three different angle in the diagram

Adding the additive inverse of B (that is, “-B” which must exist in every Vector Space . A+B-B = 0 -B A= -B

What is magnitude vector?

The magnitude of a vector is the square root of the scalar product of that vector by itself.

l A l  = [tex]\sqrt{A . A}[/tex]

l B l = [tex]\sqrt{B . B}[/tex]

l A + B l  = [tex]\sqrt{( A + B ) . ( A + B ) } = \sqrt{AA + AB + BA + BB}[/tex]

l A + B l = [tex]\sqrt{l A l^{2} + l B I ^{2} + 2 .l A l l B l cos (angle) }[/tex]

In case |A+B| = |A| + |B|

That is the same as the condition that the squares are equal.

l A + B l² = ( l A l + l B l)² =  l A l²  + l B l²  + 2|A||B|

|A||B| cos(angle)=|A||B|

|A||B|[cos(angle)−1]=0

This happens if and only if :

Either |A| or |B| or [cos(angle) - 1] must be zero.

In case |A| and |B| are not zero then cos(angle) = 1, that is : angle = 0

That means B = c*A where c is a positive constant. And, also when either A or B (or both) are the zero vector.

The second question is easier:

|A+B| = 0 implies A+B must be the zero vector.

A+B = 0

then, adding the additive inverse of B (that is, “-B” which must exist in every Vector Space)

A+B-B = 0 -B

A= -B

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The simplified form is = 3.53553

Simplify is to make something less complicated or much less cluttered. An instance of simplification is when you provide an explanation for a difficult math idea in truly smooth phrases for a kid to understand. An example of simplifying is whilst you chop out a whole lot of the activities that were making you busy and careworn. To turn out to be less difficult.

The phrase simplifies the approach to make something simpler to do or understand. So, by lowering or simplifying the fractions approach we make the fraction as simple as viable. We do this by dividing the numerator and the denominator via the biggest range that may divide into each number precisely.

Converting mixed numbers to an improper fraction,

[tex]5^{1}. 0.5^{1/2}[/tex]

= 5* 0.70710

= 3.53553

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If the function be f(x) = 1.5 x² - 4 then the inverse of the function is [tex]f^{-1}[/tex](x) = 3x - 4

An inverse function, also known as an anti function, is a function that can be reversed into another function. Simply put, if any function " f " takes x to y, then its inverse will take y to x. The inverse function is denoted by f-1 or F-1 if the function is denoted by ' f ' or ' F '.

Let the given function be f(x) = 1.5 x² - 4

To find the inverse of the function

[tex]f^{-1}[/tex](x)

To differentiate f(x) with respect to x

[tex]f^{-1}[/tex](x) = 2x(1.5) - 4

simplifying the above equation, we get

[tex]f^{-1}[/tex](x) = 3x- 4

Therefore, the inverse of f(x) = 3x - 4

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The statements and reasons to prove that line j is parallel to line k include the following:

Parallel lines can be defined as two (2) lines that are always the same (equal) distance apart and never meet.

A supplementary angle can be defined as two (2) angles or arc whose sum is equal to 180 degrees (180°).

Mathematically, a supplementary angle can be calculated by using this formula:

Q + R = 180

Where:

Q and R are measure of the angles subtended.

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The probability that a cartridge is classified as not underfilled is 0.85, while the probability that a cartridge is either overfilled or underfilled is 0.55.

Probability is defined as the likeliness of an event to occur, and ranges from 0 to 1 or can be expressed as percentage.

(a) If a cartridge is classified as not underfilled, it means that it is either medium filled or overfilled. Then the probability of that is the sum of the probabilities of the two.

P(underfilled) = P(overfilled) + P(medium filled)

P(underfilled) = 0.40 + 0.45

P(underfilled) = 0.85

(b) If a cartridge is either overfilled or underfilled, then its probability is the sum of their probabilities.

P(either overfilled or underfilled) = P(overfilled) + P(underfilled)

P(either overfilled or underfilled) = 0.40 + 0.15

P(either overfilled or underfilled) = 0.55

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The missing value of direct variation is 32

The direct variation property states that , whenever the division of values of two quantities is a constant value then there is same variation between these two corresponding values. Or we can also say that if the divisible  nature of two corresponding values is constant, then one quantity is directly varies with second quantity.

Let say if x and y are two values and k is a constant value, then

According to direct variation

k = y / x or y = k x

In the given question,

Take the order pair (4,1.5)

As it is an direct variation, we can substitute it

y=k x  

1.5 = k × 4

K=0.375

Now find the direct variation equation

y=k x

put value of 2nd pair and k in the eq.

12 = 0.375 × x

x = 32

Therefore, missing value is 32

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i) y = 5x + 6

Gradient = 5 and intercept = 6

ii) y = 5 - 6x

Gradient = -6 and Intercept = 5

1)(bi)x + y = 6

Gradient = - 1 and intercept = 6

ii) 5y = 10x + 5

Gradient = 2 and Intercept = 1

iii) 6x - 10y = 2

Gradient = 0.6 and intercept = - 0.2

2)ai) 4x + y + 2 = 0

The line that depicts it is line A

ii) 4x + 2y = 8

The line that depicts it is line C

iii) 2x - 4y = 8

The line that depicts it is line B

1a) The general equation for a line in slope intercept form is;

y = mx + c

where;

m is slope or gradient

c is y-intercept

i) y = 5x + 6

Gradient = 5 and intercept = 6

ii) y = 5 - 6x

y = -6x + 5

Gradient = -6 and Intercept = 5

1)(bi)x + y = 6 can be rearranged in slope intercept form as;

y = -x + 6

Gradient = - 1 and intercept = 6

ii) 5y = 10x + 5

y = 2x + 1

Gradient = 2 and Intercept = 1

iii) 6x - 10y = 2

10y = 6x - 2

y = 0.6x - 0.2

Gradient = 0.6 and intercept = - 0.2

2)ai) 4x + y + 2 = 0

y = -4x - 2

Slope = -4 and intercept = -2 and the line that depicts it is line A

ii) 4x + 2y = 8

y = -2x + 4

Slope = -2 and intercept = 4 and the line that depicts it is line C

iii) 2x - 4y = 8

y = 0.5x - 2

Slope = 0.5 and intercept = -2 and the line that depicts it is line B

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The equivalent expression of the expression given as −3/4÷(−8/12) is 9/8

The expression is given as

−3/4÷(−8/12)

Rewrite the expression properly as

−3/4 ÷ (−8/12)

Express the expression as a product

So, we have

−3/4 ÷ (−8/12) = −3/4 x −12/8

Divide the common multiples

So, we have

−3/4 ÷ (−8/12) = −3/4 x −3/2

Evaluate the products

−3/4 ÷ (−8/12) = 9/8

Hence, the equivalent expression of the expression given as −3/4÷(−8/12) is 9/8

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The length of PR is 12cm using the Pythagorean theorem.

So, Pythagorean formula: [tex]c^{2} ={a^2+b^2}[/tex]

Now, insert the values in the formula as follows:

[tex]c^{2} ={a^2+b^2}\\37^{2} ={35^2+PR^2}\\37^{2} - 35^2={PR^2}\\PR^2 = 1369 - 1225\\PR^2 =144\\\\PR\sqrt{144} \\PR = 12\\[/tex]

Therefore, using the Pythagorean theorem the length of PR is 12cm.

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Answer:

x  = 3/10   OR WRITE   x = 0.3

Step-by-step explanation:

2 (8 x + 5)  - 11  +9x =  (5x + 2)     regroup  (16 x +10) -11 + 9 x = (15x + 2) = 25x -1       as ( we did 10 -11 also)    then    25x -1  =  15x + 2    =  25x  = 15x + 3 = 10x = 3   =  x  = 3/10  x = 0.3

If 6 different colored tiles are available to make a pattern in a row of floor tile. The number of  possible different 4-color patterns that are possible if no colors may be repeated is: 360.

Given:

Different colored tiles=6

Different color patterns=4

There are 6 different colored tiles if one colored tiles is pick there will be 5 colored tiles, if two colored tiles is pick there will be 4 colored tiles, if  three colored tiles is pick there will be only 3 colored tiles left.

Hence,

Possible different color patterns=(6×5×4×3)

Possible different color patterns=360 ways

Therefore If 6 different colored tiles are available to make a pattern in a row of floor tile. The number of  possible different 4-color patterns that are possible if no colors may be repeated is: 360.

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Nine hundred Fifty-Six and three hundreths

900 + 50 + 6 + 3/100

956.03

900+ 50+ 6+ .03

According to the question, the given rational expression can be simplified as below:

Rational Expression = [tex]\frac{3x-3}{x^{2} -x}[/tex]

To simplify the given rational expression which are in the form of polynomial expression, calculate the factors of numerator and denominator. The divide it from each other.

Rational expression = [tex]\frac{3x-3}{x^{2} -x}=\frac{3(x-1)}{x(x-1)}=\frac{3}{x}[/tex]

Therefore, the final answer of the given expression is [tex]\frac{3}{x}[/tex].

What is polynomial expression?

Polynomial expression are those equation whose highest degree is two. It is also called as quadratic equation. And it is simplified by performing the factors on the given expressions.

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Answer:

x² + 9x + 20

Step-by-step explanation:

f(x) = x² + 5x + 6

g(x) = x + 2

Now, let us find (f o g)(x)

 (f  o g)(x) =  f(g(x))

               = f(x+2)

               = (x + 2)² + 5(x +2) + 6

{Identity used: (a + b)² = a² + 2ab +b²}

               = x² + 2*x*2 + 2² +5*x +5*2 + 6

               =x² + 4x + 4 + 5x + 10 + 6        

               = x² + 4x + 5x + 4 + 10 + 6

Combine like terms,

               = x² + 9x + 20          

The linear equation passing through (1, 4) and (6, -1) is y = -x + 5.

Linear equation can be represented in slope intercept form.

The line passes through the points (1, 4) and (6, -1).

The equation of the line can be represented as follows:

y = mx + b

where

Therefore, let's find slope using (1, 4) and (6, -1)

m = -1  - 4 / 6 - 1

m = - 5 / 5

m = -1

Therefore, let's find the y-intercept using (1, 4)

y = -x + b

4 = -1 + b

b = 4 + 1

b = 5

Therefore, the linear equation passing through (1, 4) and (6, -1) is y = -x + 5.

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Answer:

See attached graph for the rotated figure

The incorrect image point from the choices given is the first choice  T'(-3, -2). T' should have coordinates (-3, 2)

Step-by-step explanation:

When a point A(x, y) on the XY coordinate system is rotated 180° (either clockwise or anticlockwise) the coordinates of the new point A' will be (-x, -y). Thus the new coordinates will be the negative of the original coordinates

Looking at the figure and following the above rule we get the following results for original and rotated coordinates by 180°

Original Coordinates(x, y)             Rotated 180 Coordinates (-x, -y)

K(-1, -1)                                                 K'(1, 1)                          (-1 becomes 1)
P(0, -1)                                                 P'(0, 1)                           (0 stays at 0)
T(3, -2)                                                T'(-3, 2)
Q(0, -4)                                                Q'(0, 4)

To get the rotated image draw the new point coordinates and connect them

For the pair of functions, (g⁰f)(x) is - 6 x ² - 3 x + 22 and (f ⁰g)(x) is - 18 x ² + 7 x - 5

The given questions asks to find (g⁰f)(x) and (f ⁰g)(x), that is, function of g in terms of f(x) and function of f in terms of g(x).

To find : (g⁰f)(x)

Step 1 : Write g (x)

=> g (x) = - 3x -1

Step 2 : Put f (x) in place of x in g (x)

=> g (f(x) = -3 ( 2 x ² + x - 7 ) + 1

=> g (f(x) = - 6 x ² - 3 x + 21  + 1

=> g (f(x) = - 6 x ² - 3 x + 22

=> (g⁰f)(x) = - 6 x ² - 3 x + 22

Similarly,

To find : (f ⁰g)(x)

Step 1 : Write f (x)

=> f (x) = 2 x ² + x - 7

Step 2 : Put g (x) in place of x in f (x)

=> f (g(x) = 2 ( -3 x - 1 ) ² + x - 7

=> f (g(x) = 2 - ( 9 x ² + 1 + 3 x ) + x - 7

=> f (g(x) = - 18 x ² + 2 + 6 x + x - 7

=> f (g(x) = - 18 x ² + 7 x - 5

=> (f ⁰g)(x) = - 18 x ² + 7 x - 5

Therefore, we get for the pair of functions, (g⁰f)(x) as - 6 x ² - 3 x + 22 and (f ⁰g)(x) as - 18 x ² + 7 x - 5

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Answer:

Step-by-step explanation:

Answer:Vamos a convertir 420 kcal/h a una tasa de cal/min con estas tasas de conversion: Hay 60min en 1h Hay 1000cal en 1 kcal entonces 1000 x 420 = 420000 / 60 = 7000

For the pair of functions, (g⁰f)(x) is 4 x ² - 7 and (f ⁰g)(x) is 16 x ² + 8 x - 1

The given questions asks to find (g⁰f)(x) and (f ⁰g)(x), that is, function of g in terms of f(x) and function of f in terms of g(x).

To find : (g⁰f)(x)

Step 1 : Write g (x)

=> g (x) = 4 x + 1

Step 2 : Put f (x) in place of x in g (x)

=> g (f(x) = 4 ( x ² - 2 ) + 1

=> g (f(x) = 4 x ² - 8 + 1

=> g (f(x) = 4 x ² - 7

=> (g⁰f)(x) = 4 x ² - 7

Similarly,

To find : (f ⁰g)(x)

Step 1 : Write f (x)

=> f (x) = x ² - 2

Step 2 : Put g (x) in place of x in f (x)

=> f (g(x) = ( 4 x + 1 ) ² - 2

=> f (g(x) = 16 x ² + 1 + 8 x -2

=> f (g(x) = 16 x ² + 8 x - 1

=> (f ⁰g)(x) = 16 x ² + 8 x - 1

Therefore, we get for the pair of functions, (g⁰f)(x) as 4 x ² - 7 and (f ⁰g)(x) as 16 x ² + 8 x - 1

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Answer:

5y

Step-by-step explanation:

if you add 5+y then it'll be 5y, you just add the two terms together

Answer to this question is:

Domain of given function is [0,∞),{x|x≥0}

Range of given function is (-∞,0],{y|y≤0}

Inverse of given function is x²

The domain is all values of x that make the expression defined.The entire range of independent variable values is the domain of a function.

This definition can be stated simply as follows: The domain is the collection of all x-values that can be used to make the function "work" and produce actual y-values.

When choosing the domain, keep in mind that a fraction's denominator (bottom) cannot be 0.

Interval Notation: [0,∞)

Set-Builder Notation:{x|x≥0}

The set of all valid y values constitutes the range.

Interval Notation:(-∞,0]

Set-Builder Notation:{y|y≤0}

For calculation of inverse:

∴Domain of given function is  [0,∞),{x|x≥0}

Range of given function is (-∞,0],{y|y≤0}

Inverse of given function is x²

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Answer:

The length of a rectangle is 6 meters longer than three times its width. If the total perimeter is 468 meters, what are the dimensions of the rectangle?

the answer would be 177

If you randomly choose two cards from a thoroughly shuffled deck and if you put the first card back in the deck before you draw the second then the probability that the first card is an ace and the second card is a king is 0.59%.

The probability that the first card is an ace and the second card is a king can be calculated as follows,

Initially, divide the expected outcomes by the total possible outcomes

As a deck has a total of 52 cards, the total possible outcomes are considered to be 52

A deck consists of 4 aces and 4 king cards

Hence,

the probability that the first card is an ace = 4 / 52

the probability that the second card is a heart = 4 / 52

The probability that the first card is an ace and the second card is a heart can be determined by multiplying the probability of an ace with the probability of heart,

Probability = (4/52)(4/52)

Probability = 0.0059

Converting into percentage,

Probability = 0.0059 × 100

Probability = 0.59%

Hence the probability that the first card is an ace and the second card is a king is calculated to be 0.59%.

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When a number is 3 times a first number. A third number is 100 more than the first number and the sum of the three numbers is 450, the numbers are 70, 170 and 210.

Let the first number = x

Second number = 3x

Third number = x + 100

Sum = 450

The numbers will be:

x + 3x + x + 100 = 450

5x + 100 = 450

5x = 450 - 100

5x = 350

Divide

x = 350 / 5

x = 70

First number = 70

Second number = 3x = 3 × 70 = 210

Third number = x + 100 = 70 + 100 = 170

Therefore, the numbers are 70, 170 and 210.

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One number is 3 times a first number. A third number is 100 more than the first number. If the sum of the three numbers is 450 , find the numbers.

[tex](-8)^\frac{1}{2} \neq -(8^\frac{1}{2})[/tex] because 8 is an odd power of 2 and [tex](-27)^\frac{1}{3} = -(27)^\frac{1}{3}[/tex] because 27 can be written with the power which a multiple of 3.

Consider [tex](-8)^\frac{1}{2}[/tex].

We know that 8 = 2³

So [tex](-8)^\frac{1}{2}[/tex] = [tex](-2^3)^\frac{1}{2}[/tex]

Also there is no real value for this expression. If 8 had an even power, then we would have got a real value. Because here 3 does not cancel with 2 completely.

Now for [tex]-(8^\frac{1}{2})[/tex] , this expression means the negative sign of the square root of 8. Even though 8 is not an perfect square, 8 has a real square root value. So this expression gives a real value. But the previous one does not.

Hence, [tex](-8)^\frac{1}{2} \neq -(8^\frac{1}{2})[/tex]

Consider [tex](-27)^\frac{1}{3}[/tex] = [tex](-3^3)^\frac{1}{3}[/tex] = -3, canceling out the 3 on the denominator with the power of (-3).

Also,  [tex]-(27)^\frac{1}{3}[/tex] =  [tex]-(3^3)^\frac{1}{3}[/tex] = -3 again.

Hence  [tex](-27)^\frac{1}{3} = -(27)^\frac{1}{3}[/tex] .

This is because the power was a multiple of 3.

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