The number of ways to permute three correct answers among five questions is 5Choose3 which is 5!/(3!*2!) which equals 10. We must then have the correct answer three times which happens .25 of the time, and two wrong answers 75 of the time. So the probability is 10*0.25^3*.75^2 which is 0.087890625 or roughly an 8.8% chance.

5 0

3 years ago

1. What are the solutions (coordinate points) to the system of equations?

Leto [7]

Question 1

The given system of equations is:

$y = {x}^{2} + 5x + 6 \\ y = 3x + 6$

Equate the two equations:

${x}^{2} + 5x + 6 = 3x + 6$

Rewrite in standard form:

${x}^{2} + 5x - 3x + 6 - 6 = 0$

${x}^{2} + 2x = 0$

$x(x + 2) = 0$

$x = 0 \: or \: x = - 2$

When we put x=0, in y=3x +6, we get:

$y = 3(0) + 6 = 6$

One solution is (0,6)

When we put x=-2, into y=3x+6, we get:

$y = 3( - 2) + 6 = 0$

Another solution is (-2,0)

The solutions are; (0,6) and (-2,0)

Question 2:

The function is

$f(x) = {x}^{6} - {x}^{4}$

Let us put x=-x,

$f( - x) = {( - x)}^{6} - {( - x)}^{4}$

This gives:

$f( - x) = {x}^{6} - {x}^{4}$

We can observe that:

$f(x) = f( - x)$

This is the property of an even function.

Question 3:

The given function is

$f(x) = {x}^{2} + 3x - 2$

The average rate of change of f(x) from x=a to x=b is given as:

$\frac{f(b) - f(a)}{b - a}$

This is the slope of the secant line connecting the two points on f(x)

From x=2 to x=6, the average rate of change

$= \frac{f(6) - f(2)}{6 - 2} \\ = \frac{ {6}^{2} + 3 \times 6 - 2 - {2}^{2} - 3 \times 2 + 2 }{4} \\ = \frac{36 + 18 - 4 - 6}{4} \\ = \frac{44}{4} \\ = 11$

The average rate of change is 11

3 0

3 years ago