Numerical Analysis Questions and Answers – Gauss’s Backward Interpolation Formula

This set of Numerical Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Gauss’s Backward Interpolation Formula”.

1. What will be the solution for the following table using Gauss’s backward interpolation formula, where x = 3?

x f(x)
1 1
2 2
3 1

a) 1
b) 0.328
c) 0.327
d) 0.322
View Answer

Answer: a
Explanation:
Gauss's backward difference interpolation method to find solution
h=2-1=1
Now the central difference table is

x p=(x-2)/1 y Δy Δ2y
1 -1 1
1
2 0 2 -2
-1
3 1 1

Solution of Gauss's backward interpolation is y(3)=1.

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2. What will be the solution for the following table using Gauss’s backward interpolation formula, where x = 3?

x f(x)
1 2
2 3
3 4

a) 0.2452
b) 0.2864
c) 0.2862
d) 4
View Answer

Answer: d
Explanation:
Gauss's backward method to find solution
h=2-1=1
Now the central difference table is

x p=(x-2)/1 y Δy Δ2y
1 -1 2
1
2 0 3 0
1
3 1 4
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Solution of Gauss's backward interpolation is y(3)=4.

3. What will be the solution for the following table using Gauss’s backward interpolation formula, where x = 3?

x f(x)
1 2
2 3
3 4
4 8
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a) 11.152
b) 4
c) 11.122
d) 11.128
View Answer

Answer: b
Explanation: Given,
Gauss's backward method to find solution
h=2-1=1
Now the central difference table is

x p=(x-2)/1 y Δy Δ2y Δ3y
1 -1 2
1
2 0 3 0
1 3
3 1 4 3
4
4 2 8

Solution of Gauss's backward interpolation is y(3)=4.

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4. What will be the solution for the following table using Gauss’s backward interpolation formula, where x = 3?

x f(x)
1 2
2 6
3 4
4 8
5 10
6 12

a) 1
b) 4.1774
c) 4
d) 2
View Answer

Answer: c
Explanation:
Gauss's backward difference interpolation method to find solution
h=2-1=1
Now the central difference table is

x p=(x-3)/1 y Δy Δ2y Δ3y Δ4y Δ5y
1 -2 2
1
2 -1 3 0
1 3
3 0 4 3 -8
4 -5 15
4 1 8 -2 7
2 2
5 2 10 0
2
6 3 12

Solution of Gauss's backward interpolation is y(3)=4.

5. What will be the solution for the following table using Gauss’s backward interpolation formula, where x = 3?

x f(x)
1 2
2 6
3 4
4 8
5 10
6 12
7 24
8 26
9 28

a) 4
b) 9
c) 7
d) 6
View Answer

Answer: a
Explanation:
Gauss's backward method to find solution
h=2-1=1
Now the central difference table is

x p=(x-5)/1 y Δy Δ2y Δ3y Δ4y Δ5y Δ6y Δ7y Δ8y
1 -4 2
1
2 -3 3 0
1 3
3 -2 4 3 -8
4 -5 15
4 -1 8 -2 7 -14
2 2 1 -25
5 0 10 0 8 -39 162
2 10 -38 137
6 1 12 10 -30 98
12 -20 60
7 2 24 -10 30
2 10
8 3 26 0
2
9 4 28

Solution of Gauss's backward interpolation is y(3)=4.

6. What will be the solution of the equation 4x-1 using Gauss’s backward interpolation formula where x1 = 2 and x2 = 4 and x = 2.1, step value (h) = 0.25?
a) 8.4
b) 7.4
c) 8
d) 5
View Answer

Answer: b
Explanation:
Gauss's backward method to find solution
h=2.25-2=0.25
Now the central difference table is

x p=(x-3)/0.25 y Δy Δ2y
2 -4 7
1
2.25 -3 8 0
1
2.5 -2 9 0
1
2.75 -1 10 0
1
3 0 11 0
1
3.25 1 12 0
1
3.5 2 13 0
1
3.75 3 14 0
1
4 4 15

Solution of Gauss's backward interpolation is y(2.1)=7.4.

7. What will be the solution of the equation -2x2 using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 2 & step value (h) = 0.25?
a) 2
b) 7
c) 5
d) -8
View Answer

Answer: d
Explanation:
Gauss's backward method to find solution
h=1.25-1=0.25

Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y Δ3y
1 -2 -2
-1.125
1.25 -1 -3.125 -0.25
-1.375 0
1.5 0 -4.5 -0.25
-1.625 0
1.75 1 -6.125 -0.25
-1.875
2 2 -8

Solution of Gauss's backward interpolation is y(2)=-8.

8. What will be the solution of the equation 3x using Gauss’s backward interpolation formula, where x1 = 2 and x2 = 4 & x = 2 & step value (h) = 0.25?
a) 8
b) -6
c) 5
d) 1
View Answer

Answer: b
Explanation:
Gauss's backward method to find solution
h=2.25-2=0.25
Now the central difference table is

x p=(x-3)/0.25 y Δy Δ2y
2 -4 -6
-0.75
2.25 -3 -6.75 0
-0.75
2.5 -2 -7.5 0
-0.75
2.75 -1 -8.25 0
-0.75
3 0 -9 0
-0.75
3.25 1 -9.75 0
-0.75
3.5 2 -10.5 0
-0.75
3.75 3 -11.25 0
-0.75
4 4 -12

Solution of Gauss's backward interpolation is y(2)=-6.

9. What will be the solution of the equation x – 1 using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 1
b) 3
c) 0
d) 4
View Answer

Answer: c
Explanation:
Gauss's backward method to find solution
h=1.25-1=0.25
Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y
1 -2 0
0.25
1.25 -1 0.25 0
0.25
1.5 0 0.5 0
0.25
1.75 1 0.75 0
0.25
2 2 1

Solution of Gauss's backward interpolation is y(1)=0.

10. What will be the solution of the equation -x using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 4
b) 2
c) 3
d) 1
View Answer

Answer: d
Explanation:
Gauss's backward method to find solution
h=1.25-1=0.25
Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y
1 -2 -1
-0.25
1.25 -1 -1.25 0
-0.25
1.5 0 -1.5 0
-0.25
1.75 1 -1.75 0
-0.25
2 2 -2

Solution of Gauss's backward interpolation is y(1)=-1.

11. What will be the solution of the equation -sin(x) using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 4
b) 2
c) 3
d) -0.8415
View Answer

Answer: d
Explanation:
Gauss's backward method to find solution
h=1.25-1=0.25
Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y
1 -2 -0.8415
-0.1075
1.25 -1 -0.949 0.059
-0.0485 0.003
1.5 0 -0.9975 0.062 -0.0039
0.0135 -0.0008
1.75 1 -0.984 0.0612
0.0747
2 2 -0.9093

Solution of Gauss's backward interpolation is y(1)=-0.8415.

12. What will be the solution of the equation -cos(x) using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) -0.5403
b) 2
c) 3
d) 0.0167
View Answer

Answer: a

Explanation:
Gauss's backward method to find solution
h=1.25-1=0.25
Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y
1 -2 -0.5403
0.225
1.25 -1 -0.3153 0.0196
0.2446 -0.0152
1.5 0 -0.0707 0.0044 -0.0003
0.249 -0.0155
1.75 1 0.1782 -0.0111
0.2379
2 2 0.4161

Solution of Gauss's backward interpolation is y(1)=-0.5403.

13. What will be the solution of the equation -tan(x) using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 0.9496
b) -1.5574
c) 3
d) 0.0167
View Answer

Answer: b
Explanation:

Gauss's backward method to find solution
h=1.25-1=0.25
Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y
1 -2 -1.5574
-1.4522
1.25 -1 -3.0096 -9.6397
-11.0919 40.3533
1.5 0 -14.1014 30.7137 -94.0241
19.6218 -53.6708
1.75 1 5.5204 -22.9571
-3.3353
2 2 2.185

Solution of Gauss's backward interpolation is y(1)=-1.5574.

14. What will be the solution of the equation cot(x) using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) 0.9496
b) 4190.0569
c) -0.6421
d) 0.0167
View Answer

Answer: c
Explanation:
Gauss's backward method to find solution

h=1.25-1=0.25
Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y
1 -2 -0.6421
0.3098
1.25 -1 -0.3323 -0.0485
0.2614 0.0392
1.5 0 -0.0709 -0.0093 -0.0054
0.2521 0.0337
1.75 1 0.1811 0.0244
0.2765
2 2 0.4577

Solution of Gauss's backward interpolation is y(1)=-0.6421.

15. What will be the solution of the equation -sec(x) using Gauss’s backward interpolation formula, where x1 = 1 and x2 = 2 & x = 1 & step value (h) = 0.25?
a) -1.8508
b) 4190.0569
c) 3.3389
d) 0.0167
View Answer

Answer: a
Explanation:
Gauss's backward method to find solution
h=1.25-1=0.25

Now the central difference table is

x p=(x-1.5)/0.25 y Δy Δ2y Δ3y Δ4y
1 -2 -1.8508
-1.3205
1.25 -1 -3.1714 -9.6449
-10.9655 40.3575
1.5 0 -14.1368 30.7125 -94.0243
19.7471 -53.6668
1.75 1 5.6102 -22.9543
-3.2072
2 2 2.403

Solution of Gauss's backward interpolation is y(1)=-1.8508.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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