Consider the function represented by 9x + 3y = 12 with x as the independent variable. How can this function be written
using function notation?
f(x)=
x+4
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Answer:

regular

Step-by-step explanation:

1. look at table

Answer:

The correct option is A)

P(Win|Regular)=0.76

P(Win|Prevent )=0.58

You are more likely to win by playing regular defense.

Step-by-step explanation:

Consider the provided table.

We need to find which strategy is better.

If team play regular defense then they win 38 matches out of 50.

[tex]Probability=\frac{\text{Favorable outcomes}}{\text{Total number of outcomes}}[/tex]

[tex]P(Win|Regular)=\frac{38}{50}[/tex]

[tex]P(Win|Regular)=0.76[/tex]

If team play prevent defense then they win 29 matches out of 50.

Thus, the probability of win is:

[tex]P(Win|Prevent )=\frac{29}{50}[/tex]

[tex]P(Win|Prevent )=0.58[/tex]

Since, 0.76 is greater than 0.58

That means the probability of winning the game by playing regular defense is more as compare to playing prevent defense.

Hence, the conclusion is: You are more likely to win by playing regular defense.

Thus, the correct option is A)

P(Win|Regular)=0.76

P(Win|Prevent )=0.58

You are more likely to win by playing regular defense.

Answer: 25 litres

Step-by-step explanation: If it takes 5 litres of diesel to cover 25km, it means it will take more litres to cover 125km. So you have to divide 125km by 25km then you will get 5 which you will then multiply by 5 litres to get 25 litres of diesel.

Answer:

x+6

Step-by-step explanation:

Let's see if the numerator is factorable.

Since the coefficient of x^2 is 1 (a=1), all you have to do is find two numbers that multiply to be -6  (c) and add up to be 5 (b).

Those numbers are 6 and -1.

So the factored form of the numerator is (x+6)(x-1)

So when you divide (x+6)(x-1) by (x-1) you get (x+6) because (x-1)/(x-1)=1 for number x except x=1 (since that would lead to division by 0).

Anyways, this is what I'm saying:

[tex]\frac{(x+6)(x-1)}{(x-1)}=\frac{(x+6)\xout{(x-1)}}{\xout{(x-1)}}[/tex]

[tex]x+6[/tex]

To find (f/g)(x) with f(x) = 6x + 2 and g(x) = x - 7, one must divide f(x) by g(x). The domain restriction occurs because division by zero is not defined, so we exclude the x value that makes g(x) zero, which is x = 7.

To perform the indicated operation (f/g)(x) with the given functions f(x) = 6x + 2 and g(x) = x - 7, we need to divide the function f(x) by the function g(x). This operation is equivalent to finding the quotient of the two functions, which is expressed as:

(f/g)(x) = f(x)/g(x) = (6x + 2)/(x - 7)

The domain restriction occurs when the denominator, g(x), is equal to zero since division by zero is undefined. So we must find the value of x for which g(x) = 0. Since g(x) = x - 7, setting this equal to zero gives us:

x - 7 = 0 → x = 7

Therefore, the domain of the function (f/g)(x) is all real numbers except for x = 7, because at x = 7 the function is undefined. The domain of (f/g)(x) can be expressed as ℜ - {7}, where ℜ represents the set of all real numbers.

Find the answer in the attachment.

The hyperbola's equation is x² / 3600 - y² / 121 = 1, centered at the origin (0,0). Its vertices are at (60,0), (-60,0) on the x-axis, and (0,11), (0,-11) on the y-axis.

To find the equation of the hyperbola, we need to determine its center and the distances from the center to the vertices along the x and y axes. The general equation of a hyperbola centered at (h, k) is given by:

(x - h)² / a² - (y - k)² / b² = 1

Where (h, k) is the center of the hyperbola, and 'a' and 'b' are the distances from the center to the vertices along the x and y axes, respectively.

In this case, since the hyperbola is symmetric along the x and y axes, the center is at the origin (0, 0). Also, we know the distance from the center to the vertices along the x-axis is 60 units (60 and -60) and along the y-axis is 11 units (11 and -11).

So, a = 60 and b = 11.

Now we can plug these values into the equation:

x² / (60)² - y² / (11)² = 1

Simplifying further:

x² / 3600 - y² / 121 = 1

And that's the equation of the hyperbola.

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Answer:

Step-by-step explanation:

There are 3 ways to find the other x intercept.

1) Polynomial Long Division.

Divide x^2 - 3x + 2 by the binomial x - 2, because by the Factor Theorem if a is a root of a polynomial then x - a is a factor of said polynomial.

2) Just solving for x when y = 0, by using the quadratic formula.

[tex]x^2 - 3x + 2 = 0\\x_{12} = \frac{3 \pm \sqrt{9 - 4(1)(2)}}{2} = \frac{3 \pm 1}{2} = 2, 1[/tex].

So the other x - intercept is at (1, 0)

3) Using Vietta's Theorem regarding the solutions of a quadratic

Namely, the sum of the solutions of a quadratic equation is equal to the quotient between the negative coefficient of the linear term divided by the coefficient of the quadratic term.

[tex]x_1 + x_2 = \frac{-b}{a}[/tex]

And the product between the solutions of a quadratic equation is just the quotient between the constant term and the coefficient of the quadratic term.

[tex]x_1 \cdot x_2 = \frac{c}{a}[/tex]

These relations between the solutions give us a brief idea of what the solutions should be like.

Answer:

129.8 approximately

Step-by-step explanation:

So this sounds like a problem for the Law of Cosines. The largest angle is always opposite the largest side in a triangle.

So 11 is the largest side so the angle opposite to it is what we are trying to find. Let's call that angle, X.

My math is case sensitive.

X is the angle opposite to the side x.

Law of cosines formula is:

[tex]x^2=a^2+b^2-2ab \cos(X)[/tex]

So we are looking for X.

We know x=11, a=4, and b=8 (it didn't matter if you called b=4 and a=8).

[tex]11^2=4^2+8^2-2(4)(8)\cos(X)[/tex]

[tex]121=16+64-64\cos(X)[/tex]

[tex]121=80-64\cos(X)[/tex]

Subtract 80 on both sides:

[tex]121-80=-64\cos(X)[/tex]

[tex]41=-64\cos(X)[/tex]

Divide both sides by -64:

[tex]\frac{41}{-64}=\cos(X)[/tex]

Now do the inverse of cosine of both sides or just arccos( )

[these are same thing]

[tex]\arccos(\frac{-41}{64})=X[/tex]

Time for the calculator:

X=129.8 approximately

Answer:

0.00008 ÷ 640,000,000 means

8*10^-5 ÷ 6.4*10^8

so let's collect to simplify the operation

(8÷6.4)*(10^-12) -5-7=-12

then the answer becomes 1.25×10^-14 that is B

Answer:

option C

Step-by-step explanation:

Evaluate 0.00008 ÷ 640,000,000.

0.00008 can be written in standard notation

Move the decimal point to the end

so it becomes  [tex]8 \cdot 10^{-5}[/tex]

for 640,000,000 , remove all the zeros and write it in standard form

[tex]64 \cdot 10^7[/tex]

Now we divide both

[tex]\frac{8 \cdot 10^{-5}}{64 \cdot 10^7}[/tex]

Apply exponential property

a^m divide by a^n  is a^ m-n

[tex]\frac{8}{64} =0.125[/tex]

[tex]\frac{10^{-5}}{10^7}=10^{-12}[/tex]

[tex]0.125 \cdot 10^{-12}= 1.25 \cdot 10^{-13}[/tex]

Answer:

Option B. g(x) is greater

Step-by-step explanation:

step 1

Find the value of f(x) when the value of x is equal to 3

we have

f(x)=2x-1

substitute the value of x=3

f(3)=2(3)-1=5

step 2

Find the value of g(x) when the value of x is equal to 3

Observing the graph

when x=3

g(3)=7

step 3

Compare the values

f(x)=5

g(x)=7

so

g(x) > f(x)

g(x) is greater

Answer:

Correct option is:

B. g(x) is greater

Step-by-step explanation:

Firstly, we find the value of f(x) when x=3

f(x)=2x-1

substitute the value of x=3

f(3)=2×3-1=5

On observing the graph, we see that g(x)=7 when x=3

Now, on Comparing the values of f(x) and g(x) when x=3

f(3)=5

g(3)=7

so, g(x) > f(x) when x=3

So, Correct option is:

B. g(x) is greater

ANSWER

[tex]{(x + 10)}^{2} + {(y + 6)}^{2} = 121[/tex]

EXPLANATION

The equation of the circle in general form is given as:

[tex] {x}^{2} + {y}^{2} + 20x + 12y + 15 = 0[/tex]

To obtain the standard form, we need to complete the squares.

We rearrange the terms to obtain:

[tex] {x}^{2} + 20x + {y}^{2} + 12y = - 15 [/tex]

Add the square of half the coefficient of the linear terms to both sides to get:

[tex]{x}^{2} + 20x +100 + {y}^{2} + 12y + 36 = - 15 + 100 + 36[/tex]

Factor the perfect square trinomial and simplify the RHS.

[tex]{(x + 10)}^{2} + {(y + 6)}^{2} = 121[/tex]

This is the equation of the circle in standard form.

Answer:

Step-by-step explanation:

[tex]4.5x-100>125\qquad\text{add 100 to both sides}\\\\4.5x-100+100>125+100\\\\4.5x>225\qquad\text{divide both sides by 4.5}\\\\\dfrac{4.5x}{4.5}>\dfrac{225}{4.5}\\\\x>50[/tex]

The solution to the inequality 4.5x - 100 > 125 is x > 50.

The given inequality is 4.5x - 100 > 125.

To solve it, first, add 100 to both sides to get 4.5x > 225. Then, divide by 4.5 to find x > 50. Therefore, the solution to the inequality is x > 50.

ANSWER

C) 4x-3y=-12

EXPLANATION

The intercept form of a straight line is given by:

[tex] \frac{x}{x - intercept} + \frac{y}{y - intercept} = 1[/tex]

From the the x-intercept is -3 and the y-intercept is 4.

This is because each box is one unit each.

We substitute the intercepts to get:

[tex] \frac{x}{ - 3} + \frac{y}{4} = 1[/tex]

We now multiply through by -12 to get

[tex] - 12 \times \frac{x}{ - 3} + - 12 \times \frac{y}{4} = 1 \times - 12[/tex]

[tex]4x - 3y = -12[/tex]

The correct choice is C.

Answer:

a)120

b)6.67%

Step-by-step explanation:

Given:

No. of digits given= 6

Digits given= 1,2,3,5,8,9

Number to be formed should be 3-digits, as we have to choose 3 digits from given 6-digits so the no. of combinations will be

6P3= 6!/3!

      = 6*5*4*3*2*1/3*2*1

      =6*5*4

      =120

Now finding the probability that both the first digit and the last digit of the three-digit number are even numbers:

As the first and last digits can only be even

then the form of number can be

a)2n8 or

b)8n2

where n can be 1,3,5 or 9

4*2=8

so there can be 8 three-digit numbers with both the first digit and the last digit even numbers

And probability = 8/120

                          = 0.0667

                          =6.67%

The probability that both the first digit and the last digit of the three-digit number are even numbers is 6.67% !

1.

[tex]6\cdot5\cdot4=120[/tex]

2.

[tex]|\Omega|=120\\|A|=2\cdot4\cdot1=8\\\\P(A)=\dfrac{8}{120}=\dfrac{1}{15}\approx6.7\%[/tex]

Answer:

[tex]-8 < x < 8[/tex]

Step-by-step explanation:

Your compound inequality will include two inequalities.

These are:

x > -8

x < 8

Put your lowest number first, ensuring that your sign is pointed in the correct direction.

[tex]-8 < x[/tex]

Next, enter your higher number, again making sure that your sign is pointing in the correct direction.

[tex]-8 < x < 8[/tex]

Answer:

-8 < r < 8

Step-by-step explanation:

Let r = real number

Greater than  >

r>-8

less than  <

r <8

We want a compound inequality so we combine these

-8 < r < 8

Answer:

x ≤ 3764.72

Step-by-step expl anation:

The Montanez family cannot use more than 7250.50 gallons, this means that they can use less than or equal to 7250.50 gallons, this tells you which sign to use.  The variable x can be used to describe how much water they have left to use and then you add 3,485.78 gallons to x.

x + 3485.78 ≤ 7250.50   This inequality means that the amount of water the family has yet to use added to 3485.78 gallons cannot exceed 7250.50 gallons.

Next, you simplify the inequality using the property of inequalities.

x ≤ 7250.50 - 3485.78

x ≤ 3764.72

The required inequality is x<941.18 for the Montanez family if they have used 3485.78 gallons of water.

It is also a relationship between variables but this is in greater than , less than, greater than or equal to , less than or equal to is used.

Let the water used by a member be x

so

x*4=7250.50-3485.78

4x=3764.72

x=941.18

Hence one member can only use 941.18 gallons of water in the month.

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[tex]\bf \begin{array}{ccll} \$&hour\\ \cline{1-2} 5.5&1\\ x&3\frac{1}{3} \end{array}\implies \cfrac{5.5}{x}=\cfrac{1}{3\frac{1}{3}}\implies \cfrac{5.5}{x}=\cfrac{1}{\frac{3\cdot 3+1}{3}}\implies \cfrac{5.5}{x}=\cfrac{1}{\frac{10}{3}}\implies \cfrac{5.5}{x}=\cfrac{\frac{1}{1}}{\frac{10}{3}} \\\\\\ \cfrac{5.5}{x}=\cfrac{1}{1}\cdot \cfrac{3}{10}\implies \cfrac{5.5}{x}=\cfrac{3}{10}\implies 55=3x\implies \stackrel{\textit{about 18 bucks and 33 cents}}{\cfrac{55}{3}=x\implies 18\frac{1}{3}=x}[/tex]

Answer:

$18.3

Step-by-step explanation:

If you are paid $5.50/hour for mowing yards, and you take 3 1/3 hours to mow a yard, you should earn $18.3.

3 1/3 hours

$5.50 and hour

$5.50 x 3 = $16.5

$5.50 / 3 = $1.8

$16.5 + $1.8 = $18.3

Therefore, you are owed $18.3.

Answer:

B

Step-by-step explanation:

Given

(5 + 7i) + (- 2 + 6i ) ← remove parenthesis and collect like terms

= 5 + 7i - 2 + 6i

= 3 + 13i → B

The sum of the complex number is 3 + 13i.

Option B is the correct answer.

We have,

To find the sum of the complex numbers (5+7i) and (-2+6i), you can simply add the real parts together and add the imaginary parts together separately.

Real part: 5 + (-2) = 3

Imaginary part: 7i + 6i = 13i

Combining the real and imaginary parts, we get:

Sum = 3 + 13i

Therefore,

The sum of the complex number is 3 + 13i.

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Answer:

The time of a commercial airplane is 280 minutes

Step-by-step explanation:

Let

x -----> the speed of a commercial airplane

y ----> the speed of a jet plane

t -----> the time that a jet airplane takes  from Vancouver to Regina

we know that

The speed is equal to divide the distance by the time

y=2x ----> equation A

The speed of a commercial airplane is equal to

x=1,730/(t+140) ----> equation B

The speed of a jet airplane is equal to

y=1,730/t -----> equation C

substitute equation B and equation C in equation A

1,730/t=2(1,730/(t+140))

Solve for t

1/t=(2/(t+140))

t+140=2t

2t-t=140

t=140 minutes

The time of a commercial airplane is

t+140=140+140=280 minutes

Answer:

x = - 15

Step-by-step explanation:

The equation is  [tex]\frac{5}{x}=\frac{4}{x+3}[/tex]

We now cross mulitply and do algebra to figure the value of x (shown below):

[tex]\frac{5}{x}=\frac{4}{x+3}\\5(x+3)=4(x)\\5x+15=4x\\5x-4x=-15\\x=-15[/tex]

Hence x = -15

Answer:

D

Step-by-step explanation:

[tex]\bf \sqrt{3x^2}\cdot \sqrt{4x}\implies \sqrt{3x^2\cdot 4x}\implies \sqrt{12x^2x}\implies \sqrt{4\cdot 3\cdot x^2x} \\\\\\ \sqrt{2^2\cdot 3\cdot x^2x}\implies 2x\sqrt{3x}[/tex]

Answer:

h=7

Step-by-step explanation:

[tex]h+(-3)=4[/tex]

may be rewritten as

[tex]h-3=4[/tex]

as adding a negative is the same as subtracting a positive.

To solve, add 3 to both sides.

[tex]h-3=4\\h=7[/tex]

Answer:

h=7

Step-by-step explanation:

1) Add three to both sides

2) You should get h=7

Answer:

Assume that [tex]a[/tex] and [tex]p[/tex] are constants. The slope of the line will be equal to

Step-by-step explanation:

Rewrite the expression of the line to express [tex]y[/tex] in terms of [tex]x[/tex] and the constants.

Substract [tex]x\cdot \cos{(a)}[/tex] from both sides of the equation:

[tex]y \sin{(a)} = p - x\cos{(a)}[/tex].

In case [tex]\sin{a} \ne 0[/tex], divide both sides with [tex]\sin{a}[/tex]:

[tex]\displaystyle y = - \frac{\cos{(a)}}{\sin{(a)}}\cdot x+ \frac{p}{\sin{(a)}}[/tex].

Take the first derivative of both sides with respect to [tex]x[/tex]. [tex]\frac{p}{\sin{(a)}}[/tex] is a constant, so its first derivative will be zero.

[tex]\displaystyle \frac{dy}{dx} = - \frac{\cos{(a)}}{\sin{(a)}}[/tex].

[tex]\displaystyle \frac{dy}{dx}[/tex] is the slope of this line. The slope of this line is therefore

[tex]\displaystyle - \frac{\cos{(a)}}{\sin{(a)}} = -\cot{(a)}[/tex].

In case [tex]\sin{a} = 0[/tex], the equation of this line becomes:

[tex]y \sin{(a)} = p - x\cos{(a)}[/tex].

[tex]x\cos{(a)} = p[/tex].

[tex]\displaystyle x = \frac{p}{\cos{(a)}}[/tex],

which is the equation of a vertical line that goes through the point [tex]\displaystyle \left(0, \frac{p}{\cos{(a)}}\right)[/tex]. The slope of this line will be infinity.

Answer:

m=7

Remainder =4

If q=1 then r=3 or r=-1.

If q=2 then r=3.

They are probably looking for q=1 and r=3 because the other combinations were used earlier in the problem.

Step-by-step explanation:

Let's assume the remainders left when doing P divided by (x-1) and P divided by (2x+3) is R.

By remainder theorem we have that:

P(1)=R

P(-3/2)=R

[tex]P(1)=2(1)^3+m(1)^2-5[/tex]

[tex]=2+m-5=m-3[/tex]

[tex]P(\frac{-3}{2})=2(\frac{-3}{2})^3+m(\frac{-3}{2})^2-5[/tex]

[tex]=2(\frac{-27}{8})+m(\frac{9}{4})-5[/tex]

[tex]=-\frac{27}{4}+\frac{9m}{4}-5[/tex]

[tex]=\frac{-27+9m-20}{4}[/tex]

[tex]=\frac{9m-47}{4}[/tex]

Both of these are equal to R.

[tex]m-3=R[/tex]

[tex]\frac{9m-47}{4}=R[/tex]

I'm going to substitute second R which is (9m-47)/4 in place of first R.

[tex]m-3=\frac{9m-47}{4}[/tex]

Multiply both sides by 4:

[tex]4(m-3)=9m-47[/tex]

Distribute:

[tex]4m-12=9m-47[/tex]

Subtract 4m on both sides:

[tex]-12=5m-47[/tex]

Add 47 on both sides:

[tex]-12+47=5m[/tex]

Simplify left hand side:

[tex]35=5m[/tex]

Divide both sides by 5:

[tex]\frac{35}{5}=m[/tex]

[tex]7=m[/tex]

So the value for m is 7.

[tex]P(x)=2x^3+7x^2-5[/tex]

What is the remainder when dividing P by (x-1) or (2x+3)?

Well recall that we said m-3=R which means r=m-3=7-3=4.

So the remainder is 4 when dividing P by (x-1) or (2x+3).

Now P divided by (qx+r) will also give the same remainder R=4.

So by remainder theorem we have that P(-r/q)=4.

Let's plug this in:

[tex]P(\frac{-r}{q})=2(\frac{-r}{q})^3+m(\frac{-r}{q})^2-5[/tex]

Let x=-r/q

This is equal to 4 so we have this equation:

[tex]2u^3+7u^2-5=4[/tex]

Subtract 4 on both sides:

[tex]2u^3+7u^2-9=0[/tex]

I see one obvious solution of 1.

I seen this because I see 2+7-9 is 0.

u=1 would do that.

Let's see if we can find any other real solutions.

Dividing:

1     |   2    7     0     -9

     |         2      9      9

       -----------------------

          2    9     9      0

This gives us the quadratic equation to solve:

[tex]2x^2+9x+9=0[/tex]

Compare this to [tex]ax^2+bx+c=0[/tex]

[tex]a=2[/tex]

[tex]b=9[/tex]

[tex]c=9[/tex]

Since the coefficient of [tex]x^2[/tex] is not 1, we have to find two numbers that multiply to be [tex]ac[/tex] and add up to be [tex]b[/tex].

Those numbers are 6 and 3 because [tex]6(3)=18=ac[/tex] while [tex]6+3=9=b[/tex].

So we are going to replace [tex]bx[/tex] or [tex]9x[/tex] with [tex]6x+3x[/tex] then factor by grouping:

[tex]2x^2+6x+3x+9=0[/tex]

[tex](2x^2+6x)+(3x+9)=0[/tex]

[tex]2x(x+3)+3(x+3)=0[/tex]

[tex](x+3)(2x+3)=0[/tex]

This means x+3=0 or 2x+3=0.

We need to solve both of these:

x+3=0

Subtract 3 on both sides:

x=-3

----

2x+3=0

Subtract 3 on both sides:

2x=-3

Divide both sides by 2:

x=-3/2

So the solutions to P(x)=4:

[tex]x \in \{-3,\frac{-3}{2},1\}[/tex]

If x=-3 is a solution then (x+3) is a factor that you can divide P by to get remainder 4.

If x=-3/2 is a solution then (2x+3) is a factor that you can divide P by to get remainder 4.

If x=1 is a solution then (x-1) is a factor that you can divide P by to get remainder 4.

Compare (qx+r) to (x+3); we see one possibility for (q,r)=(1,3).

Compare (qx+r) to (2x+3); we see another possibility is (q,r)=(2,3).

Compare (qx+r) to (x-1); we see another possibility is (q,r)=(1,-1).

For this case we must represent the following expression algebraically, in addition to indicating its result:

"2 plus 9"

So, we have:

[tex]2 + 9 =[/tex]

By law of the signs of the sum, we have that equal signs are added and the same sign is placed:

[tex]2 + 9 = 11[/tex]

ANswer:

11

Answer:

Choice A:

[tex]a_1=5[/tex]

[tex]a_{n}=a_{n-1}+2[/tex]

Step-by-step explanation:

[tex]a_n=5+(n-1)2[/tex]

means we looking for first term 5 and the sequence is going up by 2.

In general,

[tex]a_n=a_1+(n-1)d[/tex]

means you have first term [tex]a_1[/tex] and the sequence has a common difference of d.

So it is between the first two choices.

The explicit form of an arithmetic sequence is: [tex]a_n=a_1+(n-1)d[/tex]

An equivalent recursive form is [tex]a_n=a_{n-1}+d \text{ where } a_1 \text{ is the first term}[/tex]

So d again here is 2.

So choice a is correct.

[tex]a_1=5[/tex]

[tex]a_{n}=a_{n-1}+2[/tex]

Answer: Option a

[tex]\left \{ {{a_1=5} \atop {a_n=a_{(n-1)}+2}} \right.[/tex]

Step-by-step explanation:

The arithmetic sequences have the following explicit formula

[tex]a_n=a_1 +(n-1)*d[/tex]

Where d is the common difference between the consecutive terms and [tex]a_1[/tex] is the first term of the sequence:

The recursive formula for an arithmetic sequence is as follows

[tex]\left \{ {{a_1} \atop {a_n=a_{(n-1)}+d}} \right.[/tex]

Where d is the common difference between the consecutive terms and [tex]a_1[/tex] is the first term of the sequence:

In this case we have the explicit formula [tex]a_n=5+(n-1)*2[/tex]

Notice that in this case

[tex]a_1 = 5\\d = 2[/tex]

Then the recursive formula is:

[tex]\left \{ {{a_1=5} \atop {a_n=a_{(n-1)}+2}} \right.[/tex]

The answer is the option a.

Answer:

corresponding

Step-by-step explanation:

Answer:

Corresponding

Step-by-step explanation:

I like to call corresponding angles, the copy and paste angles because you can copy and paste the top intersection over the bottom intersection; the angles that lay down on top of each other are the corresponding angles. 1 and 5 do this.

Answer:

Option B

Step-by-step explanation:

we know that

To find the reciprocal of a fraction, flip the fraction.

Remember that

A number multiplied by its reciprocal is equal to 1

In this problem we have

[tex]\frac{10b}{2b+8}[/tex]

Flip the fraction

[tex]\frac{2b+8}{10b}[/tex] -----> reciprocal

therefore

The reciprocal is

The quantity 2 multiplied by b plus 8 end of quantity divided by the quantity 10 multiplied by b end of quantity.

2048 different ways can the sequence of heads and tails appear.

The definition of a possibility is something that may be true or might occur, or something that can be chosen from among a series of choices.

According to the question

Fair coin is tossed 11 times

There are two possibilities for each flip (Heads or Tails). You multiply those together to get the total number of unique sequences.

Here’s an example for 2 flips: HH - HT - TT - TH. (That’s 2 × 2, or [tex]2^{2}[/tex])

Here’s 3 flips: HHH - HHT - HTH - HTT - THH - THT - TTH - TTT. (That’s

2× 2 × 2, or [tex]2^{3}[/tex]).

For ten flips, it’s [tex]2^{10}[/tex]… which is 1024.

For 11 flips, it's [tex]2^{11}[/tex].......Which is 2048

2048 different ways can the sequence of heads and tails appear.

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Final answer:

There are 2048 different ways that a sequence of heads and tails can appear after tossing a fair coin 11 times, because each coin toss has two possible outcomes and the events are independent.

Explanation:

If a fair coin is tossed 11 times, the number of different sequences of heads and tails that can appear is calculated using the formula for the number of outcomes of binomial events. Since each toss of the coin has two possible outcomes (either a head or a tail), and we are tossing the coin 11 times, we use the power of 2 raised to the 11th power, which gives us 211 = 2048. Therefore, there are 2048 different ways a sequence of heads and tails can appear after tossing a coin 11 times.

The reason why there are so many combinations is that each coin toss is independent of the previous one, with a 50 percent chance of landing on either side. This principle is used in probability theory to calculate the possible outcomes of repeated binary events. As an interesting note, if we were to look at possibilities of a specific number of heads and tails, such as 10 heads and 1 tail or vice versa, this would be represented in Pascal's triangle, which reflects the coefficients in the binomial expansion.

Answer:

k = 2

Step-by-step explanation:

If the roots are real and equal then the condition for the discriminant is

b² - 4ac = 0

For 2x² - (k + 2)x + k = 0 ← in standard form

with a = 2, b = - (k + 2) and c = k, then

(- (k + 2))² - (4 × 2 × k ) = 0

k² + 4k + 4 - 8k = 0

k² - 4k + 4 = 0

(k - 2)² = 0

Equate factor to zero and solve for k

(k - 2)² = 0 ⇒ k - 2 = 0 ⇒ k = 2

Answer:

Step-by-step explanation:

A quadratic equation has two equal real roots if a discriminant is equal 0.

[tex]ax^2+bx+c=0[/tex]

Discriminant [tex]b^2-4ac[/tex]

We have the equation

[tex]2x^2-(k+2)x+k=0\to a=2,\ b=-(k+2),\ c=k[/tex]

Substitute:

[tex]b^2-4ac=\bigg(-(k+2)\bigg)^2-4(2)(k)\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\=k^2+2(k)(2)+2^2-8k=k^2+4k+4-8k=k^2-4k+4\\\\b^2-4ac=0\iff k^2-4k+4=0\\\\k^2-2k-2k+4=0\\\\k(k-2)-2(k-2)=0\\\\(k-2)(k-2)=0\\\\(k-2)^2=0\iff k-2=0\qquad\text{add 2 to both sides}\\\\k=2[/tex]