Editorial - Matlan - TO TKA 1

Problem 1

the absolute value of P equals the absolute value of A minus B times the absolute value of C plus D

A minus B equals the 2 by 2 matrix Row 1: 3 4 Row 2: 2 negative 2 minus the 2 by 2 matrix Row 1: negative 5 negative 3 Row 2: 3 2 equals the 2 by 2 matrix Row 1: 8 7 Row 2: negative 1 negative 4

the absolute value of A minus B equals negative 32 plus 7 equals negative 25

C plus D equals the 2 by 2 matrix Row 1: 2 4 Row 2: 3 5 plus the 2 by 2 matrix Row 1: 3 4 Row 2: 2 negative 1 equals the 2 by 2 matrix Row 1: 5 8 Row 2: 5 4

the absolute value of C plus D equals 20 minus 40 equals negative 20

therefore the absolute value of P equals negative 25 times negative 20 equals 500

(A)

Problem 2

the 2 by 2 matrix Row 1: Column 1, 6 x plus 3 plus y minus 2 Column 2, 8 x plus 4 plus 3 Row 2: Column 1, 9 minus 2 y plus 4 Column 2, 12 minus 6 plus the 2 by 2 matrix Row 1: 1 negative 3 Row 2: 8 negative 8 equals the 2 by 2 matrix Row 1: Column 1, 3 Column 2, x minus 3 Row 2: Column 1, y Column 2, negative 2

9 minus 2 y plus 4 plus 8 equals y

therefore y equals 7

8 x plus 4 plus 3 minus 3 equals x minus 3

therefore x equals negative 1

Subproblem 2.1

x plus y equals negative 7

x plus y equals negative 1 plus 7 equals 6

The statement is false.

Subproblem 2.2

x squared minus y squared equals negative 48

x squared minus y squared equals 1 minus 49 equals negative 48

The statement is true.

Subproblem 2.3

the 2 by 2 matrix Row 1: Column 1, 2 x minus 1 Column 2, 4 Row 2: Column 1, 2 Column 2, 1 times the 2 by 2 matrix Row 1: Column 1, y minus 6 Column 2, 2 Row 2: Column 1, 4 Column 2, 1 equals the 2 by 2 matrix Row 1: 13 negative 2 Row 2: 6 5

the 2 by 2 matrix Row 1: negative 3 4 Row 2: 2 1 times the 2 by 2 matrix Row 1: 1 2 Row 2: 4 1 equals the 2 by 2 matrix Row 1: 16 negative 2 Row 2: 6 5

The statement is true.

Problem 3

f with hook of x plus h of x equals x squared minus 3 x plus 2 plus 3 x plus 5 equals x squared plus 7

f with hook of x times g of x equals open paren x squared plus 7 close paren times open paren 2 x squared minus x close paren equals 2 x to the fourth power minus x cubed plus 14 x squared minus 7 x

(B) key different?

Problem 4

f with hook of x equals open paren x plus 8 close paren times open paren x minus 2 close paren times Q of x plus 2 x minus 1

f with hook of 2 equals 3

f with hook of negative 8 equals negative 17

f with hook of x equals open paren 2 x plus 1 close paren times R of x minus 2

f with hook of negative one half equals negative 2

Problem 4.1

Divide f with hook of x by x minus 2 and the remainder is 3

True, as f with hook of 2 equals 3

Problem 4.2

Divide f with hook of x by 2 x squared minus 3 x minus 2 and the remainder is 1 minus 2 x

2 x squared minus 3 x minus 2 equals open paren 2 x plus 1 close paren times open paren x minus 2 close paren

That means f with hook of 2 equals 1 minus 4 equals negative 3

Which is not true, so the statement is false.

Problem 4.3

Divide f with hook of x by 2 x squared plus 17 x plus 8 and the remainder is 2 x minus 1

2 x squared plus 17 x plus 8 equals open paren 2 x plus 1 close paren times open paren x plus 8 close paren

That means f with hook of negative one half equals negative 2 times one half minus 1 equals negative 2

Which is true, so the statement is true.

Problem 5

I hate Horner's method of synthetic division, so I won't use it.

f with hook of 2 equals 0 equals 16 plus 8 a minus 4 plus 2 b minus 12 equals 2 b plus 8 a

f with hook of negative 3 equals 0 equals 81 minus 27 a minus 9 minus 3 b minus 12 equals 60 minus 27 a minus 3 b

therefore a equals 4 comma b equals negative 16

a squared minus b squared equals 16 minus 256 equals negative 240

(D)

Problem 6

Fast.

1. If this year it's 87 percent, then next year at worst it's 87 minus 8 equals 79 percent

2. Impossible, as 78 is less than 79

3. Impossible, 78 is less than 79

4. Yeah, the maximum is 87 plus 8 equals 95 percent

5. the absolute value of T minus 0.87 is less than or equal to 0.08 yeah

This problem is quite bad as it's quite ambiguous.

Problem 7

Limit problems are free if you know how to solve them.

2 x squared plus 3 x minus 2 is greater than 0

Whatever's inside the log cannot be smaller or equal to 0

open paren 2 x minus 1 close paren times open paren x plus 2 close paren is greater than 0

Test at x equals 0

negative 2 is greater than 0

x is greater than one half or x is less than negative 2

The first statement is automatically invalidated.

f with hook of 2 equals the log base 3 of open paren 2 times 4 plus 6 minus 2 close paren equals the log base 3 of 12

The second statement is clearly false.

The graphic never touches the y axis as x equals 0is indeterminate. Therefore, the third statement is true.

The fourth statement is false, as logs can have negative values if we just test out the inner function.

Therefore the REAL answer is that of just the third statement is true.

Problem 8

Personally, I used functional analysis (made-up term)

2 to the x-th power for x is less than or equal to 1 will approach y equals 0 as it approaches negative infinity

3 x minus 1 will approach y equals infinity as it approaches infinity

Therefore, the minimum value is most likely at the second part of the function, at x squared minus 1

x sub min equals negative b over 2 a equals 0

y sub min equals 0 minus 1 equals negative 1

The first statement is true.

The second statement is also true, as the maximum value is at infinity

The third statement can be proven false just by testing, which I won't do as it's tedious to write here.

Problem 9

g of x equals p x plus q

f with hook of x equals a raised to the p x plus q power plus b

Well, since there are 4 unknowns, we need to utilize all 4 points.

f with hook of 0 equals four thirds equals a to the q-th power plus b

b equals four thirds minus a to the q-th power

f with hook of 1 equals 4 equals a raised to the p plus q power plus b

a to the p-th power times a to the q-th power plus four thirds minus a to the q-th power equals 4

a to the q-th power times open paren a to the p-th power minus 1 close paren equals 4 minus four thirds equals eight thirds

a to the q-th power equals 8 over 3 times open paren a to the p-th power minus 1 close paren

f with hook of negative 1 equals 28 over 27 equals a raised to the negative p plus q power plus b

the fraction with numerator a to the q-th power and denominator a to the p-th power plus four thirds minus a to the q-th power equals 28 over 27

a to the q-th power times open paren the fraction with numerator 1 and denominator a to the p-th power minus 1 close paren equals 28 over 27 minus four thirds equals negative 8 over 27

8 over 3 times open paren a to the p-th power minus 1 close paren times open paren the fraction with numerator 1 minus a to the p-th power and denominator a to the p-th power close paren equals negative 8 over 27

therefore a to the p-th power equals 9

therefore a to the q-th power equals 8 over 3 times open paren 9 minus 1 close paren equals one third

therefore b equals four thirds minus one third equals 1

f with hook of x equals a to the p-th power to the x-th power times a to the q-th power plus b equals 9 to the x-th power times one third plus 1 equals 3 raised to the 2 x minus 1 power plus 1

There we go. The first statement is true, g of x equals 2 x minus 1

f with hook of negative 2 equals 3 to the negative 5 power plus 1 equals 1 over 243 plus 1

The second statement is clearly false.

The graph also never touches y equals 1, as the original asymptote of y equals 0 has been moved up 1 unit. So, the third statement is true!

Problem 10

Subproblems 10.1 to 10.3

the cosine of open paren 2 x minus pi close paren equals the cosine of open paren pi minus 2 x close paren equals negative cosine 2 x

The best way to judge transformations is via comparing the distance between 2 minima and/or maxima.

the sine of open paren pi over 4 times 2 close paren equals 1

negative the cosine of open paren pi over 2 times 2 close paren equals 1

pi over 2 minus pi over 4 equals pi over 4

Therefore, the first three statements are true. If you move sine 2 x by pi over 4 in any direction, you get negative cosine 2 x

Subproblem 10.4

sine 2 x equals negative cosine 2 x

sine 2 x over cosine 2 x equals negative 1

tangent 2 x equals negative 1

2 x equals negative pi over 4 plus n pi

x equals negative pi over 8 plus n pi over 2

Try n equals 1

x equals negative pi over 8 plus pi over 2 equals three eighths pi

Substitute that to sine 2 x

the sine of open paren 2 times three eighths pi close paren equals the fraction with numerator the square root of 2 and denominator 2

Wait. This subproblem's statement is actually true, as both graphs go through open paren three eighths pi comma the fraction with numerator the square root of 2 and denominator 2 close paren

Subproblem 10.5

f with hook of x and g of x both fall at three eighths pi is less than or equal to x is less than or equal to pi over 2

Clearly false, and you can prove that just by knowing that negative cosine 2 x peaks at pi over 2, so in this interval it's actually rising.

Problem 11

Well, only the first one is actually needed. If you look at the image, you can find the amplitude of the first function, because we know the peak is at pi over 4

From there, we can guess the shift of the function, therefore making the second statement obsolete.

Problem 12

AE right arrow equals BD right arrow

BD right arrow equals a plus b

AE right arrow plus BD right arrow equals 2 a plus 2 b

I saw another problem like this once, using hexagons. If you encounter something like that but you can't solve it outright, try tiling the shape!

Problem 13

Just do it.

open paren 2 i minus 3 j plus 4 i plus j close paren times open paren 3 i minus 2 j plus i minus 2 j close paren

open paren 6 i minus 2 j close paren times open paren 4 i minus 4 j close paren equals 24 plus 8 equals 32

Problem 14

First statement is obvious.

Second statement is obvious.

Third statemnt is obvious.

Fourth statement is obvious, just extend one vector after another.

Fifth statement can be found via pythagoras.

Problem 15

Can't draw. But, if you visualize, it's easy.

Problem 16

First statement is clearly false.

Second statement is clearly true.

Third statement can be checked easily to be false

Fourth statement is obviously true.

(C)

Problem 17

the fraction with numerator p times q and denominator the metric of q

The first statement's probably useless, let's try using the second.

the absolute value of p minus q squared equals the absolute value of p squared plus the absolute value of q squared minus 2 times open paren p times q close paren

p times q equals the fraction with numerator the absolute value of p squared plus 48 and denominator 2

We can find the rest. The answer is B.

Problem 18

Subproblem 18.1

the absolute value of u right arrow equals 3

the absolute value of u right arrow equals the square root of 1 plus 4 plus 4 equals 3

True.

Subproblem 18.2

u right arrow times v right arrow equals 7

the absolute value of u right arrow minus v right arrow squared equals the absolute value of u right arrow squared plus the absolute value of v right arrow squared minus 2 times open paren u right arrow times v right arrow close paren

2 times open paren u right arrow times v right arrow close paren equals the absolute value of v right arrow squared minus 27

the absolute value of u right arrow plus v right arrow squared equals the absolute value of u right arrow squared plus the absolute value of v right arrow squared plus 2 times open paren u right arrow times v right arrow close paren

negative 2 times open paren u right arrow times v right arrow close paren equals the absolute value of v right arrow squared minus 55

2 times open paren u right arrow times v right arrow close paren equals negative the absolute value of v right arrow squared plus 55

4 times open paren u right arrow times v right arrow close paren equals 28

u right arrow times v right arrow equals 7

True.

Subproblem 18.3

the absolute value of v right arrow equals seven thirds

64 equals 9 plus the absolute value of v right arrow squared plus 14

the absolute value of v right arrow squared equals 41

False.

Subproblem 18.4

False.

seven thirds

Subproblem 18.5

seven thirds times one third times open paren negative i plus 2 j minus 2 k close paren equals negative fourteen ninths k minus seven ninths i plus fourteen ninths j

True.

Problem 19