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Grade 11 Trigonometry CCSS: HSG-SRT.D.10

Which of the following is true concerning the law of sines and right triangles?

The law of sines is not valid for right triangles.
The law of sines can only be proved for the acute angles of a right triangle.
The law of sines can be easily proved for a right triangle, using trig ratios and the fact that $sin(90°)=1$ .
The law of sines can proved for a right triangle, and the easiest proof involves the use of the Pythagorean Theorem and the formula for the area of a triangle.

Grade 11 Trigonometry CCSS: HSG-SRT.D.10

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Is this proof valid for all triangles? Explain your answer.

Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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Letting

$b^2 = c^2 - a^2$ , the equation no becomes:

$b^2 x^2 - y^2 = b^2 a^2$

$x^2 / a^2 - y^2 / b^2 = 1$
This is the equation of a hyperbola, centered at the origin with foci

$(-c,0)$ and

$(c,0)$ . Where does the substitution

$b^2 = c^2 - a^2$ come from?

Since $b$ usually appears in the equation for a hyperbola, it must be included. Using the Pythagorean theorem, $a^2 +b^2 = c^2$ , simply rearrange the equation.
It has to be done, to ensure the asymptotes are related to the equation. The equations of the asymptotes are $y= pm b/a$ , and knowing that $|c| > |a|$ , squaring and rearranging results in $b^2 = c^2 - a^2$ .
It's done to simply the equation. $b$ is not defined yet, and since $|c| > |a|$ , $c^2 > a^2$ , and so there must be a positive number, $b^2$ such that $b^2 = c^2 - a^2$ .
Knowing that $b$ is the length of the semi-minor axis, a right triangle can be formed with the center of the hyperbola and either foci, with $b$ as the length of one leg of this triangle. Applying the Pythagorean theorem results in $b^2 + c^2 = a^2$ , and simply rearrange.

Grade 12 Polynomials and Rational Expressions CCSS: HSA-APR.C.5

Euler's number,

$e$ , is an irrational number (like

$pi$ ). You can approximate

$e$ with the formula

$(1+1/n)^n$ , such that the approximation improves as

$n$ approaches infinity. Which of the following is an equivalent formula, using the Binomial Theorem and the fact that

$({:(n),(k):}) (1/n)^k = 1/(k!)$ as

$n$ approaches infinity?

$sum_{k=0}^n 1/(k!)$
$sum_{k=0}^n (k^n)/(k!)$
$sum_{k=0}^n (n!)/(k!)$
$sum_{k=0}^n (n-k)/(k!)$

College Calculus

Suppose

$f(x)=x^2-x-2$ ,

$x in [-1,2]$ . Show there exists a point

$c in (-1,2)$ with a horizontal tangent. Find c.

Grade 11 Three Dimensional Shapes CCSS: HSG-SRT.C.8, HSG-MG.A.3

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The largest cone that will fit in a sphere will be inscribed in the sphere; that is, the cone's apex will touch the sphere, and the circumference of its base will also touch the sphere.

Consider a cross section of the cone inscribed in the sphere, one that intersects the center of the sphere and is perpendicular to the base of the cone. What is the relationship between the radius of the sphere, the radius of the cone, and the height of the cone? (Hint: use the Pythagorean Theorem). Also, state the restrictions on the height and radius of the cone.

Grade 11 Sequences and Series CCSS: HSA-SSE.B.4

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The equation in the third step is found by subtracting the equation from step 2 from the equation in step 1. What happens to all the terms on the right hand side?

They are ignored, because they are all going to be much smaller than either $a$ or $ar^n$ , depending on whether $r$ is greater or less than 1.
All but two of them are eliminated by subtraction. Aside from the first term of the first equation and the last term of the second equation, each term in the first equation has an equal term in the second equation, and thus they become n - 2 zeros.
Using the factor theorem, they all cancel out except for $a$ and $ar^n$ .
Dividing both sides of the equation by $ar, ar^2, ... , ar^(n-1)$ , they all cancel out.

Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

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After some more algebra, and then moving all terms with

$x$ or

$y$ to the left side of the equation (and all others to the right), the resulting equation is:

$a^2 x^2 - c^2 x^2 + a^2 y^2$	$=$	$a^4 -a^2 c^2$
$(a^2 - c^2)x^2 + a^2 y^2$	$=$	$(a^2 - c^2) a^2$
$b^2 x^2 + a^2 y^2$	$=$	$a^2 b^2$
$x^2 / a^2 + y^2 / b^2$	$=$	$1$

This is the formula for the equation of an ellipse centered at

$(0,0)$ . Why can the substitution

$b^2 = a^2-c^2$ be made?

Since $b$ is undefined so far, it can be defined as any value. Then, using the Pythagorean theorem, $a^2 + b^2 = c^2$ , simply rearrange to solve for $b^2$ .
Knowing that the semi-minor axis is $b$ units long, one can substitute the square of this value for $a^2-c^2$ .
Since $b$ is not yet defined, it can be used to simplify the equation by defining $b^2 = a^2 - c^2$ . A positive value for $a^2-c^2$ exists since $|a| > |c|$ .
Projecting $b$ , the length of the semi-minor axis, onto the semi-major axis it is seen that $b=a-c$ . Then, simply square both sides of the equation.

Grade 12 Statistics and Probability Concepts

A standardized test is taken by all American public school 11th graders. 80% of the test-takers passed the exam. Forty-five 11th grade public school students who took the exam were randomly selected, and 67% of them passed the exam.
Which of the following is true about the sampling distribution?

The sampling distribution cannot be assumed to be approximately Normal since 45(1-0.80)=9 <10.
The sampling distribution can be assumed to be approximately Normal since 45(0.80)=36 > 10. The mean is 0.80 and the standard deviation is 0.0596.
The sampling distribution can be assumed to be approximately Normal since 45(1-0.70)=30 > 10. The mean is 0.80 and the standard deviation is 0.0596.
The sampling distribution cannot be assumed to be approximately Normal since the population size is not given.
The sampling distribution can be assumed to be approximately Normal since n>30, according to the Central Limit Theorem. The mean score is 80% and the standard deviation is 0.671%.
The sampling distribution cannot be assumed to be approximately Normal since the scores are not normally distributed.

Grade 11 Trigonometry CCSS: HSF-TF.C.8

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Using the information in the previous questions, write a proof for the Pythagorean Identity,

$sin^2(theta) + cos^2(theta) = 1$ .

Grade 9 Study Skills and Strategies

Answer Explanations:

Question 2:
Pay close attention to the words, "must," "may", and "cannot" in the answer choices. Eliminate the choices you know are incorrect or only partly correct. Using the diagram, angle 6 is a right angle, so eliminate choice "C". Also, eliminate choice "D" because the words, "must," "may", and "cannot" from the first three options cover all conditions. From here, guess between "A" and "B" or apply your knowledge of geometry theorems to select the correct answer "A".

Question 3:
Circle the word "EXCEPT." Three of the answers will be correct, so be careful not to select the first answer that looks right, you are looking for a wrong answer. If you are not sure of the formula or units for density, look for the answer choice that is different from the others. In this case, "mL," "L," and "cc" are all units for volume, but "cm" is a unit for length. Select answer "D" because it is different from the other choices.

Question 4:
Circle the phrase, "most correct," it implies there is more than one correct answer so beware of selecting the first answer that seems correct. Based on the answer choices, there are two decisions you need to make, whether the eclipse would be lunar or solar and whether the eclipse will be seen or may be seen. Even if you can only decide on one of those two factors, you can then eliminate two choices and guess.

Question 5:
Eliminate choice "D" as it is the least like the other three choices since it compares the sample to the original rock and not the other sample. From here you will need to use your understanding of density to select the correct answer.

Question 6:
Be on the look out for common errors. Do a quick logic check or estimation. If you increase the size of something, it gets larger, so as a percentage the answer must be greater than 100%. Cross off choices "A" and "B". A common error would be adding 50 and 25 to get 75, so even if you are not sure how to do the math, cross off choice "C" and select "D."

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