Free Printable Midterm_Numerical Methods

1.

f(x)=x3+2x, [−1,1]

a)

1/√3

b)

±1/√3

c)

-1/√3

d)

The MVT doesn't apply

2.

Find the root for the equation $3x+\cos x=3$ between $x=0$ and $x=1$ correct to four decimal places.

a)

0.5

b)

0.7579

c)

0.758

d)

0.7469

3.

Figure shows the equation $2x^3=2-x^2$ . Find the root of the equation $2x^3+x^2-2=0$ in $\left[-2,2\right]$ .

a)

0

b)

1.414

c)

-1.414

d)

0.858

4.

Find how many intersection are there between $y=2x$ and $y=\ln x$ ?

a)

1

b)

0

c)

2

5.

Find the interval where the root of the equation $x^3=-x-1$ lies.

a)

$\left[0,0.5\right]$

b)

$\left[-1,-0.5\right]$

c)

$\left[-0.5,0\right]$

d)

$\left[0.5,1\right]$

6.

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

7.

By using graphical method, there is a real root between [a,b] for $\ln\left(x-2\right)+x^2-4=0$ Hence, state the values of a and b.

a)

$a=1,\ b=4$

b)

$a=2,\ b=3$

c)

$a=0,\ b=2$

d)

$a=0,\ b=4$

8.

By taking 0.2 as the first approximation, evaluate the real root of the equation $x^2-\frac{1}{x}+4=0$ correct to 3 s.f.

a)

0.246

b)

0.635

c)

0.153

d)

0.724

9.

Show that the equation $2x^3+x^2=33$ has a root in the interval $2<x<2.5$ .

Find this root correct to 3 s.f.

a)

2.17

b)

2.45

c)

2.39

d)

2.28

10.

By taking $x=2$ as the first approximation, calculate using Newton-Raphson method, the third approximation to $7^{\frac{1}{3}}$ (3 s.f.)

a)

1.91

b)

1.92

c)

1.89

d)

1.90

11.

Estimate $\int_0^8\sqrt[]{1+x^2}dx$ by using trapezoidal rule with 5 ordinates correct to 3 d.p. What is this value?

a)

33.946; approximated value

b)

33.946; absolute value

c)

32.246; approximated value

d)

32.246; absolute value

12.

By taking 0.2 as the first approximation, evaluate the real root of the equation $x^2-\frac{1}{x}+4=0$ correct to 3 s.f.

a)

0.246

b)

0.635

c)

0.153

d)

0.724

13.

Show that the equation $2x^3+x^2=33$ has a root in the interval $2<x<2.5$ .

Find this root correct to 3 s.f.

a)

2.17

b)

2.45

c)

2.39

d)

2.28

14.

By taking $x=2$ as the first approximation, calculate using Newton-Raphson method, the third approximation to $7^{\frac{1}{3}}$ (3 s.f.)

a)

1.91

b)

1.92

c)

1.89

d)

1.90

15.

The convergence of which of the following method is sensitive to starting value?

a)

False position

b)

Gauss seidal method

c)

Newton-Raphson method

d)

All of these

16.

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

17.

Which of the following statements applies to the bisection method used for finding roots of functions?

a)

Converges within a few iterations

b)

Guaranteed to work for all continuous functions

c)

Is faster than the Newton-Raphson method

d)

Requires that there be no error in determining the sign of the function

18.

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

19.

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

20.

Using Bisection method, negative root of x3 - 4x + 9 = 0 correct to three decimal places is

a)

-2.506

b)

-2.706

c)

- 2.406

d)

None of these

21.

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

22.

The equation f(x) is given as x2-4=0. Considering the initial approximation at x=6 then the value of x1 is given as

a)

10/3

b)

4/3

c)

7/3

d)

13/3

23.

How many roots are there in the equation

x=\frac{x^4+2}{5}

?

a)

1

b)

2

c)

3

d)

4

24.

Find the root for the equation $3x+\cos x=3$ between $x=0$ and $x=1$ correct to four decimal places.

a)

0.5

b)

0.7579

c)

0.758

d)

0.7469

25.

Find the interval where the root of the equation $x^3=-x-1$ lies.

a)

$\left[0,0.5\right]$

b)

$\left[-1,-0.5\right]$

c)

$\left[-0.5,0\right]$

d)

$\left[0.5,1\right]$

26.

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

27.

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

28.

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

a)

Interval halving method

b)

Bolzano's method

c)

Method of False Position

d)

Newton method

29.

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)

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

30.

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

a)

Secant

b)

Normal

c)

Chord

d)

Tangent

31.

It is a branch of applied mathematics, which studies the methods and algorithms to find approximate solutions of equations when exact solutions cannot be determined via algebraic methods.

a)

Numerical Methods

b)

Physics

c)

Chemistry

d)

Engineering Economics

32.

These are concerned with the value of a variable or a parameter that satisfies a single non-linear equation and are especially valuable in engineering design contexts where it is open impossible to explicitly solve design equations for parameters.

a)

Systems of Linear Algebraic Equations

b)

Ordinary Differential Equations

c)

Roots of Equations

d)

Integration

33.

It refers to how closely individual computer or measured values agree with each other.

a)

Precision

b)

Accuracy

c)

Truncation

d)

Round-off

34.

It refers to how closely a computed or measured value agrees with the true value of the parameter being measured.

a)

Precision

b)

Accuracy

c)

Truncation

d)

Round-off

35.

These errors happened when number having limited significant figures are used to represent exact numbers.

a)

Concatenation Errors results

b)

Truncation Errors results

c)

Round-off Errors results

d)

Relative True Errors results

36.

These happened when approximations are used to represent exact mathematical procedures.

a)

Truncation Errors results

b)

Round-off Errors results

c)

Relative Errors results

d)

Concatenation Errors results

Midterm_Numerical Methods

Total questions: 36

Worksheet time: 21mins

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