Question 1. Let the points A(2,-1), B(-3,4), and C(-5,-7) be the vertices of triangle A B C.
(a) Let P and Q be the midpoints of the sides AB and AC respectively. Find the coordinates of P and Q. (2 marks)
(b) Show that PQ is parallel to BC. (2 marks)

In linear equation, 16.27 is the length of the longest straw that can fit into the can (so that the straw is not bent and fits entirely inside the can).

Describe a linear equation using an example.

An equation in which there is only one variable is known as a linear equation in one variable. It has the mathematical formula Ax + B = 0, where A and B can be any two real numbers, and x is an unknowable variable with just one possible value.

                    A linear equation in one variable would be 9x + 78 = 18, for instance.

l² = r² + h²

l² =  3² + 16²

l² =   9 + 256

     =   265

l =  √265

l = 16.27

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Answer:

Hi,  the correct answer is A) m∠2 = 60°.

let me put it in this way.

Step-by-step explanation:

Since parallel lines M and N are cut by transversals A B and B C, corresponding angles are formed. Corresponding angles are pairs of angles that are formed by intersecting lines and are on opposite sides of the transversal. In this case, angles 2 and C are corresponding angles, as are angles 3 and A.

We know that angles 2 and C form a straight angle along line M, and angles C and 60 degrees form a straight angle along line N. Since corresponding angles are congruent (have the same measure), this means that angle 2 must have the same measure as the 60-degree angle

Answer:

b)  7

Step-by-step explanation:

Given equation:

[tex](\sin x + \csc x)^2+(\cos x + \sec x)^2=k+\tan^2 x + \cot^2 x[/tex]

Expand the left side of the equation:

[tex]\implies \sin^2x+2 \sin x \csc x + \csc^2x + \cos^2 x + 2 \cos x \sec x + \sec^2 x[/tex]

Simplify:

[tex]\implies \sin^2x+2 \sin x \cdot \dfrac{1}{\sin x} + \csc^2x + \cos^2 x + 2 \cos x \cdot \dfrac{1}{\cos x} + \sec^2 x[/tex]

[tex]\implies \sin^2x+2 + \csc^2x + \cos^2 x + 2 + \sec^2 x[/tex]

[tex]\implies \sec^2 x+\csc^2x+ \sin^2x + \cos^2 x +4[/tex]

Use the identity:   sin²x + cos²x = 1

[tex]\implies \sec^2 x+ \csc^2x+1+4[/tex]

[tex]\implies \sec^2 x+ \csc^2x+5[/tex]

Use the identities:  sec²x = tan²x + 1  and  csc²x = cot²x + 1

[tex]\implies \tan^2x +1 +\cot^2x +1 +5[/tex]

[tex]\implies 7+\tan^2x + \cot^2x[/tex]

Therefore:

[tex](\sin x + \csc x)^2+(\cos x + \sec x)^2=7+\tan^2 x + \cot^2 x[/tex]

[tex] \huge{ \bold{Answer -: }}[/tex]

Option b is correct

[tex] \large \bf{given}[/tex]

[tex] \large{ \sf{ \: (sin \: x \: + cosec \: x) {}^{2} + (cos \: x + sec \: x) {}^{2} }}[/tex]

[tex]\large{ \sf{ = k + { \tan^{2} x + \cot^{2} x}}}[/tex]

[tex] \large{ \sf{\implies \: sin {}^{2} x + cosec {}^{2} x + 2 + {cos}^{2} x + {sec}^{2} x + 2}}[/tex]

[tex]\large{ \sf{ = k + {tan}^{2} x + {cot}^{2} x}}[/tex]

[tex]\large{ \sf{ \implies \: 1 + {cosec}^{2} x - {cot}^{2} x + {sec}^{2} x - {tan}^{2} x + 4 = k}}[/tex]

[tex]\large{ \sf{ \pink{ \implies \: 1 + 1 + 1 + 4 = k \implies \: k = 7}}}[/tex]

[tex] \therefore \: k \: = 7[/tex]

Answer: 61.75

Step-by-step explanation:

The value of the expression 15/r -1 when r = 5 will be equal to 2.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the expression is 15/r -1. The value of r is 5. The value of the expression at r=5 will be calculated as,

E = 15/r - 1

E = 15 / 5- 1

E = 3 - 1

E = 2

Therefore, the value of the expression 15/r -1 when r = 5 will be equal to 2.

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The solution is Option D.

The value of a from the equation of slope is a = 6

What is an Equation of a line?

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second poin

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the value to be determined be = a

Now , the slope of the line be = m

The value of m = 3/2

So , Slope m = 3/2

And , the value of a is calculated by

Slope = rise / run

Slope = y/x

Since , the value of x = 4

3/2 = y/4

Multiply by 4 on both sides of the equation , we get

y = 3 x 2

y = 6

Therefore , the value of a = 6

Hence , The value of a from the equation of slope is a = 6

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The required value of h in the equation 4/9 + h = 11/9 is 7/9 or seven ninths.

An equation is a combination of different variables, in which two mathematical expressions are equal to each other.

The given equation is,

4/9 + h = 11/9    (1)

To determine the value of h, solve the equation,

Subtract 4/9 from both the sides,

4/9 + h - 4/9 = 11/9 - 4/9

h = 7/9

The value of h is 7/9 or seven ninths.

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Standard deviation for 6,8,10,4 will be;

⇒ 2.58

What is Standard deviation?

Standard deviation is the measure of dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.

Given that;

The data set is,

⇒ 6, 8, 10, 4

Now,

Since, The data set is,

⇒ 6, 8, 10, 4

Hence, Mean = 6 + 8 + 10 + 4 / 4

                     = 7

So, The standard deviation= √ (6 - 7)² + (8 - 7)² + (10 - 7)² + (4 - 7)² / (4 - 1)

                                        = √20/3

                                        = 2.58

Thus, Standard deviation for 6,8,10,4 will be;

⇒ 2.58

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Answer:

12.56 km

Step-by-step explanation:

The circumference of a circle is given by

[tex]c = 2\pi \: r[/tex]

let's find the radius

the area of a circle is given by

[tex]a = \pi \: {r}^{2} [/tex]

12.56 = 3.14 r²

r² = 12.56/3.14

r² = 4

r = √4

r = 2

let's find the circumference

C = 2πr

C = 2 × 3.14 × 2

C = 12.56 km

I hope this helps

Step-by-step explanation:

For this case suppose that we have a quadratic equation of the form:

 The solution to the quadratic recuacion is given by:

 Where,

 The discriminant is:

 When the discriminant is greater than zero, then the root is positive, and therefore, we have two positive real solutions.

 Answer:

 B. it has two real solutions

Answer: $17.77 per pass

Step-by-step explanation:

$835.19 / (45 + 2)

$835.19 / 47

= $17.77 price per pass

Step-by-step explanation:

At x=0, we have a zero and it bounces off so one of our answers must include

x^2.

Our other zeroes have multiplicities of 1, so our anly reasonable answer is A

Using ratios we know that the number of guides on Sunday was 6.

In mathematics, a ratio shows how frequently one number appears in another.

For instance, the ratio of oranges to lemons in a fruit plate would be eight to six if there were eight oranges and six lemons.

Oranges make up 8:14 of the total fruit, whereas lemons make up 6:8 of the total fruit.

So, we know that the proportion of tourists to guides was the same on both days of the weekend.

On Saturday, the ratio was:

120:18

120/18

20/3

Ratio: 20:3

So, a number of guides on Sunday will be:

40:x

40/20 = x/3

20x = 40*3

x = 2*3

x = 6

Therefore, using ratios we know that the number of guides on Sunday was 6.

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Complete question:
The City Sights tour company had the same ratio of tourists to guides on both days this weekend. On Saturday, there were 120 tourists and 18 guides. On Sunday, there were 40 tourists. Find  number of guides on sunday.

From the diagram,  the values of  <CAB=60⁰, <ACB=90⁰, <ABC=30⁰,BC=15, DA=32 and BA= 32

You should know that an angle is the intersection of two or more straight lines.

Considering the two tringles

ΔACD≡ΔABC

lines DC=BC

<ACD=<ACB=90⁰

Line DA=AB

This implies that 5x+12=8x

Collecting like terms

8x-5x=12

3x=12

Therefore x=4

This means that 5*4+12=32

This implies that BA=AD=32

In triangle ABC;

<ACB=90⁰

<ADC=<ABC=30⁰

This means that

<BAC=90-30=60

Therefore <CAB=60⁰, <ACB=90⁰, <ABC=30⁰,BC=15, DA=32 and BA= 32 respectively

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The range of the five numbers 0 , 10 , a , b and c is a + b + c < 13

According to the question,

Statement (1) Its given that a + b + c < 13

Since, a, b and c are positive integers So,  a,b,c > 0

Now, note that the maximum value that any one of a, b, c can take is 10

Range will always be = 10 - 0 = 10

Statement (2)  a < b < c < 13

There are multiple values of a, b, and c that can be possible.

eg; a = 1, b = 5, c = 12

=> Range = 12 - 0 = 12

or, a = 1, b = 2, c = 3

=> Range = 10 - 0 = 10

which can't possible

Therefore , Statement (1) is correct about range

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The word problem have the following answers:

Word problems are mathematical problems which involves the use ofordinary words, instead of mathematical symbols.

From the question, if y is the balance after x weeks, this implies Johnson took out $45 multiplied by x in total including ATM fees, and their balance will be the equation;

y = $650 - $45x

After 8weeks, their balance will be;

y = $650 - $45(8)

y = $650 - $360

y = $290

After 8 weeks, Jackson will pay in ATM fees alone;

$5 × 8 = $40

Therefore, the equation y = $650 - $45x is to express their balance, y, after x weeks and $290 will be their balance after 8 weeks. While Johnson would have paid in $40 in ATM fees alone after 8 weeks.

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ΔBAD is congruent to ΔBCD by the ASA rule.

What is the ASA congruency of triangles?

If any two angles and sides included between the angles of one triangle are equivalent to the corresponding two angles and sides included between the angles of the second triangle, then the two triangles are said to be congruent by the ASA rule.

In the given figure ABCD is a square.

In ΔBAD and ΔBCD

∠B ≅ ∠B ----Common Angle

∠A ≅ ∠A  ------Each of 90° angle

Side BD ≅ Side BD ----Common Side

So we can say that ΔBAD ≅ ΔBCD -----(ASA rule)

Hence, ΔBAD is congruent to ΔBCD by the ASA rule.

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Answer:

Step-by-step explanation:

The number of boxes of pies that the Chess Club must sell to reach their goal of $150 can be represented by the variable x. We can create the following equation to represent this relationship:

1.25x + 16.25 = 150

To find the number of boxes of pies that the Chess Club must sell, we can solve this equation by subtracting 16.25 from both sides and then dividing both sides by 1.25:

1.25x = 150 - 16.25

1.25x = 133.75

x = 133.75 / 1.25

x = 107.

Therefore, the Chess Club must sell 107 more boxes of pies to reach their goal of $150.

By Factorise the equations we get 2 points for each equation:

for a) we get (x+4),(x-1)

for b) we get (x-3),(x+1)

for c) we get (x+4),(x-2)

a) x^2 + 3x-4

for this equation we have to take factors of -4:

x^2+4x-x-4

Taking common in the equation we get:

x(x+4)-1(x-4)

After taking common, Now we get two points:(x+4,)(x-1)

b) x^2 +-2x-3

for this equation we have to take factors of -3:

x^2-3x+x-3

Taking common in the equation we get:

x(x-3)+1(x-3)

After taking common, Now we get two points:(x-3,)(x+1)

c) x^2 + 2x-8

for this equation we have to take factors of -8:

x^2+4x-2x-8

Taking common in the equation we get:

x(x+4)-2(x+4)

After taking common, Now we get two points:(x+4,)(x-2)

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"The correct question is":

Factorise the following:

a) x ^2+ 3x-4

b) x^2 - 2x - 3

c) x ^2+ 2x - 8

The value of the logarithm expression logₙ (n)⁻² will be negative 2.

When the relevant components and basic processes of a numerical method are given values, the expression's result is the result of the computation it depicts.

The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.

The expression is given below.

⇒ logₙ (n)⁻²

Simplify the expression, then we have

⇒ logₙ (n)⁻²

⇒ -2 logₙ (n)

⇒ - 2

The value of the logarithm expression logₙ (n)⁻² will be negative 2.

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Part a:  f(x) = -|x+ 2| - 1, is the modulus function.

part b: f(x) = 2x² - 5, the graph for each function is plotted along with the points.

For the given equation of the polynomial.

Part a: f(x) = -|x+ 2| - 1, is the modulus function.

Put the values for x and find y.

For x = -3, y = 0

For x = -2, y = 1

For x = 0, y = -1

For x = 3, y = -4

Thus, the graph for the obtained points are plotted.

part b: f(x) = 2x² - 5

Put the values for x and find y.

For x = -1.581, y = 0

For x = 1.581, y = 1

For x = 0, y = -5

Thus, the graph for the obtained points are plotted.

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a) The expression that represents the boiling temperature (in degree celsius) of platinum is of: y = 4x + 199.

b) The boiling temperature of platinum is of 3827 ºC.

The boiling temperature of the platinum is the output of a function, hence it is represented by the variable y.

The boiling temperature of zinc is the input variable of the function, hence it is represented by variable x.

The boiling temperature of platinum is 199 more than four times the boiling temperature of zinc, hence the expression is given as follows:

y = 4x + 199.

Since the boiling temperature of zing is of 907ºC, the boiling temperature of platinum is obtained as follows:

y = 4(907) + 199 = 3827 ºC.

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is it two questions?

if it is then.....

Is answer correct?

From the statement, Anthony is correct because 3*6=18

We should know that the problem is to determine the number of pencils he bought

The given parameters are that

You can draw a 3 packages of pencils with 6 pencils in each to represent t package.

Now let us do the following:

For Neville

3*6 = 18

And also,

18÷3 = 6

For Anthony

6*3 =18 and

18÷6 = 3

In conclusion, looking at the two analysis, it seems that Anthony is more correct knowing fully well that each of the multiplications and divisions give us 18 or 6

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Complete question:

Critique Reasoning Neville and Anthony are solving this problem: Barbara bought You can draw a 3 packages of pencils with 6 pencils in each to represent t package. How many pencils did she buy in all? Neville says, “I add because of the words in all. The answer is  9 pencils.” Anthony says, “I multiply because there are equal groups. The answer is  18 pencils.” Who is correct? Explain.

Answer: x = -9

Step-by-step explanation: 4

x

+

2

=

3

x

−

7

⇒

4x−3x=−7−2⇒x=−9

The solutions of the quadratic equation x³+ 16x -3x² = 48 are 3, –4i, 4i

A quadratic equation is the second-order degree algebraic expression in a variable. the standard form of this expression is  ax² + bx + c = 0 where a. b are coefficients and x is the variable and c is a constant.

A mathematical statement with two expressions that have the same value and the sign "equal to" between them is called an equation.

We are given the quadratic equation as;

x³+ 16x -3x² = 48

Subtract 48 from both sides;

x³+ 16x -3x² - 48= 48 -48

x³+ 16x -3x² - 48= 0

Factor left side of equation.

(x - 3)( x² + 16) = 0

Therefore, the roots are 3, –4i, 4i

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The minimum sample size required is 108.

What is the standard deviation?

A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. A low standard deviation means data are clustered around the mean, and a high standard deviation indicates data are more spread out.

To find the minimum sample size required, we can use the formula for the margin of error for a confidence interval for a mean:

margin of error = (z*s) / (√n)

where z is the z-score corresponding to the desired confidence level (in this case, z = 1.28 for an 80% confidence interval), s is the standard deviation of the population, and n is the sample size.

We are given that the margin of error should be no more than 0.06 and that the standard deviation is 0.9. We can solve for n by rearranging the formula and substituting in the given values:

n = [(zs) / margin of error]²

= [(1.280.9) / 0.06]²

= approximately 107.5

Since the sample size must be an integer, we must round up to the next integer, which is 108.

Hence, the minimum sample size required is 108.

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Answer:

There are 12 nickels, 9 quarters, and 15 dimes.

I hope this turns out useful for you!

Step-by-step explanation:

Reading the problem repeatedly, I found that the nickels were the value that I could change to find out the values of the other coins and to find the answer. I started with the number of nickels as 20, calculating 20 x 5 (5 as the value of the nickel) = 100 ($1.00). Then, I subtracted three from 20 since the number of quarters was 3 less and got 17. 17 x 25 = a value far too big when we were going to add dimes as well, so 8 lowered the number of nickels a reasonable amount and tried again, following the directions in the problem. It took a bit, but I found a value that worked.

Nickel = $0.05

Quarter = $0.25

Dime = $0.10

My work (though I converted a bit using my sense of money)

12 x 5 = $0.60

12 - 3 = 9

9 x 25 = $2.25

9 + 6 = 15

15 x 10 = $1.50

$0.60 + $2.25 + $1.50 = $4.35

The simplified form of the given expression is, [tex]((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5 = (\frac{4}{9} )^7[/tex].

What is simplifying an expression?

To simplify an expression, create an equivalent expression without any terms that are similar. The expression will then be rewritten using the fewest terms possible.

Consider, the given expression,

[tex]((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5[/tex]

To simplify this using the exponent rule:

[tex]((a)^b)^c = a^b^.^c[/tex]

⇒

[tex]((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5 = (\frac{4}{9})^(^-^4^*^-^3)(\frac{4}{9})^-^5\\ ((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5 = (\frac{4}{9})^1^2(\frac{4}{9} )^-^5[/tex]

Now use the exponent rule:

[tex]a^ba^c = a^(^b^+^c^)[/tex]

⇒

[tex]((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5 = (\frac{4}{9} )^(^1^2^-^5^)\\((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5 = (\frac{4}{9} )^7[/tex]

Hence, the simplified form of the given expression is, [tex]((\frac{4}{9} )^-^4)^-^3(\frac{4}{9})^-^5 = (\frac{4}{9} )^7[/tex].

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lowest price Mike can pay for this juice is £15.8

What is an integer?

An integer (pronounced IN-tuh-jer) is an integer (not a fraction) that is positive, negative, or zero.

An example of an integer is:

-5, 1, 5, 8, 97, 3,043.

Examples of non-integers are:

-1.43, 1 3/4, 3.14, 0.09, and 5643.1.

The set of integers denoted by Z is formally defined as:

Z = {...,-3,-2,-1,0,1,2,3,...}

It is given that Mike drinks of a litre of juice each day.

Juice costs £4.40 for a 2 litre carton and £2.60 for a 1 litre carton.

Mike buys enough juice to last for 7 days.

Total litre of milk needed = 7 × 1 = 7 litre

7l = 2l + 2l + 2l + 1l

Lowest cost = 4.40 + 4.40 + 4.40 + 2.60 = 15.8

Therefore,   lowest price Mike can pay for this juice is £15.8

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