# Numerical Analysis (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 8 - 13 of 21

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## Question number: 8

» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)

Appeared in Year: 2010

### Describe in Detail

If f (x) =1/x ^{2}, find the divided differences f (a, b) and f (a, b, c).

### Explanation

we can write the Newton’s divided difference formulas as

First find f (b, c)

Then, from (1) and (2)

## Question number: 9

» Numerical Analysis » Summation Formula » Euler-Maclaurin's

Appeared in Year: 2014

### Describe in Detail

Using Euler’s method, compute the values of y correct upto 4 places of decimal for the differential equation with initial condition x _{0} = 0, y _{0} = 1, taking h = 0.05.

### Explanation

The given differential equation is

with initial condition x _{0} = 0, y _{0} = 1, taking h = 0.05.

Using Euler’s method,

where

So, putting the initial condition, when n = 0

and

Now first modification of y _{1}

## Question number: 10

» Numerical Analysis » Interpolation Formulae » Gauss

Appeared in Year: 2011

### Describe in Detail

Use mathematical induction to prove

### Explanation

Let ∆ ^{i} define the i ^{th} finite difference which is defined as

∆ =E-1 where E is a shift operator that is

E ^{n} f _{x} = f _{x + nh}

Then

## Question number: 11

» Numerical Analysis » Numerical Integration

Appeared in Year: 2012

### Describe in Detail

The speed y (in km/hr) of a car at different points of time x between 10: 00 a. m. and 10: 40 a. m. on some day was recorded as follows:

Time x (a. m. ) | 10.00 | 10.10 | 10.20 | 10.30 | 10.40 |

Speed y (in km/hr) | 24.2 | 35.0 | 41.3 | 42.8 | 39.2 |

Calculate the approximate distance covered by the car between 10: 00 a. m. and 10: 40 a. m. on that day using Simpson’s one-third formula for numerical integration.

### Explanation

Answer: T he one-third rule of the integrand using Simpson’s rule is

Here n = 4, b = 10: 40 = 40 minutes, a = 10.00 = 0 minutes, h= (b-a) /n = 10 minutes

The distance covered by the car between 10: 00 a. m. and 10: 40

## Question number: 12

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2010

### Describe in Detail

The following values of the function f (x) for values of x are given:

f (l) = 4, f (2) = 5, f (7) = 5, f (8) = 4.

Find the value of f (6) and also the value of x for which f (x) is maximum or minimum.

### Explanation

This is written in the table in this function y = f (x)

x | 1 | 2 | 7 | 8 |

y | 4 | 5 | 5 | 4 |

For find x = 6, we use Lagrange’s interpolation polynomial because the x values is not equal interval. There are four values of x which gives the

## Question number: 13

» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)

Appeared in Year: 2011

### Describe in Detail

Use Simpson’s rule with five ordinates to compute an approximation to π with the help of the integration of the function (1 + x ^{2}) ^{-1} from 0 to 1.

### Explanation

T he rule of the integrand using Simpson’s rule for five ordinates is

where , n = 5

h= (1 - 0) /5 = 1/5

We divide the range 0 to 1 in five equal part i. e.

Use Simpson’s rule