This set of Numerical Methods Multiple Choice Questions & Answers focuses on “Gauss Elimination Method – 2”.
1. Which of the following step is not involved in Gauss Elimination Method?
a) Elimination of unknowns
b) Reduction to an upper triangular system
c) Finding unknowns by back substitution
d) Evaluation of cofactors
View Answer
Explanation: Elimination of unknowns, reduction to an upper triangular system and finding unknowns by back substitution are the primary steps involved in Gauss Elimination.
2. Gauss Elimination Method is well adopted for which of the application?
a) Computer operations
b) Network circuit problems
c) MATLAB operations
d) Telecommunication operations
View Answer
Explanation: Gauss Elimination Method is well adopted for computer operations. It is an effective and fast process.
3. What are the coefficients of the equation obtained during the elimination called?
a) Joints
b) Pivots
c) Calculated coefficients
d) Operative coefficients
View Answer
Explanation: The coefficients of the equation obtained during the elimination are called as pivots.
4. How the transformation of coefficient matrix A to upper triangular matrix is done?
a) Elementary row transformations
b) Elementary column transformations
c) Successive multiplication
d) Successive division
View Answer
Explanation: The transformation of coefficient matrix A to upper triangular matrix is done through elementary row transformations. We will not lead towards the correct answer if we’ll perform transformations in rows as well as columns.
5. How many types of pivoting are there?
a) 2
b) 3
c) 4
d) 5
View Answer
Explanation: There are two types of pivoting, namely, partial and complete pivoting.
6. The modified procedure of complete pivoting is called as ____________
a) Partial
b) Additional
c) Reduced
d) Modified
View Answer
Explanation: The modified procedure of complete pivoting is called as Partial Pivoting.
7. Apply Gauss Elimination method to solve the following equations.
x + 4y – z = -5 x + y – 6z = -12 3x – y – z = 4
a) x = 1.6479, y = -1.1408, z = 2.0845
b) x = 4.0461, y = -1.1408, z = 3.254
c) x = 7.2478, y = -2.586, z = 8.265
d) x = 2.8471, y = 5.5123, z = 2.0845
View Answer
Explanation: x + 4y – z = -5 ………(i)
x + y – 6z = -12 …………………..(ii)
3x – y – z = 4 …………………….(iii)
To eliminate x, operate (ii) – (i) and (iii) – 3(i),
-3y – 5z = -7 ………………………(iv)
-13 + 2z = 19 ………………………(v)
To eliminate y, (v) – (13/3)(iv),
(71/3) z = (148/3)
Now by back substitution,
Z = \(\frac{148}{71}\) = 2.0845
Y = \(\frac{7}{3} – \frac{5}{3}(\frac{148}{71})\)
= \(\frac{-81}{71}\) = -1.1408
X = -5 -4\((\frac{-81}{71}) + (\frac{148}{71})\)
= \(\frac{117}{71}\) = 1.6479
Hence x = 1.6479, y = -1.1408, z = 2.0845.
8. Apply Gauss Elimination method to solve the following equations.
10x – 7y = 3z + 5u = 6 -6x + 8y – z – 4u = 5 3x + y + 4z + 11u = 2 5x – 9y – 2z + 4u = 7
a) u = 1, z = -7, y = 4, x = 5
b) u = 1, z = -7, y = 4, x = 5
c) u = 1, z = -7, y = 4, x = 5
d) u = 1, z = -7, y = 4, x = 5
View Answer
Explanation: 10x – 7y = 3z + 5u = 6 …….(i)
-6x + 8y – z – 4u = 5 ………………….(ii)
3x + y + 4z + 11u = 2 ………………….(iii)
5x – 9y – 2z + 4u = 7 ………………….(iv)
To eliminate x, operate
[(ii) – \((\frac{-6}{10})\) (i)], [(iii) – \(\frac{3}{10}\)(i)] and [(iv) – \(\frac{5}{10}\)(i)]
3.8y + 0.8z – u = 8.6 ……………………(v)
3.1y + 3.1z + 9.5u = 0.2 …………………(vi)
-5.5y – 3.5z + 1.5u = 4 ………………….(vii)
To eliminate y, operate [(vi) – \(\frac{3.1}{3.8}\)(v)], [(vii) – \(\frac{5.5}{3.8}\)(v)],
2.447z + 10.31u = -6.815 …………………(viii)
-2.342z + 0.052u = 16.44 …………………(ix)
To eliminate z operate [(ix) – \((\frac{-2.342}{2.447})\) (viii)] ,
9.924u = 9.924
By back substitution,
u = 1, z = -7, y = 4, x = 5.
9. Apply Gauss Elimination method to solve the following equations.
2x + y + z = 10 3x + 2y + 3z = 18 X + 4y + 9z = 16
a) X = 7, y = -4, z = 5
b) X = 7, y = -9, z = 5
c) X = 5, y = 1, z = -8
d) X = 5, y = 1, z = -3
View Answer
Explanation:
2x + y + z = 10 ………………….(i)
3x + 2y + 3z = 18 ………………..(ii)
x + 4y + 9z = 16 …………………(iii)
To eliminate x, operate (i) – 2(iii), (ii) – 3(iii)
7y + 17z = 22 ……………………(iv)
5y + 12z = 15 ……………………(v)
To eliminate y, operate [(iv) – \(\frac{7}{5}\)(v)]
0.2z = 1
By back substitution,
z = 5
7y = 22 – 85
y= -9
x = 16 + 36 – 45
x = 7
Hence, x = 7, y = -9, z = 5.
10. Apply Gauss Elimination method to solve the following equations.
2x – y + 3z = 9 x + y + z = 6 x – y + z = 2
a) X = 5, y = 14, z = 5
b) X = -13, y = 4, z = 15
c) X = -13, y = 1, z = -8
d) X = 13, y = 1, z = -8
View Answer
Explanation: 2x – y + 3z = 9 ……….(i)
x + y + z = 6 ……………………(ii)
x – y + z = 2 ……………………(iii)
To eliminate x, operate (ii) – (iii)
y = 4
Now, operate (i) – 2(ii),
-3y + z = 3
Now by back substitution,
Z = 15
X + 4 +15 = 6
X = -13.
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