Tamil Nadu Board of Secondary EducationHSC Commerce Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 5 - Numerical Methods [Latest edition]

## Chapter 5: Numerical Methods

Exercise 5.1Exercise 5.2Exercise 5.3Miscellaneous problems
Exercise 5.1 [Pages 111 - 112]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 5 Numerical MethodsExercise 5.1 [Pages 111 - 112]

Exercise 5.1 | Q 1 | Page 111

Evaluate Δ(log ax)

Exercise 5.1 | Q 2 | Page 111

If y = x3 – x2 + x – 1 calculate the values of y for x = 0, 1, 2, 3, 4, 5 and form the forward differences table

Exercise 5.1 | Q 3 | Page 111

If h = 1 then prove that (E–1Δ)x3 = 3x2 – 3x + 1

Exercise 5.1 | Q 4 | Page 112

If f(x) = x2 + 3x than show that Δf(x) = 2x + 4

Exercise 5.1 | Q 5 | Page 112

Evaluate Δ[1/((x + 1)(x + 2))] by taking ‘1’ as the interval of differencing

Exercise 5.1 | Q 6 | Page 112

Find the missing entry in the following table

 x 0 1 2 3 4 yx 1 3 9 - 81
Exercise 5.1 | Q 7 | Page 112

Following are the population of a district

 Year (x) 1881 1891 1901 1911 1921 1931 Population (y)Thousands 363 391 421 - 467 501

Find the population of the year 1911

Exercise 5.1 | Q 8 | Page 112

Find the missing entries from the following.

 x 0 1 2 3 4 5 y = f(x) 0 - 8 15 - 35
Exercise 5.2 [Pages 119 - 120]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 5 Numerical MethodsExercise 5.2 [Pages 119 - 120]

Exercise 5.2 | Q 1 | Page 119

Using graphic method, find the value of y when x = 48 from the following data:

 x 40 50 60 70 y 6.2 7.2 9.1 12
Exercise 5.2 | Q 2 | Page 119

The following data relates to indirect labour expenses and the level of output

 Months Jan Feb Mar Units of output 200 300 400 Indirect labourexpenses (Rs) 2500 2800 3100
 Months Apr May June Units of output 640 540 580 Indirect labourexpenses (Rs) 3820 3220 3640

Estimate the expenses at a level of output of 350 units, by using graphic method.

Exercise 5.2 | Q 3 | Page 119

Using Newton’s forward interpolation formula find the cubic polynomial.

 x 0 1 2 3 f(x) 1 2 1 10
Exercise 5.2 | Q 4 | Page 119

The population of a city in a censes taken once in 10 years is given below. Estimate the population in the year 1955.

 Year 1951 1961 1971 1981 Population inlakhs 35 42 58 84
Exercise 5.2 | Q 5 | Page 119

In an examination the number of candidates who secured marks between certain intervals was as follows:

 Marks 0 - 19 20 - 39 40 - 59 60 - 79 80 - 99 No. of candidates 41 62 65 50 17

Estimate the number of candidates whose marks are less than 70.

Exercise 5.2 | Q 6 | Page 119

Find the value of f(x) when x = 32 from the following table:

 x 30 5 40 45 50 f(x) 15.9 14.9 14.1 13.3 12.5
Exercise 5.2 | Q 7 | Page 119

The following data gives the melting point of a alloy of lead and zinc where ‘t’ is the temperature in degree c and P is the percentage of lead in the alloy.

 P 40 50 60 70 80 90 T 180 204 226 250 276 304

Find the melting point of the alloy containing 84 percent lead.

Exercise 5.2 | Q 8 | Page 119

Find f(2.8) from the following table:

 x 0 1 2 3 f(x) 1 2 11 34
Exercise 5.2 | Q 9 | Page 119

Using interpolation estimate the output of a factory in 1986 from the following data.

 Year 1974 1978 1982 1990 Output in 1000tones 25 60 80 170
Exercise 5.2 | Q 10 | Page 119

Use Lagrange’s formula and estimate from the following data the number of workers getting income not exceeding Rs. 26 per month.

 Income notexceeding (₹) 15 25 30 35 No. of workers 36 40 45 48
Exercise 5.2 | Q 11 | Page 120

Using interpolation estimate the business done in 1985 from the following data

 Year 1982 1983 1984 1986 Business done(in lakhs) 150 235 365 525
Exercise 5.2 | Q 12 | Page 120

Using interpolation, find the value of f(x) when x = 15

 x 3 7 11 19 f(x) 42 43 47 60
Exercise 5.3 [Pages 120 - 121]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 5 Numerical MethodsExercise 5.3 [Pages 120 - 121]

#### MCQ

Exercise 5.3 | Q 1 | Page 120

Choose the correct alternative:

Δ2y0

• y2 – 2y1 + y0

• y2 + 2y1 – y0

• y2 + 2y1 + y0

• y2 + y1 + 2y0

Exercise 5.3 | Q 2 | Page 120

Choose the correct alternative:

Δf(x) =

• f(x + h)

• f(x) – f (x + h)

• f(x + h) – f(x)

• f(x) – f (x – h)

Exercise 5.3 | Q 3 | Page 120

Choose the correct alternative:

E ≡

• 1 + Δ

• 1 – Δ

• 1 + ∇

• 1 – ∇

Exercise 5.3 | Q 4 | Page 120

Choose the correct alternative:

If h = 1, then Δ(x2) =

• 2x

• 2x – 1

• 2x + 1

• 1

Exercise 5.3 | Q 5 | Page 120

Choose the correct alternative:

If c is a constant then Δc =

• c

• Δ

• Δ2

• 0

Exercise 5.3 | Q 6 | Page 120

Choose the correct alternative:

If m and n are positive integers then Δm Δn f(x)=

• Δm+n f(x)

• Δm f(x)

• Δn f(x)

• Δm-n f(x)

Exercise 5.3 | Q 7 | Page 120

Choose the correct alternative:

If ‘n’ is a positive integer Δn-n f(x)]

• f(2x)

• f(x + h)

• f(x)

• Δf(x)

Exercise 5.3 | Q 8 | Page 120

Choose the correct alternative:

E f(x) =

• f(x – h)

• f(x)

• f(x + h)

• f(x + 2h)

Exercise 5.3 | Q 9 | Page 120

Choose the correct alternative:

∇ ≡

• 1 + E

• 1 – E

• 1 – E-1

• 1 + E-1

Exercise 5.3 | Q 10 | Page 120

Choose the correct alternative:

∇f(a) =

• f(a) + f(a – h)

• f(a) – f(a + h)

• f(a) – f(a – h)

• f(a)

Exercise 5.3 | Q 11 | Page 120

Choose the correct alternative:

For the given points (x0, y0) and (x1, y1) the Lagrange’s formula is

• y(x) = (x - x_1)/(x_0 - x_1) y_0 + (x - x_0)/(x_1 - x_0) y_1

• y(x) = (x_1 - x)/(x_0 - x_1) y_0 + (x - x_0)/(x_1 - x_0) y_1

• y(x) = (x - x_1)/(x_0 - x_1) y_1 + (x - x_0)/(x_1 - x_0) y_0

• y(x) = (x_1 - x)/(x_0 - x_1) y_1 + (x - x_0)/(x_1 - x_0) y_0

Exercise 5.3 | Q 12 | Page 120

Choose the correct alternative:

Lagrange’s interpolation formula can be used for

• Equal intervals only

• Unequal intervals only

• Both equal and unequal intervals

• None of these

Exercise 5.3 | Q 13 | Page 120

Choose the correct alternative:

If f(x) = x2 + 2x + 2 and the interval of differencing is unity then Δf(x)

• 2x – 3

• 2x + 3

• x + 3

• x – 3

Exercise 5.3 | Q 14 | Page 121

Choose the correct alternative:

For the given data find the value of Δ3y0 is

 x 5 6 9 11 y 12 13 15 18
• 1

• 0

• 2

• – 1

Miscellaneous problems [Page 121]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 5 Numerical MethodsMiscellaneous problems [Page 121]

Miscellaneous problems | Q 1 | Page 121

If f(x) = eax then show that f(0), Δf(0), Δ2f(0) are in G.P

Miscellaneous problems | Q 2. i) | Page 121

Prove that (1 + Δ)(1 – ∇) = 1

Miscellaneous problems | Q 2. ii) | Page 121

Prove that  Δ∇ = Δ – ∇

Miscellaneous problems | Q 2. iii) | Page 121

Prove that EV = Δ = ∇E

Miscellaneous problems | Q 3 | Page 121

A second degree polynomial passes though the point (1, –1) (2, –1) (3, 1) (4, 5). Find the polynomial

Miscellaneous problems | Q 4 | Page 121

Find the missing figures in the following table:

 x 0 5 10 15 20 25 y 7 11 - 18 - 32
Miscellaneous problems | Q 5 | Page 121

Find f(0.5) if f(– 1) = 202, f(0) = 175, f(1) = 82 and f(2) = 55

Miscellaneous problems | Q 6 | Page 121

From the following data find y at x = 43 and x = 84.

 x 40 50 60 70 80 90 y 184 204 226 250 276 304
Miscellaneous problems | Q 7 | Page 121

The area A of circle of diameter ‘d’ is given for the following values

 D 80 85 90 95 100 A 5026 5674 6362 7088 7854

Find the approximate values for the areas of circles of diameter 82 and 91 respectively

Miscellaneous problems | Q 8 | Page 121

If u0 = 560, u1 = 556, u2 = 520, u4 = 385, show that u3 = 465

Miscellaneous problems | Q 9 | Page 121

From the following table obtain a polynomial of degree y in x.

 x 1 2 3 4 5 y 1 – 1 1 – 1 1
Miscellaneous problems | Q 10 | Page 121

Using Lagrange’s interpolation formula find a polynominal which passes through the points (0, –12), (1, 0), (3, 6) and (4, 12)

## Chapter 5: Numerical Methods

Exercise 5.1Exercise 5.2Exercise 5.3Miscellaneous problems

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 5 - Numerical Methods

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Concepts covered in Class 12th Business Mathematics and Statistics Answers Guide chapter 5 Numerical Methods are Finite Differences, Interpolation.

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