Gauss Backward Interpolation Formula Questions and Answers

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.

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

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.

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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