#### Chapters

Chapter 2: Integral Calculus – 1

Chapter 3: Integral Calculus – 2

Chapter 4: Differential Equations

Chapter 5: Numerical Methods

Chapter 6: Random Variable and Mathematical expectation

Chapter 7: Probability Distributions

Chapter 8: Sampling techniques and Statistical Inference

Chapter 9: Applied Statistics

Chapter 10: Operations Research

## Chapter 5: Numerical Methods

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

Evaluate Δ(log ax)

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

If h = 1 then prove that (E^{–1}Δ)x^{3} = 3x^{2} – 3x + 1

If f(x) = x^{2} + 3x than show that Δf(x) = 2x + 4

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

Find the missing entry in the following table

x |
0 | 1 | 2 | 3 | 4 |

y_{x} |
1 | 3 | 9 | - | 81 |

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

Find the missing entries from the following.

x |
0 | 1 | 2 | 3 | 4 | 5 |

y = f(x) |
0 | - | 8 | 15 | - | 35 |

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

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 |

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.

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

x |
0 | 1 | 2 | 3 |

f(x) |
1 | 2 | 1 | 10 |

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 |

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.

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 |

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.

Find f(2.8) from the following table:

x |
0 | 1 | 2 | 3 |

f(x) |
1 | 2 | 11 | 34 |

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 |

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 |

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 |

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

x |
3 | 7 | 11 | 19 |

f(x) |
42 | 43 | 47 | 60 |

### 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

Choose the correct alternative:

Δ^{2}y_{0} =

y

_{2}– 2y_{1}+ y_{0}y

_{2}+ 2y_{1}– y_{0}y

_{2}+ 2y_{1}+ y_{0}y

_{2}+ y_{1}+ 2y_{0}

Choose the correct alternative:

Δf(x) =

f(x + h)

f(x) – f (x + h)

f(x + h) – f(x)

f(x) – f (x – h)

Choose the correct alternative:

E ≡

1 + Δ

1 – Δ

1 + ∇

1 – ∇

Choose the correct alternative:

If h = 1, then Δ(x^{2}) =

2x

2x – 1

2x + 1

1

Choose the correct alternative:

If c is a constant then Δc =

c

Δ

Δ

^{2}0

Choose the correct alternative:

If m and n are positive integers then Δ^{m} Δ^{n} f(x)=

Δ

^{m+n}f(x)Δ

^{m}f(x)Δ

^{n}f(x)Δ

^{m-n}f(x)

Choose the correct alternative:

If ‘n’ is a positive integer Δ^{n}[Δ^{-n} f(x)]

f(2x)

f(x + h)

f(x)

Δf(x)

Choose the correct alternative:

E f(x) =

f(x – h)

f(x)

f(x + h)

f(x + 2h)

Choose the correct alternative:

∇ ≡

1 + E

1 – E

1 – E

^{-1}1 + E

^{-1}

Choose the correct alternative:

∇f(a) =

f(a) + f(a – h)

f(a) – f(a + h)

f(a) – f(a – h)

f(a)

Choose the correct alternative:

For the given points (x_{0}, y_{0}) and (x_{1}, y_{1}) 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`

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

Choose the correct alternative:

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

2x – 3

2x + 3

x + 3

x – 3

Choose the correct alternative:

For the given data find the value of Δ^{3}y_{0} is

x |
5 | 6 | 9 | 11 |

y |
12 | 13 | 15 | 18 |

1

0

2

– 1

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

If f(x) = e^{ax} then show that f(0), Δf(0), Δ^{2}f(0) are in G.P

Prove that (1 + Δ)(1 – ∇) = 1

Prove that Δ∇ = Δ – ∇

Prove that EV = Δ = ∇E

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

Find the missing figures in the following table:

x |
0 | 5 | 10 | 15 | 20 | 25 |

y |
7 | 11 | - | 18 | - | 32 |

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

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 |

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

If u_{0} = 560, u_{1} = 556, u_{2} = 520, u_{4} = 385, show that u_{3} = 465

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

x |
1 | 2 | 3 | 4 | 5 |

y |
1 | – 1 | 1 | – 1 | 1 |

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

## 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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