Students can download 12th Business Maths Chapter 5 Numerical Methods Additional Problems and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

One Mark Questions

Question 1.
Match the following.

(a) ∆f(x) (i) 2x + 1
(b) E2 f(x) (ii) 1 + ∆
(c) E (iii) f(x + h) – f(x)
(d) ∆x2, h = 1 (iv) f(x + 2h)

Answer:
(a) – (iii), (b) – (iv), (c) – (ii), (d) – (i)

Question 2.
E-n f(x) is ______
(a) f(x + nh)
(b) f(x – nh)
(c) f(-nh)
(d) f(x – n)
Answer:
(b) f(x – nh)

Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

Question 3.
E is a _____
(a) shifting operator
(b) Displacement operator
(c) 1 + ∆
(d) all of these
(e) none of these
Answer:
(d) all of these

Question 4.
4 y3 = _______
(a) (E – 1)4 y3
(b) (E3 – 1) y3
(c) (E – 1)3 y0
(d) (E – 1)4 y0
Answer:
(a) (E – 1)4 y3

Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

Question 5.
Fill in the blanks.

  1. The two methods of interpolation are _______ and _______
  2. If values of x are not equidistant we use _______ method.
  3. ∆(f(x) + g(x)) = ______
  4. k yn = ______
  5. The first three terms in Newton’s method will give a ________ interpolation.

Answer:

  1. graphical method, algebraic method
  2. Lagrange’s method
  3. ∆f(x) + ∆g(x)
  4. k-1 yn+1 – ∆k-1 yn
  5. Parabolic

Question 6.
Say true or false

  1. ∇y2 = y1 – y0
  2. 2 yn = ∇yn – ∇yn+1
  3. When 5 values are given, the polynomial which fits the data is of degree 4
  4. E ∆ = ∆ E
  5. f(2) + ∆f(2) = f(3)

Answer:

  1. False
  2. True
  3. True
  4. True
  5. True

II. 2 Mark Questions

Question 1.
Find the missing term from the following data.
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q1
Solution:
Since three values of y = f(x) are given, the polynomial which fits the data is of degree two.
Hence third differences are zero.
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q1.1

Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

Question 2.
From the following data estimate the export for the year 2000
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q2
Solution:
Consider a polynomial of degree two.
Hence third differences are zero.
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q2.1

Question 3.
For the tabulated values of y = f(x), find ∆y3 and ∆3y2
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q3
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q3.1

Question 4.
If f(x) = x2 + ax + b, find ∆r f(x)
Solution:
∆f(x) = f(x + h) – f(x)
= [(x + h )2 + a(x + h) + b] – [x2 + ax + b]
= 2xh + h2 + ah
2 f(x) = [2(x + h) h + h2 + ah] – [2xh + h2 + ah] = 2h2
3 f(x) = 0
Thus ∆r f(x) = 0 for all r ≥ 3

Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

Question 5.
Show that ∆3 y4 = ∇3 y7
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems II Q5
Hence proved

III. 3 and 5 Marks Questions

Question 1.
If f(0) = 5, f(1) = 6, f(3) = 50, f(4) = 105, find f(2) by using Lagrange’s formula.
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q1
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q1.1

Question 2.
Find y when x = 0.2 given that
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q2
Solution:
Since the required value of y is near the beginning of the table, we use Newton’s forward difference formula
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q2.1
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q2.2

Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

Question 3.
Find the number of men getting wages between Rs.30 and Rs.35 from the following table:
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q3
Solution:
The difference table
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q3.1
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q3.2
No. of men getting wages less than 35 is 24. Therefore the number of men getting wages between Rs.30 and Rs.35 is y (35) – y (30)
(i.e) 24 – 9 = 15

Question 4.
Using Newton’s formula estimate the population of town for the year 1995:
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q4
Solution:
1995 lies in (1991, 2001). Hence we use Newton’s backward interpolation formula.
Here x = 1995, xn = 2001, h = 10
1995 = xn + nh
⇒ 1995 = 2001 + 10n
⇒ n = \(\frac{1995-2001}{10}\)
⇒ n = -0.6
The backward difference table is given below
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q4.1
y = 101 – 4.8 + 0.48 + 0.056 + 0.1008
y = 96.8368
Hence the population for the year 1995 is 96.837 thousands.

Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems

Question 5.
Using Lagrange’s formula find y(11) from the following table
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q5
Solution:
Given
x0 = 6, y0 = 13
x1 = 7, y1 = 14
x2 = 10, y2 = 15
x3 = 12, y3 = 17
x = 11
Using Lagrange’s formula,
Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Additional Problems III Q5.1

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