(x, why?): September 2021

Thursday, September 30, 2021

(x, why?) Mini: Sloppy

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Sloppy, smudgy. Math is messy, even when you use a pencil!

There were times in Park Slope that we'd call it Park Slop. Granted, that part is called Gowanus these days because it sounds trendy, even though it's the named of a clogged, congested expressway and a toxic canal. (Serious, it's a Superfund site.)

I also write Fiction!

Come back often for more funny math and geeky comics.

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2013

Part I: Each correct answer will receive 2 credits.

21. The quantities p and q vary inversely. If p = 20 when q = -2, and p = x when q = -2x + 2, then x equals

1) -4 and 5
2) 20/19
3) -5 and 4
4) -1/4

Answer: 1) -4 and 5

If p and q vary inversely, then (p)(q) = k or p = kq.

Since k = pq, then k = (20)(-2) = -40

This means that k = (x)(-2x + 2) = 40

-2x² + 2x = 40
-2x² + 2x - 40 = 0
x² - x + 20 = 0
(x - 5)(x + 4) = 0
x = 5 or x = 4

22. What is the solution set of the equation -SQRT(2) sec x = 2, when 0° < x < 360°?

1) {45°, 135°, 225°, 315°}
2) {45°, 315°}
3) {135°, 225°}
4) {225°, 315°}

Answer: 3) {135°, 225°}

Use the Identity sec x = 1 / cos x

-SQRT(2) sec x = 2
sec x = -2 / SQRT(2)
1/cos x = -2 / SQRT(2)
cos x = -SQRT(2) / 2

cos x = -SQRT(2)/2 when x = 135° or 225°.

23. The discriminant of a quadratic equation is 24. The roots are

1) imaginary
2) real, rational, and equal
3) real, rational, and unequal
4) real, irrational, and unequal

Answer: 4) real, irrational, and unequal

When the discriminant is LESS THAN 0, the roots are imaginary. Eliminate Choice (1)

When the discriminant is EQUAL TO 0, the roots are real, rational and equal. Eliminate Choice (2)

When the discriminant is GREATER THAN 0, there are two real unequal roots. If the discriminant is a PERFECT SQUARE, then the roots are rational. But 24 is not a perfect square, so eliminate Choice (3).

The roots of this equation are real, unequal and irrational, which is Choice (4).

24. How many different six-letter arrangements can be made using the letters of the word “TATTOO”?

1) 60
2) 90
3) 120
4) 720

Answer: 1) 60

The number of ways to rearrange six letters is 6! However, we have to account for repeated letters. There are two Os and three Ts.

So 6! must be divided by 2! and by 3! or

6 * 5 * 4 * 3 * 2 * 1
3 * 2 * 1 * 2 * 1

Canceling common factors leaves 5 * 4 * 3 = 60.

25. Expressed in simplest form, 3y/(2y - 6) + 9/(6 - 2y) is equivalent to
1) (-6y² + 36y - 54) / ( (2y-6)(6-2y) )
2) (3y - 9)/(2y - 6)
3) 3/2
4) -3/2

Answer: 3) 3/2

Note that the demoninators are opposites. You can reverse the second denominator by multiplying the second fraction by (-1)/(-1). Then the fractions can be combined.

3y/(2y - 6) + (-1/-1) (9/(6 - 2y)
= 3y/(2y - 6) - 9/(2y - 6)
= (3y - 9)/(2y - 6)

After the fractions are combined, you can continue to simplify:
(3y - 9)/(2y - 6) = (3)(y - 3)/( (2)(y - 3) ) = 3/2

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!

Geometry Problems of the Day (Geometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.

21. In the diagram of trapezoid ABCD below, AB || DC, AD ≅ BC, m∠A = 4x + 20, and m∠C = 3x - 15.

What is m∠D?

1) 25
2) 35
3) 60
4) 90

Answer: 3) 60

There is no relation between angles A and C unless you know that the trapezoid is isosceles. You know this is the case because AD ≅ BC.

You know that AB || CD, and therefore the consecutive angles A and D are interior angles on the same side of a transversal. That makes them supplementary. Since the trapezoid is isosceles, the base angles are congruent.

So 4x + 20 + 3x - 15 = 180
7x + 5 = 180
7x = 175
x = 25

Angle D is congruent to angle C, which is 3(25) - 15 = 75 - 15 = 60.

22. In circle R shown below, diameter DE is perpendicular to chord ST at point L

Which statement is not always true?

1) SL ≅ TL
2) RS = DR
3) RL ≅ LE
4) (DL)(LE) = (SL)(LT)

Answer: 3) RL ≅ LE

When a diameter is perpendicular to a chord, it will bisect the chord. Also, when any two chords intersect (including a diameter) the product of the lengths of the two segments of the chords will be equal.

Choice (1) is always true because ST is bisected.

Choice (2) is always true because both RS and DR are radii. All radii in a circle are equal in length.

Choice (3) is NOT always true. Most of the time it will NOT be true. The chord does not have to bisect the radius. It could but doesn't have to.

Choice (4) is always true for any two chords that intersect, perpendicular or not.

23. Which equation represents circle A shown in the diagram below?

1) (x - 4)² + (y - 1)² = 3
2) (x + 4)² + (y + 1)² = 3
3) (x - 4)² + (y - 1)² = 9
4) (x + 4)² + (y + 1)² = 9

Answer: 4) (x + 4)² + (y + 1)² = 9

The equation of a circle is (x - h)² + (y - k)² = r², where (h,k) is the center of the circle and r is the radius. Note that there are MINUS signs in the formula.

It should be obvious that the radius is 3 and 3² = 9, so eliminate choices (1) and (2).

The signs for h and k are flipped because of the minus signs, and the center is (-4, -1) so h = 4 and k = 1 NOT -4 and -1, respectively.

Note that there was another circle equation problem Five Questions ago, in yesterday's post.

24. Which equation represents a line that is parallel to the line whose equation is 3x - 2y = 7?

1) y = -3/2x + 5
2) y = -2/3 x + 4
3) y = 3/2 x - 5
4) y = 2/3 x - 4

Answer: 3) y = 3/2 x - 5

Parallel lines have the same slope, so we need to know the slope of the given line. The choices are all in slope-intercept form.

The slope of a line in Standard Form is -A/B, or -3/-2 = 3/2.

Alternatively, rewrite 3x - 2y = 7
-2y = -3x + 7
y = 3/2x - 7/2

The slope of the two parallel lines must be 3/2.

25. In the diagram below of circle O, PAC and PBD are secants.

If mCB = 70 and mAB = 20, what is the degree measure of ∠P?

1) 25
2) 35
3) 45
4) 50

Answer: 1) 25

The formula for angle P is (CD - AB) / 2.

(70 - 20) / 2 = 50 / 2 = 25

More to come. Comments and questions welcome.

More Regents problems.

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You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
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Wednesday, September 29, 2021

Geometry Problems of the Day (Geometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.

16. Which set of numbers could not represent the lengths of the sides of a right triangle?

1) {1, 3, SQRT(10)}
2) {2, 3, 4}
3) {3, 4, 5}
4) {8, 15, 17}

Answer: 2) {2, 3, 4}

You should know a number of Pythagorean Triples, where each number is a rational number. The smallest one containing three whole numbers is 3,4,5. This means that 2,3,4 could NOT possibly be a right triangle.

You can use Pythagorean Theorem to check, but you should have known already.

2² + 3² = 4 + 9 = 13 =/= 4²

1² + 3² = 1 + 9 = 10 = SQRT(10)²

3² + 4² = 9 + 16 = 25 = 5²

8² + 15² = 64 + 225 = 289 = 17²

17. How many points are 5 units from a line and also equidistant from two points on the line?

1) 1
2) 2
3) 3
4) 0

Answer: 2) 2

Look at the image below.

18. The equation of a circle is (x - 2)² + (y + 5)² = 32. What are the coordinates of the center of this circle and the length of its radius?

1) (-2,5) and 16
2) (2,-5) and 16
3) (-2,5) and 4 SQRT(2)
4) (2,-5) and 4 SQRT(2)

Answer: 4) (2,-5) and 4 SQRT(2)

The equation of a circle is (x - h)² + (y - k)² = r², where (h,k) is the center of the circle and r is the radius. Note that there are MINUS signs in the formula.

It should be obvious that the square root of 32 is NOT 16, so eliminate choices (1) and (2).

The signs for h and k are flipped because of the minus signs, so the center is (2, -5) NOT (-2, 5).

The square root of 32 = SQRT(16) * SQRT(2) = 4 * SQRT(2)

14. The equation of a line is y = 2/3 x + 5. What is an equation of the line that is perpendicular to the given line and that passes through the point (4,2)?

1) y = 2/3 x - 2/3
2) y = 3/2 x - 4
3) y = -3/2 x + 7
4) y = -3/2 x + 8

Answer: 4) y = -3/2 x + 8

The slope of a line perpendicular to a given line with be the negative reciprocal of the slope of the given line. That means that the line we are looking for has a slope of -3/2. Eliminate Chpices (1) and (2).

The correct equation will have (4,2) as a sollution, so check that point in each equation:

y = -3/2 (4) + 7 = -6 + 7 = 1 =/= 2, not a solution

y = -3/2 (4) + 8 = -6 + 8 = 2, which is a solution

20. Consider the relationship between the two statements below.

If SQRT(16 + 9) =/= 4 + 3, then 5 =/= 4 + 3.
If SQRT(16 + 9) = 4 + 3, then 5 = 4 + 3.
These statements are

1) inverses
2) converses
3) contrapostives
4) biconditionals

Answer: 1) inverses

An inverse of a conditional has each part of the statement negated -- either a "NOT" is added or an existing "NOT" is removed.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Tuesday, September 28, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2013

Part I: Each correct answer will receive 2 credits.

16. The area of triangle ABC is 42. If AB = 8 and m∠B = 61, the length of BC is approximately

1) 5.1
2) 9.2
3) 12.0
4) 21.7

Answer: 3) 12.0

The area of the triangle is given by A = 1/2 ab sin C, the product of the length of two sides times the sin of the included angle.

42 = 1/2 (8) (x) sin 61
x = 2 * 42 / (8 sin 61) = 12.005...

17. When factored completely, the expression 3x³ - 5x² - 48x + 80 is equivalent to

1) (x² - 16)(3x - 5)
2) (x² + 16)(3x - 5)(3x - 5)
3) (x + 4)(x - 4)(3x - 5)
4) (x + 4)(x - 4)(3x - 5)(3x - 5)

Answer: 3) (x + 4)(x - 4)(3x - 5)

It should be obvious that CHoices (2) and (4) would result in polynomials of order 4. That is, the first tem will be x⁴, not x³. Also, Choice (1) is NOT factored completely, so the only possible answer is Choice (3).

Factor by grouping:

3x³ - 5x² - 48x + 80
= 3x³ - 48x - 5x² + 80
= 3x(x² - 16) - 5(x² - 16)
= (3x - 5)(x² - 16)
This is Choice (1), but it is not factored completely.
= (3x - 5)(x + 4)(x - 4)

18. The value of sin (180 + x) is equivalent to

1) -sin x
2) -sin (90 - x)
3) sin x
4) sin (90 - x)

Answer: 1) -sin x

When a point was rotated 180 degrees about the origin, the point (x, y) becomes (-x, -y). Since any point (x, y) on the unit circle is (cos x, sin x), then adding 180 to sin x would change it to -sinx.

19. The sum of CBRT(6a⁴b²) and CBRT(162a⁴b²) expressed in simplest radical form, is

1) SixthRoot(6a⁴b²)
2) 2a²b CBRT(21a²b)
3) 4a CBRT(6ab²)
4) 10a²b CBRT(8)

Answer: 3) 4a CBRT(6ab²)

The two expressions cannot be added unless they are like terms. Simplify the radicals by finding cube roots.

CBRT(6a⁴b²) + CBRT(162a⁴b²)
= CBRT(6a³a b²) + CBRT(2 * 3⁴ a³a b²)
= CBRT(6a³a b²) + CBRT(2 * 3 3³ a³a b²)
= a * CBRT(6 a b²) + 3a * CBRT(6a b²)
= 4a * CBRT(6 a b²)

Choice (1) with its Sixth root was ridiculous and should have been immediately eliminated.

20. Which equation is represented by the graph below?

1) y = 2 cos 3x
2) y = 2 sin 3
3) y = 2 cos 2π/3 x
4) y = 2 cos 2π/3 x

Answer: 1) y = 2 cos 3x

Cos (0) = 1 and 2 Cos (0) = 2. However 2 Sin (0) = 0, so eliminate choices (3) and (4).

The 3 in front of the x increases the frequency by a factor of 3, meaning that waves will be compressed and each cycle will be smaller, not bigger.

A frequency of 2π/3 would stretch out the graph. And the coefficient of x does not correspond to the number on the x-axis.

The correct choice is (1).

More to come. Comments and questions welcome.

More Regents problems.

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(x, why?) Mini: Favorite Days

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Happy triangular number day

Actually, I could've waited for next month to do this because 10/03/21, 10/06/21, 10/15/21 and 10/28/21 each have three of them. (Then again, I could've done this in two other months ... but I didn't.) And that's leaving off the trivial case of 1.

It's nice to have something to look forward to each month. Then again, I keep looking forward to those weekends and just hope I not too tired to enjoy them.

I also write Fiction!

Come back often for more funny math and geeky comics.

Monday, September 27, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2013

Part I: Each correct answer will receive 2 credits.

11. If sin A = 1/3 , what is the value of cos 2A?

1) -2/3
2) 2/3
3) -7/9
4) 7/9

Answer: 4) 7/9

If you put cos(2 * sin^-1(1/3)) into your calculator, you will get .7777..., which is 7/9.

Once of the double angle formulas for cosine is cos 2A = 1 - 2 sin² A.
So cos 2A = 1 - 2(1/3)² = 1 - 2(1/9) = 1 - 2/9 = 7/9

12. In the interval 0° < x < 360°, tan x is undefined when x equals

1) 0° and 90°
2) 90° and 180°
3) 180° and 270°
4) 90° and 270°

Answer: 4) 90° and 270°

Tan x = sin x / cos x, so Tan x is undefined when cos x = 0, which is when x is 90° and 270°.

On the Cartesian plane, tangent is undefined at the y-axis, which is the top and bottom of the unit circle.

13. If f(x) = SQRT(9 - x²), what are its domain and range?

1) domain: {x |-3 < x < 3}; range: {y|0 < y < 3}
2) domain: {x | x =/= +3}; range: {y|0 < y < 3}
3) domain: {x |-3 x < -3}; range: {y| y =/= 0}
4) domain: {x | x =/= +3}; range: {y|y > 0}

Answer: 1) domain: {x |-3 < x < 3}; range: {y|0 < y < 3}

The quadratic function g(x) = 9 - x² opens downward from (0, 9) with zeroes at (-3,0) and (3,0).

However, any values of g(x) < 0 would cause f(g(x)) to be undefined. So the only valid values of x are -3 <. When x = 0, f(x) will be 3, so the range of the function will be between 0 and 3, inclusive. This is Choice (1).

Choice (2) says x cannot be + 3, which is not true because those are valid values. If the values of greater magnitude that aren't allowed.

Choice (3) contains the invalid numbers in its domain and excludes all the valid ones, except 3 and -3.

Choice (4) exclues 3, which is valid, but nothing else.

This could be solved by graphing the function in your calculator and then comparing the results to the notation in the choices.

14. When x² + 3x - 4 is subtracted from x³ + 3x² - 2x, the difference is

1) x³ + 2x² - 5x + 4
2) x³ + 2x² + x - 4
3) -x³ + 4x² + x - 4
4) -x³ - 2x² + 5x + 4

Answer: 1) x³ + 2x² - 5x + 4

When subtracting the "FROM" clause goes on top. If I take $20 FROM my wallet, the amount of money that had been in my wallet would go on top.

Likewise, like subtracting a 3-digit number from a 4-digit number, you have to line the columns up correctly.

x³ + 3x² - 2x + 0
0x³ + x² + 3x - 4

x³ + 2x² - 5x + 4

Subtract each pair of terms, keeping subtraction of signed numbers in mind.

15. In the diagram below, the length of which line segment is equal to the exact value of sin θ?

1) TO
2) TS
3) OR
4) OS

Answer: 2) TS

In the Unit Circle, each point (x, y) on the circle has the coordinates (cos θ, sin θ), which makes sin θ the y-coordinate. The line segment ST runs from the x-axis (y = 0) to the point T on the circle (y = sin θ), and has a length of sin θ - 0 = sin θ.

The length of TO is 1, as it is the radius of the unit circle.

The length of OR is 1, as it goes from the origin to (1, 0).

The length of OS is cos θ, the x-coordinate of point T.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!

Sunday, September 26, 2021

Geometry Problems of the Day (Geometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.

11. Triangle ABC is shown in the diagram below. true?

If DE joins the midpoints of ADC and AEB, which statement is not true?

1) DE = 1/2 CB
2) DE || CB
3) AD/DC = DE/CB
4) Triangle ABC ~ AAED

Answer: 3) AD/DC = DE/CB

DE is a midsegment of triangle AB. That means that DE || CB and DE = 1/2 CB. Since the lines are parallel the corresponding angles along the transverals are congruent, which means that the triangles are similar. Eliminate Choices (1) and (2) and (4) because these are ALWAYS true by definition.

Choice (3) looks correct, but it's deceiving. What is true is that AD/AC = DE/CB, which is 1/2. The ratio AD/DC = 1/1 because D is the midpoint of AC.

12. The equations x² + y² = 25 and y = 5 are graphed on a set of axes.
What is the solution of this system?

1) (0,0)
2) (5,0)
3) (0,5)
4) (5,5)

Answer: 3) (0,5)

The second equation literally says that y = 5, so the y-coordinate MUST BE 5. Eliminate Choices (1) and (2).

(0)² + (5)² = 0 + 25 = 25. Choice (3) is the solution.

(5)² + (5)² = 25 + 25 = 50 =/= 25. Choice (4) is incorrect.

13. Square ABCD has vertices A(-2,-3), B(4,-1), C(2,5), and D(-4,3). What is the length of a side of the square?

1) 2 SQRT(5)
2) 2 SQRT(10)
3) 4 SQRT(5)
4) 10 SQRT(2)

Answer: 2) 2 SQRT(10)

Use the Distance Formula of Pythagorean Theorem. Pick any two consecutive points. I'll pick the two with the fewest minus signs.

SQRT( (4 - 2)² + (-1 - 5)²) = SQRT( 2² + (-6)²)
= SQRT(40) = SQRT(4) * SQRT(10) = 2 SQRT(10)

14. The diagram below shows ABD, with ray ABC, BE ⊥ AD, and ∠EBD ≅ ∠CBD.
If m∠ABE = 52, what is m∠D?

1) 26
2) 38
3) 52
4) 64

Answer: 1) 26

Since ∠EBD ≅ ∠CBD and ∠ABE + ∠EBD + ∠CBD = 180, we can solve for ∠EBD.

52 + x + x = 180
52 + 2x = 180
2x = 128
x = 64

Since EBD = 64 and BED is a right angle (90), then we can solve for D:

64 + 90 + y = 180
154 + y = 180
y = 26

15. As shown in the diagram below, FD and CB intersect at point A and ET is perpendicular to both FD and CB at A.

Which statement is not true?

1) ET is perpendicular to plane BAD.
2) ET is perpendicular to plane FAB.
3) ET is perpendicular to plane CAD.
4) ET is perpendicular to plane BAT.

Answer: 4) ET is perpendicular to plane BAT.

ET is shown as a vertical line, so it will be perpendicular to the horizontal plane. That plane can be named using any three of the following points: A, B, C, D, F.

Choice (4) using the point T. Line AT is the same as line ET, so ET is contained in that plane BAT and cannot be perpendicular to it.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Friday, September 24, 2021

Sub Plan Delivery

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Maybe he didn't do anything. And maybe that's the problem.

Or maybe it's just the usual new parent stuff.

I also write Fiction!

Come back often for more funny math and geeky comics.

Geometry Problems of the Day (Geometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.

6. 6 Plane A and plane B are two distinct planes that are both perpendicular to line ℓ. Which statement about planes A and B is true?

1) Planes A and B have a common edge, which forms a line
2) Planes A and B are perpendicular to each other.
3) Planes A and B intersect each other at exactly one point.
4) Planes A and B are parallel to each other.

Answer: 4) Planes A and B are parallel to each other.

Think of a floor and a ceiling and an elevator shaft that is perpendicular to the two of them. The floor and ceiling are parallel to each other and perpendicular to the elevator shaft.

Planes A and B will not intersect. If they were perpendicular to each other, they would have a common edge, which would form a line. So (2) couldn't be true without (1) also being true.

Choice (3) makes no sense. Two planes cannot meet at a single point.

7. Triangle ABC is similar to triangle DEF. The lengths of the sides of ABC are 5, 8, and 11. What is the length of the shortest side of DEF if its perimeter is 60?

1) 10
2) 12.5
3) 20
4) 27.5

Answer: 2) 12.5

The sides of similar triangles are proportional and so are the perimeters. If one triangle has a perimter of 5 + 8 + 11 = 24 and the other triangle has a perimeter of 60, then the scale factor between the two is 60/24 or 5/2.

The shortest side of the smaller triangle is 5 so the shortest side of the bigger triangle must be 5 * 5/2 = 12.5

3. In the diagram below of right triangle ABC, altitude CD is drawn to hypotenuse AB.

If AD = 3 and DB = 12, what is the length of altitude CD?

1) 6
2) 6 SQRT(5)
3) 3
4) 3 SQRT(5)

Answer: 1) 6

The Right Triangle Altitude Theorem says that (AD)(DB) = (CD)².

(3)(12) = (AD)²
36 = (AD>²
AD = 6

9. The diagram below shows the construction of an equilateral triangle.

Which statement justifies this construction?

1) ∠A + ∠B + ∠C = 180
2) m∠A = m∠B = m∠C
3) AB = AC = BC
4) AB + BC > AC

Answer: 3) AB = AC = BC

The construction is completed using the length of AB to create AC and BC.

The other choices are all true statements but have nothing to do with the construction of the equilateral triangle.

10. What is the slope of the line perpendicular to the line represented by the equation 2x + 4y = 12?

1) -2
2) 2
3) -1/2
4) 1/2

Answer: 2) 2

To find the slope of the line perpendicular to the given line, you first need the slope of the given line.

The given line is in Standard Form, Ax + By = C, and the slope is -A/B, or -2/4 = -1/2.

If you didn't realize this, you can rewrite it in slope-intercept form:
2x + 4y = 12
4y = -2x + 12
y = -1/2 x + 3

A line perpendicular to a line with a slope of -1/2 would have a slope of 2 because that is the inverse reciprocal and because (-1/2)(2) = -1.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2013

Part I: Each correct answer will receive 2 credits.

6. Which expression is equivalent to (9x²y⁶)^(-1/2)

1) 1/(3xy³)
2) 3xy³
3) 3/(xy³)
4) xy³/3

Answer: 1) 1/(3xy³)

The exponent (-1/2) is the product of (-1)(1/2). The (-1) tells you to take the reciprocal, and the (1/2) tells you to take the square root. When taking the square root, apply it to the coefficient the normal way and halve all the exponents on the variables.

(9x²y⁶)^(-1/2)
= (9x²y⁶)^((-1)(1/2)
= (3xy³)^(-1)
= 1/(3xy³)

7. In a certain high school, a survey revealed the mean amount of bottled water consumed by students each day was 153 bottles with a standard deviation of 22 bottles. Assuming the survey represented a normal distribution, what is the range of the number of bottled waters that approximately 68.2% of the students drink?

1) 131-164
2) 131-175
3) 142-164
4) 142-175

Answer: 2) 131-175

The 68.2% of the population refers to one standard deviation below and above the mean.

153 - 22 = 131 and 153 + 22 = 175.

8. What is the fourth term in the binomial expansion (x - 2)⁸?

1) 448x⁵
2) 448x⁴
3) -448x⁵
4) -448x⁴

Answer: 3) -448x⁵

Refer to the Binomial Theorem in the back of the booklet.

The first term will have x⁸ and each term after will have the exponent reduced by 1. So the 2nd will be x⁷, the 3rd will be x⁶ and the 4th will be x⁵. Eliminate Choices (2) and (4).

The first term will have (-2)⁰ -- that is, it's not there because it only has factors of x. The second term with have (-2)¹, then (-2)², then (-2)³. This is (-8), which means that the coefficient will be negative. The answer must be Choice (3).

The last part of the term is _nC₃. If you use your calculator, or if you can doodle that far along Pascal's triangle, you will see that

₈C₃ = 8 * 7 * 6 / (3 * 2 * 1) = 56

Multiply (-8)(56) = -448

9. Which value of k satisfies the equation 8^3k+4 = 4^2k-1?

1) -1
2) 9/4
3) -2
4) -14/5

Answer: 4) -14/5

8 is 2³ and 4 is 2². You can rewrite each expression as a power of 2 and then compare the exponents.

8^3k+4 = (2³)^{3k + 4} = 2^{9k + 12}

4^2k-1 = (2²)^{2k - 1} = 2^{4k - 2}

9k + 12 = 4k - 2
5k = -14
k = -14/5

10. There are eight people in a tennis club. Which expression can be used to find the number of different ways they can place first, second, and third in a tournament?

1) ₈P₃
2) ₈C₃
3) ₈P₅
4) ₈C₅

Answer: 1) ₈P₃

Since the order matters, you want a Permutation not a Combination. Eliminate Choices (2) and (4). Also, you should have noticed that (2) and (4) have the SAME value.

There are 8 possibilities for first, then 7 for second and then 6 for third. That is 8P3.

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Thursday, September 23, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

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Algebra 2/Trigonometry Regents, January 2013

Part I: Each correct answer will receive 2 credits.

1. What is the equation of the graph shown below?

1) y = 2^x
2) y = 2^-x
3) x = 2^y
4) x = 2^-y

Answer: 2) y = 2^-x

The graph shows exponential decay, which is given by the formula y = a^x, where 0 < a < 1.

The base in this graph cannot be 2, but it could be 1/2, which is 2^-1.

Checking the graph you can see that 2^{-(-1)^{= 2, 2^{-(-2)^{= 4, and 2^{-(-3)^{= 8.}}}}}}

2. Which ordered pair is a solution of the system of equations shown below?

x + y = 5
(x + 3)² + (y - 3)² = 53

1) (2,3)
2) (5,0)
3) (-5,10)
4) (-4,9)

Answer: 3) (-5,10)

A quick check will tell you that all four points are solutions for x + y = 5, so we can ignore that one and focus on the other equation.

(2 + 3)² + (3 - 3)² = 5² + 0² = 25

(5 + 3)² + (0 - 3)² = 8² + (-3)² = 73

(-5 + 3)² + (10 - 3)² = (-2)² + 7² = 53

(-4 + 3)² + (9 - 3)² = (-1)² + 6² = 37

3. The relationship between t, a student’s test scores, and d, the student’s success in college, is modeled by the equation d = 0.48t + 75.2. Based on this linear regression model, the correlation coefficient could be

1) between -1 and 0
2) between 0 and 1
3) equal to -1
4) equal to 0

Answer: 2) between 0 and 1

It is a positive correlation, so the correlation coefficient must be a positive number between 0 and 1.

It wouldn't be negative, so choices (1) and (3) are eliminated. A coefficient of 0 means that there is no correlation at all, so there would be no equation that could model it.

4. What is the common ratio of the geometric sequence shown below?

-2, 4, -8, 16, …

1) -1/2
2) 2
3) -2
4) -6

Answer: 3) -2

To find the common ratio divide any term by the term before it.

4/(-2) = -8/2 = 16/(-8) = -2

S_n = a₁(1 - rⁿ) / (1 - r)
= 3(1 - (-4)⁸) / (1 - (-3))
= -39321

5. Given the relation {(8,2), (3,6), (7,5), (k,4)}, which value of k will result in the relation not being a function?

1) 1
2) 2
3) 3
4) 4

Answer: 3) 3

In a function, each input can have one and only one output. If k = 3, then the relation would contain (3,6) and (3,4), and would fail the vertical line test.

When you enter 3 into a function, it can't sometimes have 6 as its output and sometimes have 4 as its output.

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Geometry Problems of the Day (Geometry Regents, January 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

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Geometry Regents, June 2013

Part I: Each correct answer will receive 2 credits.

1. If triangle MNP ≅ triangle VWX and PM is the shortest side of triangle MNP, what is the shortest side of VWX?

1) XV
2) WX
3) VW
4) NP

Answer: 1) XV

If two triangles are congruent, then the corresponding sides are congruent. When listing two congruent triangles, the order of the letters matters. In this example, angle M corresponds to angle V, etc.

2. In circle O shown in the diagram below, chords AB and CD are parallel.

If m AB = 104 and m CD = 168, what is m BD ?

1) 38
2) 44
3) 88
4) 96

Answer: 2) 44

Arcs AC and BD are equal in size because the chords are parallel. The FOUR chords together add up to 360 degrees. Don't forget the fourth chord.

104 + 168 + x + x = 360
272 + 2x = 360
2x = 88
x = 44

3. As shown in the diagram below, CD is a median of ABC.

Which statement is always true?

1) AD ≅ DB
2) AC ≅ AD
3) ∠ACD ≅ ∠CDB
4) ∠BCD ≅ ∠ACD

Answer: 1) AD ≅ DB

The definition of median is that D will be the midpoint of AB, so AD ≅ DB

Choice (2) could be true but usually will not be.

Choice (3) can never be true because ∠CDB is an exterior angle and is the sum of ∠ACD and ∠DAC.

Choice (4) would be true if CD were also an angle bisector, but it is only a median.

4. In the diagram below, under which transformation is ABC the image of ABC?

1) D₂
2) r_x-axis
3) r_y-axis
4) (x, y) --> (x-2, y)

Answer: 3) r_y-axis

The preimage was flipped over the y-axis to get the image. So it is a reflection.

Choice (1) is a Dilation of scale 2 would have doubled the size. This didn't happen.

Choice (2) is a reflection over the x-axis, which would have flipped the image upside down.

Choice (4) would move the preimage two units to the left.

5. Line segment AB is a diameter of circle O whose center has coordinates (6,8). What are the coordinates of point B if the coordinates of point A are (4,2)?

1) (1,3)
2) (5,5)
3) (8,14)
4) (10,10)

Answer: 3) (8,14)

If the center is point O, (6, 8), then the distance from the point O to B is the same as the distance from point O to A. More to the point, point O is the midpoint of AB.

To get from A to O, you go 6 - 4 = 2 units to the right and 8 - 2 = 6 units up.

Moving (+2, +6) from O means that B is (6+2, 8+6) = (8, 14).

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Factor Song, Part 2

(Click on the comic if you can't see the full image.)

Not all things in math class are equal.

It's been my experience in teaching that some approaches work better for some educators than for others.

This is a follow-up to Monday's comic.

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Come back often for more funny math and geeky comics.

Wednesday, September 22, 2021

Geometry Problems of the Day (Geometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part IV: A correct answer will receive 6 credits. Partial credit is possible.

38. In the diagram of MAH below, MH ≅ AH and medians AB and MT are drawn.

Prove: ∠MBA ≅ ∠ATM

Answer:

To prove that ∠MBA and ∠ATM are congruent, you may first wamt to prove that traingles MBA and ATM are congruent. However, you don't have enough information. I remember this problem first appearing, and I remember several math teachers trying to reason it out. When they got to around 11 steps, which they felt certain couldn't be the best approach, another teacher pointed out a different method.

A better approach is to show that triangle HTM ≅ triangle HBA.

We know that this is an isosceles triangle, which gives us a pair of sides. We have medians which give us a second side. And angle H is congruent to itself with the reflexive property.

Statement	Reason
1. MH ≅ AH, AB and MT are medians	Given
2. B and T are midpoints	Definition of median
3. BH ≅ TH	Division Property of Equality
4. ∠H ≅ ∠H	Reflexive Property
4. Triangle HTM ≅ triangle HBA	SAS
5. ∠HBA ≅ ∠HTM	CPCTC
6. ∠MBA ≅ ∠ATM	Angles supplementary to congruent angles are congruent.

End of Exam.

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Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, June 2013

Part IV: A correct answer will receive 6 credits. Partial credit is available.

39.Solve algebraically for all values of x:

x⁴ + 4x³ + 4x² = -16x

Answer:

Set the equation equal to 0. Factor out the common factors (one x) and then factor by grouping. After that, solve for each x.

x⁴ + 4x³ + 4x² = -16x
x⁴ + 4x³ + 4x² + 16x = 0
(x) (x³ + 4x² + 4x + 16) = 0

Factor by grouping

(x) (x³ + 4x + 4x² + 16) = 0
(x) ( (x)(x² + 4) + 4(x² + 4)) = 0
(x) (x + 4) (x² + 4) = 0

x = 0 or x + 4 = 0 or x² = 4
x = 0 or x = -4 or x² = -4
x = 0 or x = -4 or x = +2i

According to the grading rubric, if you factored correctly and stopped there, you only earned half of the credit. The four solutions are worth the other 3 credits.

As always, finding the solutions in a method other than algebraic will only get half the score, or three credits.

37. A ranch in the Australian Outback is shaped like triangle ACE [No image given], with m∠A = 42, m∠E = 103, and AC = 15 miles. Find the area of the ranch, to the nearest square mile.

Answer:

The area of a triangle can be found using 1/2 a b sin C, but we need a second side.

Use the Law of Sines:

sin 103 / 15 = sin 42 / x
x * sin 103 = 15 * sin 42
x = 15 * sin 42 / sin 103 = 10.3

180 - 42 - 103 = 35

A = 1/2 a b sin C
= 1/2 (10.3) (15) sin 35
= 44.30... = 44

38. Ten teams competed in a cheerleading competition at a local high school. Their scores were 29, 28, 39, 37, 45, 40, 41, 38, 37, and 48.

How many scores are within one population standard deviation from the mean?

For these data, what is the interquartile range?

Answer:

Put the numbers into a List in your calculator and run One Variable Stats.

The Mean is 38.2. The standard deviation is 5.9. This means that the range for scores that are one standard deviation from the mean is 38.2 - 5.9 = 32.3 through 38.2 + 5.9 = 44.1. There are 6 scores within that range.

THe interquartile range is Q3 - Q1. From your calculator, you know that Q3 = 41 and Q1 = 37, so the IQR is 4.

End of Exam.

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Tuesday, September 21, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, June 2013

Part III: Each correct answer will receive 4 credits. Partial credit is available.

36. Solve the equation below algebraically, and express the result in simplest radical form:

13/x = 10 - x

Answer:

Multiply both sides by x and solve the quadratic equation.

13/x = 10 - x
13 = 10x - x²
x² - 10x + 13 = 0

x = -b/(2a) + SQRT(b² - 4ac)/(2a)
= -(-10)/(2) + SQRT((-10)² - 4(13))/(2)
= 5 + SQRT(100 - 52)/2
= 5 + SQRT(48)/2
= 5 + SQRT((16)(3))/2
= 5 + 4 * SQRT(3)/2
= 5 + 2 * SQRT(3)

37. A ranch in the Australian Outback is shaped like triangle ACE [No image given], with m∠A = 42, m∠E = 103, and AC = 15 miles. Find the area of the ranch, to the nearest square mile.

Answer:

The area of a triangle can be found using 1/2 a b sin C, but we need a second side.

Use the Law of Sines:

sin 103 / 15 = sin 42 / x
x * sin 103 = 15 * sin 42
x = 15 * sin 42 / sin 103 = 10.3

180 - 42 - 103 = 35

A = 1/2 a b sin C
= 1/2 (10.3) (15) sin 35
= 44.30... = 44

38. Ten teams competed in a cheerleading competition at a local high school. Their scores were 29, 28, 39, 37, 45, 40, 41, 38, 37, and 48.

How many scores are within one population standard deviation from the mean?

For these data, what is the interquartile range?

Answer:

Put the numbers into a List in your calculator and run One Variable Stats.

THe interquartile range is Q3 - Q1. From your calculator, you know that Q3 = 41 and Q1 = 37, so the IQR is 4.

End of Part III.

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Geometry Problems of the Day (Geometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part III: Each correct answer will receive 4 credits. Partial credit is possible.

35. The coordinates of the vertices of parallelogram SWAN are S(2,-2), W(-2,-4), A(-4,6), and N(0,8). State and label the coordinates of parallelogram S"W"A"N", the image of SWAN after the transformation T_4,–2 ° D _1/2.
[The use of the set of axes below is optional.]

Answer:

When you have a Composition of Transformations, read the dot as "of the". You want to find the Translation OF THE Dilation. The order matters. If you do the transformations in the incorrect order, you will lose Half of the credit immediately.

You can sketch the points on a coordinate plane as a visual aid, but it isn't necessary. It can be done algebraically.

A dilation of 1/2, centered on the origin, means that all of the coordinates will be cut in half.

S(2,-2) --> S'(1,-1)
W(-2,-4) --> W'(-1,-2)
A(-4,6) --> A'(-2,3)
N(0,8) --> N'(0,4)

Next, add 4 to every x-coordinate and subtract 2 from every y-coordinate.

S(2,-2) --> S'(1,-1) --> S"(5, -3)
W(-2,-4) --> W'(-1,-2) --> W"(3, -4)
A(-4,6) --> A'(-2,3) --> A"(2, 1)
N(0,8) --> N'(0,4) --> N"(4, 2)

These all must be labeled.

36. In circle O shown below, chords AB and CD and radius OA are drawn, such that AB ≅ CD, OE ⊥ AB, OF ⊥ CD, OF = 16, CF = y + 10, and CD = 4y - 20.

Determine the length of DF.

Determine the length of OA.

Answer:

There is a lot going on in this image. Write down the things you know.

You need to find the length of DF, which you can do because DF ≅ CD, which is half the length of CD. You know this is true because OF is part of a radius which intersects CD at a right angle, which means that it bisects the chord.

4y - 20 = 2(y + 10)
4y - 20 = 2y + 20
2y = 40
y = 20

DF = CF = y + 10 = 20 + 10 = 30

OA is a radius. It is also the hypotenuse of a right triangle. Since chord AB ≅ to CD, then the two chords must be the same distance from the center of the circle. That means that OE ≅ OF, which is 16, so OE = 16. We know that AE ≅ DF, so AF = 30.

Use Pythagorean Therorem:

16² + 30² = (OA)²
256 + 900 = (OA)²
1156 = (OA)²
OA = 34

Not surprising since 16-30-34 is double 8-15-17, if you know your Triples.

37. If triangle RST ∼ triangle ABC, m∠A = x² - 8x, m∠C = 4x - 5, and m∠R = 5x + 30, find m∠C.

[Only an algebraic solution can receive full credit.]

Answer:

If the two triangles are similar than the corresponding angles are the congruent. That means that m∠A = m∠R.

Write a quadratic equation and solve it.

x² - 8x = 5x + 30
x² - 13x - 10 = 0
(x - 15)(x + 2) = 0
x - 15 = 0 or x + 2 = 0
x = 15 or x = -2

m∠C = 4x - 5
m∠C = 4(15) - 5 = 60 - 5 = 55, or
m∠C = 4x - 5 = 4(-2) - 5 = -13, discard this answer.

End of Part III.

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