Free Printable numerical methods 2

1.

Newton-Raphson method is used to find the root of the equation x2 - 2 = 0

If iterations are started from - 1, then iterations will be

a)

converge to -1

b)

converge to √2

c)

converge to -√2

d)

No converge

2.

We wish to solve x2 - 2 = 0 by Newton Raphson technique. If initial guess is x0 = 1.0, subsequent estimate of x (i.e. x1) will be

a)

1.414

b)

1.5

c)

2.0

d)

None of these

3.

Using Newton-Raphson method, find a root correct to three decimal places of the equation x3 - 3x - 5 = 0

a)

2.275

b)

2.279

c)

2.222

d)

None of these

4.

In the Gauss elimination method for solving a system of linear algebraic equations, triangularzation leads to

a)

Diagonal matrix

b)

Lower triangular matrix

c)

Upper triangular matrix

d)

Singular matrix

5.

If Δf(x) = f(x+h) - f(x), then a constant k, Δk equals

a)

1

b)

0

c)

f(k)- f(0)

d)

f(x + k) - f(x)

6.

The root of x3 - 2x - 5 = 0 correct to three decimal places by using Newton-Raphson method is

a)

2.0946

b)

1.0404

c)

1.7321

d)

0.7011

7.

By Trapezoidal rule,

\int_{x_0}^{x_n}ydx=---

a)

$\frac{h}{2}\left(y_0+2\left(y_1+y_2+...+y_{n-1}\right)+y_{n_{ }}\right)$

b)

$\frac{h}{2}\left(y_0-2\left(y_1+y_2+...+y_{n-1}\right)+y_{n_{ }}\right)$

c)

$\frac{h}{2}\left(y_0+2\left(y_1+y_2+...+y_{n-1}\right)-y_{n_{ }}\right)$

d)

$\frac{h}{3}\left(y_0+2\left(y_1+y_2+...+y_{n-1}\right)+y_{n_{ }}\right)$

8.

18. Trapezoidal formula is also known as __________.

a)

A. Simpson’z rule

b)

B. Co-ordinate method

c)

C. Prismoidal method

d)

D. Average end area method.

9.

Given above is the general formula for fourth order Runge-Kutta method. What is w?

a)

w denotes the evaluation (step) of RK method

b)

w denotes the consistency of RK method

c)

w are contants, any real numbers R

10.

Numerical integration techniques can be applied to find

a)

Area

b)

Volume of a solid of revolution

c)

sum of the functional values

d)

all of these

11.

$2\frac{dy}{dx}+5y^2=\sin2x$

To solve the ordinary differential equation by Euler’s method, you need to rewrite the equation as

a)

$2\frac{dy}{dx}=5y^2-\sin2x$

b)

$\frac{dy}{dx}=\frac{1}{2}\left[\sin2x-5y^2\right]$

c)

$\frac{dy}{dx}=\frac{1}{2}\left[\sin2x+5y^2\right]$

d)

$\frac{dy}{dx}=\frac{5}{2}\left[\sin2x+y^2\right]$

12.

In Trapezoidal rule, for the interval (2,5) if h=0.5 then the value of n is_________

a)

5

b)

2

c)

6

d)

4

13.

Simpson’s 1/3 rule is applicable for eight ordinates.

a)

True

b)

False

14.

Which of the following formula is a particular case of Runge – Kutta formula of first order?.

a)

Taylor's

b)

Modified Eulers

c)

Eulers

d)

Picards

15.

Taylor’s series method will be very useful to given some _______ for powerful numerical methods. (a) (b) final value (c) initial starting value (d) middle value

a)

Initial value

b)

Final value

c)

Mid value

d)

Initial starting value

16.

Simpson’s 3/8 rule is applicable for nine ordinates.

a)

True

b)

False

17.

h=0.25

a)

0.388

b)

0.188

c)

0.288

d)

0.488

18.

Which of the following are single step techniques?

a)

Taylor's

b)

Euler's

c)

RK Mehod

d)

Milne's

19.

False Position method is used to solve

a)

quadratic equations

b)

system of linear equations

c)

eigen value problems

d)

nonlinear equations

20.

The iteration formula for Newton-Raphson method is

a)

$x_{n+1}=x_n+\frac{f\left(x_n\right)}{f'\left(x_n\right)}$

b)

$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f'\left(x_n\right)}$

c)

$x_{n+1}=x_n-\frac{f'\left(x_n\right)}{f\left(x_n\right)}$

d)

$x_{n+1}=x_n+\frac{f'\left(x_n\right)}{f\left(x_n\right)}$

21.

Given that the equation $2x^3-4x^2-1=0$ has a root in the interval [2, 3]. Using Newton-Raphson method, the root correct to 2 decimal places is

a)

2.13

b)

2.11

c)

2.09

d)

2.15

22.

Which of the following iteration formula can not be derived from $x^2-6x+2=0$

a)

$x=\frac{x^2+2}{6}$

b)

$x=\sqrt{6x-2}$

c)

$x=6-\frac{2}{x}$

d)

$x=\sqrt{3\left(x-2\right)}$

23.

y p = y 0 + pC 1 ∆y o + pC 2 ∆ 2 y o + (p+1)C 3 ∆ 3 y -1 + ( p+1)C 4 ∆ 4 y -2 + ... represents the formula of

a)

Newton’s backward interpolation

b)

Gauss forward interpolation

c)

Lagrange's interpolation

d)

Newton’s forward interpolation

24.

y p = y 0 + pC 1 ∆y-1+(p+1)C 2 ∆ 2 y-1+(p+1)C 3 ∆ 3 y -2 ++(p+2)C 4 ∆ 4 y -2 + ...represents the formula

of

a)

Newton's forward interpolation

b)

Newton's backward interpolation

c)

Gauss forward interpolation

d)

Gauss backward interpolation

25.

Which of the following is a numerical equation ?

a)

aX³+bX²+c=4

b)

3X⁴-8X³+5X²-6X+7=0

c)

cos3x -4x =0

26.

Which of the following is an algebraic equation ?

a)

3X⁴-8X³+5X²-6X+7=0

b)

aX³+bX²+c=4

c)

cos3x -4x =0

27.

Which of the following is a transcendental equation ?

a)

aX³+bX²+c=4

b)

3X⁴-8X³+5X²-6X+7=0

c)

cos3x -4x =0

28.

One root of the equation

X³ +4X²-8X +2 =0 lies between........

a)

0 and 1

b)

-1 and 0

c)

-1 and 1

29.

The roots of the equation

X² - 1 =0 is ................

a)

1,-1

b)

1,1

c)

-1,-1

30.

False Position method is used to solve

a)

quadratic equations

b)

system of linear equations

c)

eigen value problems

d)

nonlinear equations

31.

Solving nonlinear equations means ..

a)

finding the x value for which f(x)=0.

b)

finding the root of the functions.

c)

finding the zero of the functions.

32.

Regula Falsi method is also known as ------

a)

Interval halving method

b)

Bolzano's method

c)

Method of False Position

d)

Newton method

33.

Bisection method is a simple but slowly convergent method.

a)

True

b)

False

34.

The equation f(x) = 0 is called

-------- if f(x) involves logarithmic, trigonometric and exponential functions.

a)

algebraic

b)

transcendental

c)

numerical

35.

In numerical integration the width of the sub intervals is given by

a)

$h=\frac{x_n-x_0}{n}$

b)

$h=\frac{x_n+x_0}{n}$

c)

$h=\frac{x_0-x_n}{n}$

d)

none of thesse

36.

In Trapezoidal rule, for the interval (2,5) if h=0.5 then the value of n is_________

a)

5

b)

2

c)

6

d)

4

37.

Simpson’s 1/3 rule is applicable for five ordinates.

a)

True

b)

False

38.

The iteration formula for Newton-Raphson method is

a)

$x_{n+1}=x_n+\frac{f\left(x_n\right)}{f'\left(x_n\right)}$

b)

$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f'\left(x_n\right)}$

c)

$x_{n+1}=x_n-\frac{f'\left(x_n\right)}{f\left(x_n\right)}$

d)

$x_{n+1}=x_n+\frac{f'\left(x_n\right)}{f\left(x_n\right)}$

39.

What line is used to find the next iteration in Newton-Raphson method?

a)

Secant

b)

Normal

c)

Chord

d)

Tangent

40.

In Newton-Raphson method, if we want to give our answer correct to 3 decimal places then we need to use how many decimal places in the calculation of the iterations?

a)

3 decimal places

b)

2 decimal places

c)

4 decimal places

d)

any number of decimal places will do

41.

In Newton-Raphson method, if we want to give our answer correct to 3 decimal places then we need to use how many decimal places in the calculation of the iterations?

a)

3 decimal places

b)

2 decimal places

c)

4 decimal places

d)

any number of decimal places will do

42.

The following are ways to verify whether there is a root in the interval [a, b] for a continuous function f(x) except

a)

f(a) < 0 and f(b) > 0

b)

f(a) x f(b) < 0

c)

f(a) x f(b) > 0

d)

The sign for f(a) and f(b) changes

43.

Bisection method is also known as ----- method.

a)

Newton Raphson

b)

Regula Falsi

c)

Interval Halving

d)

Newton

44.

What is the solution to the 4x5 Matrix ?

a)

( 3, 4, 9, 6 )

b)

( 3, 4, 9, -6 )

c)

( -6, 9, 4, 3 )

d)

( 6, 9, 4, 3 )

45.

Using back-substitution, calculate the solution for the REF matrix.

a)

(3, -6, 3)

b)

(-6, -6, 3)

c)

(6, -6, 3)

d)

(-6, -6, 3)

46.

Iteration method is a self correcting method

a)

True

b)

False

47.

Iteration will converge if the leading diagonal element are greater than the sum of other coefficients of that row.

a)

True

b)

False

48.

This is an example of a

a)

System of Quadratic Equations

b)

Reduced Row Echelon Form

c)

Augmented Matrix

d)

A Canine Doing a Backflip

49.

What is an appropriate first row command to solve this by Gaussian Elimination?

a)

2R2 + R2 -> R2

b)

-2R1 + R2 -> R2

c)

R3 +R2 --> R3

d)

2R1 + R2 --> R2

numerical methods 2

Total questions: 49

Worksheet time: 1hrs 9mins

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