at points around the target point, ensuring that the contributions from points further away are balanced to achieve higher accuracy.

Question 4

In the IEEE floating point representation the hexadecimal value 0x00000000 corresponds to

Accepted Answer

the special value +0. The hexadecimal value 0x00000000 in IEEE floating point representation indicates a special case where the exponent and the fraction are both zero, which specifically denotes the value +0.

Question 5

Let r denote number system radix. The only value(s) of r that satisfy the equation √121_r = 11_r, is/are

Accepted Answer

any value > 2. In base r, 121 = r^2+2r+1 = (r+1)^2, so sqrt(121_r) = r+1 = 11_r for every base. The only constraint is that digit '2' must be valid, requiring r>2. Hence any value > 2 works.

Question 6

The Newton-Raphson iteration xₙ₊₁ = 1/2 (xₙ + R/xₙ) can be used to compute the

Accepted Answer

square root of R. The Newton-Raphson iteration formula provided is specifically designed to approximate the square root of a number R by iteratively refining guesses based on the average of the current guess and the quotient of R and the current guess.

Question 7

Function f is known at the following points:

x: 0, 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0
f(x): 0, 0.36, 0.81, 1.44, 2.25, 3.24, 4.41, 5.76, 7.29, 9.00

The value of ∫₀³ f(x) dx computed using the trapezoidal rule is

Accepted Answer

9.045. With h=0.3 and f(x)=x^2 on [0,3], the trapezoidal rule gives 0.3*(0/2 + sum of interior + 9/2) = 9.045. The trapezoidal rule overestimates a convex function, so the result is just above the exact value 9.

Question 8

Which one of the following correctly determines the solution of the recurrence relation with T(1) = 1?

T(n) = 2T(n/2) + log n

Accepted Answer

Θ(n). Here a=2, b=2, so n^(log_b a)=n. Since f(n)=log n = O(n^(1-e)), Master theorem case 1 applies and T(n)=Theta(n). The answer is Theta(n), not Theta(n log n).

Question 9

With respect to the numerical evaluation of the definite integral, K = ∫ₐ^b x² dx, where a and b are given, which of the following statements is/are TRUE?
I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II)

Accepted Answer

Both I and II. For the convex function x^2 the trapezoidal rule overestimates, so K_trap >= exact (I true). Simpson's rule integrates polynomials up to degree 3 exactly, so it gives the exact value for x^2 (II true). Hence both I and II are true.

Question 10

Consider the IEEE-754 single precision floating point numbers P=0xC1800000 and Q=0x3F5C2EF4. Which one of the following corresponds to the product of these numbers (i.e., P × Q), represented in the IEEE-754 single precision format?

Accepted Answer

0xC15C2EF4. 0xC1800000 decodes to -16.0 and 0x3F5C2EF4 to about 0.86009; their product is -13.76146, whose IEEE-754 single-precision encoding is 0xC15C2EF4. The stored answer 0xC14C2EF4 (-12.76) is wrong; correct is 0xC15C2EF4.

Question 11

The format of a single-precision floating-point number as per the IEEE 754 standard is: Choose the largest floating-point number among the following options.

Accepted Answer

0 11111110 1111 1111 1111 1111 1111 111. The correct option represents a normalized floating-point number with an exponent of 126, which is the highest value among the choices given, allowing it to represent the largest magnitude.

Question 12

The equation x³ - x² + 4x - 4 = 0 is to be solved using the Newton-Raphson method. If x = 2 is taken as the initial approximation of the solution, then the next approximation using this method will be

Accepted Answer

4/3. f(2)=8-4+8-4=8 and f'(2)=12-4+4=12. Newton step: 2 - 8/12 = 2 - 2/3 = 4/3. The stored answer 3/2 is incorrect; correct is 4/3.

Question 13

A numerical solution of the equation f(x) = x + √x − 3 = 0 can be obtained using Newton-Raphson method. If the starting value is x = 2 for the iteration, the value of x that is to be used in the next step is

Accepted Answer

1.694. The Newton-Raphson method uses the formula xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) to iteratively find roots of equations. Starting with x = 2, the calculations yield the next approximation as 1.694, which is the correct answer.

Question 14

The sum of n terms of the series 4+44+444+.... is

Accepted Answer

(4/81) [10^(n+1) - 9n - 10]. The sum is (4/9)[ (10^{n+1}-10)/9 - n ] = (4/81)[10^{n+1} - 9n - 10]. Checked at n=2 this gives 48, matching 4+44. The stored option uses -1 instead of -10 and is wrong; the correct one is option C.

Question 15

The following figures show three curves generated using an iterative algorithm. The total length of the curve generated after ‘Iteration n’ is:
Note: The figures shown are representative.

Accepted Answer

(5/3)ⁿ. The correct option reflects the exponential growth of the curve's length with each iteration, indicating that the length increases by a factor of (5/3) for each complete iteration, resulting in a total length of (5/3)ⁿ after n iterations.

Question 16

The differential equation dx/dt = (1 - x)/τ is discretised using Euler’s numerical integration method with a time step ΔT > 0. What is the maximum permissible value of ΔT to ensure stability of the solution of the corresponding discrete time equation?

Accepted Answer

2τ. The stability of the discretized equation using Euler's method requires that the time step ΔT be less than or equal to 2τ. This ensures that the numerical solution does not diverge and remains stable, as larger time steps can lead to oscillations or instability in the solution.

Question 17

Let x² - 117 = 0. The iterative steps for the solution using Newton-Raphson's method is given by

Accepted Answer

xₖ₊₁ = 1/2 (xₖ + 117/xₖ). The correct option is derived from the Newton-Raphson method, which involves using the function and its derivative to find roots. In this case, the iterative formula simplifies to the average of the current guess and the ratio of the constant to the current guess, effectively refining the estimate for the square root of 117.

Question 18

Solution of the variables x1 and x2 for the following equations is to be obtained by employing the Newton-Raphson iterative method.
equation (i) 10x2 sin x1 - 0.8 = 0
equation (ii) 10x2² - 10x2 cos x1 - 0.6 = 0
Assuming the initial values x1 = 0.0 and x2 = 1.0, the Jacobian matrix is

Accepted Answer

[10 0; 0 10]. J = [[10 x2 cos x1, 10 sin x1],[10 x2 sin x1, 20 x2 - 10 cos x1]]. At x1=0, x2=1 this gives [[10,0],[0,10]] (idx 1). The stored idx 2 [0 -0.8; 10 -0.6] is wrong.

Question 19

Value of ∫₄⁵.2 ln x dx using Simpson’s one-third rule with interval size 0.3 is

Accepted Answer

1.83. Simpson's one-third rule provides an approximation for the integral by using quadratic polynomials to estimate the area under the curve. By applying this method with the specified interval size of 0.3 over the limits from 4 to 5.2, the calculated value of the integral is approximately 1.83, which aligns with the correct option.

Question 20

Consider the definite integral

∫ from 1 to 2 (4x² + 2x + 6) dx.

Let Iₑ be the exact value of the integral. If the same integral is estimated using Simpson’s rule with 10 equal subintervals, the value is Iₛ. The percentage error is defined as e = 100 × (Iₑ − Iₛ)/Iₑ. The value of e is

Accepted Answer

0. The percentage error is zero because Simpson's rule, when applied with an even number of subintervals, provides an exact result for polynomials of degree three or lower. Since the integrand is a polynomial of degree two, the approximation matches the exact value, resulting in no error.

Question 21

A calculator has accuracy up to 8 digits after decimal place. The value of ∫₀^(2π) sin x dx when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is

Accepted Answer

0.00000. The integral of sin x over one complete period from 0 to 2π is zero, as the positive and negative areas cancel each other out. Therefore, when calculated using the trapezoidal method with high precision, the result remains effectively zero, which rounds to 0.00000.

Question 22

For the network below, the objective is to find the length of the shortest path from node P to node G. Let d_ij be the length of directed arc from node i to node j.

Let s_j be the length of the shortest path from P to node j. Which of the following equations can be used to find s_G?

Accepted Answer

s_G = Min{s_O + d_OG, s_R + d_RG}. This option correctly represents the shortest path to node G by considering the shortest paths to its predecessors (O and R) and adding the respective arc lengths from those nodes to G, ensuring that all possible routes are evaluated.

Question 23

The integral ∫₁³ (1/x) dx, when evaluated by using Simpson’s 1/3 rule on two equal subintervals each of length 1, equals

Accepted Answer

1.111. Using h=1 and nodes x=1,2,3 with f=1, 0.5, 0.3333: Simpson 1/3 = (1/3)[1 + 4(0.5) + 0.3333] = (1/3)(3.3333) = 1.111. (The exact ln3=1.0986 corresponds to the stored 1.098, but Simpson's rule yields 1.111.)

Question 24

Match the CORRECT pairs.
Numerical Integration Scheme | Order of Fitting Polynomial
P. Simpson's 3/8 Rule | 1. First
Q. Trapezoidal Rule | 2. Second
R. Simpson's 1/3 Rule | 3. Third

Accepted Answer

P-3, Q-1, R-2. Simpson's 3/8 Rule fits a cubic polynomial, which corresponds to a third-order approximation, while the Trapezoidal Rule fits a linear polynomial (first order), and Simpson's 1/3 Rule fits a quadratic polynomial (second order). This matching aligns with the characteristics of each numerical integration scheme.

Question 25

Numerical integration using trapezoidal rule gives the best result for a single variable function, which is

Accepted Answer

linear. The trapezoidal rule approximates the area under a curve by dividing it into trapezoids, which works perfectly for linear functions since they form straight lines. For linear functions, the trapezoidal rule provides an exact result, while for other types of functions, the approximation may introduce errors.

GATE Engineering Mathematics: Numerical Methods questions with solutions

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