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Question Explanations For

Math questions (Answer Explanations, Online Math 2 Test)

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1.

The Correct Answer is (C) — Subtract 10 from both sides to get 3^x–3 = 9. From here you can see that since 3² = 9, x – 3 = 2, so x = 5.

2.

The Correct Answer is (C) — Of the possible factors given, only 10 is a product of the prime factors listed in the question. The number 28,210 is an example of a number that has the listed prime factors but is not divisible by 4, 6, 15, or 25.

3.

The Correct Answer is (A) — Slope is calculated as “rise” (change in y-value) divided by “run” (change in x-value). In this case, the rise is 4 – 3 = 1 and the run is 5 – (–2) = 7, so the slope is ¹⁄₇. As a decimal, this fraction is approximately 0.14.

If you chose (A) you may have subtracted the y-values from the x-values.
If you chose (E) you may have divided run by rise.
If you chose (D) you may have done both these things.

4.

The Correct Answer is (D) — To find the magnitude of a vector, square each component, add them together, and then take the square root. Here is how we find the magnitude in this case: $$\sqrt{5^2+0^2+7^2}=\sqrt{74}=8.6$$.

5.

The Correct Answer is (A) — This question hinges on your understanding of exponent rules. A positive exponent in a denominator is equivalent to a negative exponent in a numerator, and when multiplying two terms with the same base, simply add the exponents. With these rules in mind, first simplify the fraction: $$\frac{a^{-b}}{a^{a}} = a^{-b} a^{-a}$$ or $$a^{-a-b}$$ Next, we multiply by $a^2$ (that is, add the exponents) to get: $$a^{(-a-b+2)}$$

If you chose (B), you may have forgotten the rules for exponents in denominators.

6.

The Correct Answer is (A) — A linear function with nonzero slope will be either continuously increasing or continuously decreasing, meaning that no two $x$-values share a $y$-value (if they did, the function would either be nonlinear or its slope would be zero).

(B) is incorrect because two different $x$-values must correspond to two different $y$-values.

(C) and (D) are incorrect because even though one of them must be false, it is impossible to know which one without more information (answer (C) would be true if the function’s slope was negative; answer (D) if the slope was positive).

(E) is incorrect because we know nothing about the function’s $x$-intercepts.

7.

The Correct Answer is (E) — You can draw a right-angled triangle and label one of the acute angles θ. You can label the side opposite θ as having length 4, and the hypotenuse as having length 5, since you know that sin⁡ θ = ⁴⁄₅. You can use the Pythagorean Theorem to find that the remaining side of the triangle has length 3, and since cosine is equal to adjacent divided by hypotenuse, you know that cos⁡ θ = ³⁄₅.

8.

The Correct Answer is (D) — The function is undefined at any $x$-value that makes its denominator zero, since dividing by zero is undefined. While you can plug in the given answer choices, it is much faster to factor the denominator, which leaves you with the fraction $\frac{7}{x(x-2)(x+3)(x-6)}$ . The values of $x$ that make this denominator equal to 0 are 0, 2, –3, and 6. The only answer choice that is not one of these numbers is 3.

9.

The Correct Answer is (B) — We are given a statement of the form “If P then Q”. Any statement of this form is equivalent to the statement “If not-Q then not-P”, since if Q is false, then P cannot be true because P being true would make Q true as well—so P must be false. Statements such as “If Q then P” and “It not-P then not-Q” are not equivalent to this statement.

Instead, you can use your knowledge of numbers to find which statement is true. (A) is false if $a=b=1$. (C) and (D) are also false when $a=b=1$. (E) is false when $a=1$ and $b=2$, for example.

10.

The Correct Answer is (B) — If the mean of 7 numbers is 15, their sum must be 7 × 15 = 105. Similarly, if the mean of 8 numbers is 12, their sum must be 8 × 12 = 96. The eighth number must account for the difference between these two sums: 96 – 105 = –9.

11.

The Correct Answer is (C) — When evaluating the composition of various functions, you should work from the inside out. Start by plugging h (x) into g (x) to get g (h (x)) = 5(1 – x²) + 3 = –5x² + 8. Now, plug your result into f (x) to find the answer: 3(–5x² + 8) = –15x² + 24. If you got (D) or (E), you may have tried to multiply the functions instead of substituting them into one another.

12.

The Correct Answer is (D) — Remember that a valid function of x should pass a “vertical line test,” meaning it assigns only one output to each input, or only one possible y-value for every x-value. Answer choice (D) fails the vertical line test: for example, at x = 0, there are two possible values for y.

13.

The Correct Answer is (E) — A line is perpendicular to another line if its slope is the negative reciprocal of the first line, and vice versa. The line y = ½ x – 3 has a slope of ½, so a line with a slope of –2 would be perpendicular to it. (E) is the only answer option with this slope.

If you got (A), you found the negative slope but forgot to take the reciprocal.
If you got (C), you took the reciprocal of the entire line equation.
If you got (D), you took the reciprocal of the slope but forgot to change the sign.

14.

The Correct Answer is (E) — A reflection across the line y = x is the same thing as inverting the y and x values (for example, the point $(3,4)$ gets reflected to the point $(4,3)$). If pentagon P has vertices at $(–2, –4)$, $(–4, 1)$, $(–1, 4)$, $(2, 4)$, and $(3, 0)$, the new pentagon would have vertices at $(–4, –2)$, $(1, –4)$, $(4, –1)$, $(4, 2)$, and $(0, 3)$. Only one of these vertices is an answer choice: $(–4, –2)$.

15.

The Correct Answer is (A) — Seeing that the answers are all inverse trigonometric functions should alert you to the fact that you can create a right-angled triangle from the information in the question, with the hypotenuse going between the tops of the two poles. The poles are 12 meters apart, so the bottom side of the triangle has length 12. One is 5 meters taller, so the leg opposite to the angle of elevation has length 5. The tangent of the angle of elevation is equal to $\frac{\text{opposite}}{\text{adjacent}}$ , so the angle of elevation itself is $\tan^{-1}$ $\frac{5}{12}$ .

16.

The Correct Answer is (A) — You can use the sine law to isolate and solve for x:

$\begin{align*} \frac{\sin x}{a} &= \frac{\sin 60}{3a} \\ \sin x &= \frac{\left ( \frac{\sqrt{3}}{2} \right )}{3}\\ \sin x &= \frac1{}{2\sqrt{3}} \doteq 0.289\\ \end{align*}$

If you chose (A), you found x, not sin (x).
If you chose (E), you may have thought that the a’s wouldn’t cancel out.

17.

The Correct Answer is (A) — The easiest way to solve this question is to graph both inequalities to get a sense of the solution space. You are looking for a point that is either lower than the line $y = 9x – 8$ or higher than the line $y = –x + 8$.

The only point not strictly in the shaded area is $(–2, 15)$, which is above the line $y= –x + 8$. Therefore, it is not a solution to the inequalities and the correct answer is (A).

18.

The Correct Answer is (B) — Simply read the graph: when x is between 2 and 3 hours, y = 30. Therefore, the company would charge $30.00.

If you got (A) or (C), you forgot that the floor function will round any value to the nearest integer smaller than the value itself: 2.28 will become 2.
If you got (E), you may have taken the floor of $x$ instead of the floor of $x+1$.

19.

The Correct Answer is (D) — The important numbers for determining a trigonometric function's range are the amplitude and the vertical shift. In this case, the amplitude is 3 (the coefficient in front of the sine) and the vertical shift is 5 (the constant being added at the beginning). The range of the basic sine function is $-1 \leq x \leq 1$. Shifting this function up by 5 makes the range $4 \leq x \leq 6$, and making the amplitude 3 instead of 1 makes the range $2 \leq x \leq 8$.

If you chose (A), you forgot to shift the graph upwards and selected the range of the function y = 3 sin (2x – π).
If you chose (C), you may have confused the vertical shift and the amplitude.
If you chose (E), you picked the domain of the function.

20.

The Correct Answer is (C) — Since (3, 2) is a point on g, you know that g (3) = 2. You also know that g (x) = f (–x), so g (3) = f (–3). Therefore, f (–3) = 2, and the point (–3, 2) must lie on the graph of f.

21.

The Correct Answer is (A) — Using the CAST rule, you know that the sine of an angle in Quadrant II is positive. Since $\cos(90° - a°) = \sin (a°)$, you can see that statement I is true. Statement II is false because the tangent of any angle in Quadrant II is negative. Finally, you know that Statement III is false because there are many possible ways to arrive at an angle that is in the second quadrant; a possible value for a could be $-250$, for example. Therefore, only I is true and (A) is the correct answer.

22.

The Correct Answer is (A) — If you graph the function, you will see that the graph f (x) intersects the x-axis three times, meaning that it has three real solutions and that (A) is true.

(B) is incorrect because $f$ dips below -18 between $x=0$ and $x=2$.
(C) is incorrect because $f$ rises above -18 between $x=-18$ and $x=0$.
(D) is incorrect because $f$ is increasing over this entire range, meaning $f(x)$ gets larger as $x$ gets larger.
(E) is incorrect because $f$ is decreasing between $x=-2$ and $x=1$.

23.

The Correct Answer is (C) — By looking at the trendline on the graph, you can see that for every 10 units in the positive x-direction, the graph increases by 20 units in the positive y-direction: it has a slope of about 2. Additionally, the trendline intersects the y-axis at a value of approximately –50: it has a y-intercept of –50. Plug the given x-value of 197 into your approximated linear equation to arrive at the y-value estimate:

y = 2(197) – 50
y ≐ 344

The closest answer choice to this is (C), 352.

24.

The Correct Answer is (E) — You can write out the linear transformation as a series of equations, where y’ and x’ are the transformed values of y and x, respectively:

x' = x + y
y' = 2y – x

Now, you want to find out when x = x’ and when y = y’:

$\begin{align*} x'&=x & y' &= y \\ x+y&=x & \;\;\;\;2y-x&= y \\ y&=0 & y-x &= 0 \\ \end{align*}$

Since y = 0, you know that x must also equal 0 in order for the equation y – x = 0 to be true. Therefore, the only point that will work is (0, 0).

If you chose (B), you solved for y and stopped there.

25.

The Correct Answer is (B) — Since the retailer purchased the same amount of soybeans each year, the percent change in their expenditures will be equal to the percent change in price of the soybeans. Since the soybeans went from \$0.24 per pound to \$0.16 per pound (which is a negative change), you can set up the following equation:

$\begin{align*} (\frac{0.16-0.24}{0.24})&=\frac{x}{100}\\ x&=-33.3\end{align*}$

Therefore, the percent change was $-33\%$.

26.

The Correct Answer is (C) — The easiest way to evaluate an inverse function is to swap the variable $x$ in the function for $f(x)$, and to swap $f(x)$ for $x$. Then, solve for $f(x)$. In this question, this gives you $x=\sqrt{7(f(x))^3}$. Solving for $f(x)$ gives you $f(x)=\sqrt[3]{\frac{x^2}{7}}$. Next, substitute 6 in for $x$ to get $x=\sqrt[3]{\frac{6^2}{7}}=1.726$

27.

The Correct Answer is (E) — You can ignore the confusing terms “barn” and “cow’s grass,” simplifying them to constants such as b and c. Since b = 10^-28m² and c = 2.48 × 10⁴m², and you’re trying to find how many times b goes into c, you can write the expression bx = c. Solving for x, you get $\frac{2.48 \times 10^4}{10^{-28}}=x=2.48 \times 10^{32}$.

28.

The Correct Answer is (E) — The standard deviation of a set of numbers is equal to the square root of the mean value of the squared difference between each value and the median.

Set A is perfectly symmetrical about its mean of zero, so multiplying all its elements by 2 will not change the mean. Since the mean is zero, and all the other elements are multiplied by a factor of 2, the distance from the mean in B is twice the distance of the elements in set A. You can refer to the equation for standard deviation, where you can see that this multiplier will first be squared and then square rooted. Therefore, the standard deviation of B will be two times the standard deviation of A.

(A) and (B) are incorrect because the mean of A is the same as the mean of B, because they are both 0.

(C) is incorrect because the range of A is half the range of B (20 vs. 40).

(D) is incorrect because the standard deviation of B is exactly two times that of A.

29.

The Correct Answer is (C) — First, convert the first function from $f(2x) = x+5$ to $f(x)=\frac{1}{2}x+5$. Next, set the expression equal to 13 by writing $13=\frac{1}{2}x+5$, and solve to find that $x=16$, and so $g(6)=16$. Finally, $2g(6)=2\times 16=32$.

If you got (B), you probably forgot to multiply $g(6)$ by 2.

If you got (E), you multiplied the $g(6)$ by 4 instead of by 2.

30.

The Correct Answer is (D) — The solutions to $f(x)=0$ are found at the $x$-intercepts of the graph. Since the graph touches the $x$-axis at -2, 1, and 4, you know that the function must have $(x+2)$, $(x-1)$, and $(x-4)$ as factors, and so it must be either (A), (D), or (E). You also know that the function must have an even degree, since the graph opens upwards to both the left and right. You can find the degree of the functions in the answer choices by adding the exponents of the terms. The function in (A) has degree 3 (1+1+1), the function in (D) has degree 6 (3+2+1), and the function in (E) has degree 7 (2+3+2), so the answer must be (D). You can confirm this answer by graphing (D) into your graphing calculator.

31.

The Correct Answer is (B) — You can plug the given values into Cosine Law to find the distance between Grace and Ian:

$\begin{align*}c^2&=a^2+b^2-2ab \text{cos}(C\degree) \\ \text{distance}&=\sqrt{113-112\text{cos}38\degree}\\ \text{distance}&=4.97\sim 5\text{ feet} \end{align}$

32.

The Correct Answer is (D) — If (x - 1) is a factor of f(x), then that means that f(1) = 0. Plugging in 0 for f(x) and 1 for x, you get 0 = 2 + k – 2 – 3, or 0 = k - 3. Solving for k gives you 3.

33.

The Correct Answer is (D) — Since the order of the members of the committee doesn’t matter, you can use the formula for combinations to find out how many ways there are to form a 5-person committee out of 15 people:

$\binom{15}{5}=\frac{15!}{5!(15-5)!}=3,003$

If you got (E), you probably calculated the number of permutations instead of combinations.

34.

The Correct Answer is (B) — In general, if a given matrix $A$ has dimensions $n \times m$ and matrix $B$ has dimensions $x \times y$, the product matrix $AB$ will have dimensions $n \times y$ unless $m \neq x$, in which case the product does not exist. In this case, $n=2$, $m=7$, $x=7$ and $y=5$. Since $7 = 7$, the resultant matrix will be $2\times 5$.

If you got (C), you probably reversed the matrix’s columns and rows.

35.

The Correct Answer is (D) — A polynomial function's range is all real numbers if its degree (the highest power of $x$ that appears in it) is odd. You can find the degree of each of the answer choices by figuring out what the highest power of $x$ would be after expanding the parentheses. The polynomial in (D) has degree 3+3+3=9, which is odd, so (D) is correct.

(A) is incorrect because this polynomial has degree 1+3=4.

(B) is incorrect because this polynomial has degree 1+5=6.

(C) is incorrect because this polynomial has degree 14.

(E) is incorrect because this polynomial has degree 2+2+4.

36.

The Correct Answer is (E) — You can use log rules to rewrite the expression in the numerator in terms of log₃ 1,000, and then simplify to get the answer:

$\begin{align*} \frac{\log_{3}{1,000,000}}{\log_3 1,000}&=\frac{\log_{3}{(1,000^2)}}{\log_3 1,000}\\ \\ &=\frac{2\log_{3}{1,000}}{\log_3 1,000}\\\\ &=2 \end{align*}$

37.

The Correct Answer is (E) — You can use prime factorization to find that 392 = 2³ × 7², meaning that its prime factors are 2 and 7. Since 2² = 4 and 7² = 49 are also factors, 392 is powerful.

(A) is incorrect because 3 is a factor of 240 but 3² = 9 is not.
(B) is incorrect because 11 is a factor but 121 is not.
(C) is incorrect because 3 is a factor but 9 is not.
(D) is incorrect because 3 is a factor but 9 is not.

38.

The Correct Answer is (D) — Since the sequence is geometric, you know that every term is a certain multiple of the previous term. To get from 1 to –3, you multiply by –3, so you know that –3 is the common ratio. This allows you to eliminate (A), (B), and (E).

To decide whether the exponent should be n or n – 1, you can plug in a value for n. For example, when n = 2, a₂ = –3, so the exponent should be 1, meaning that n – 1 (and not n) is correct.

39.

The Correct Answer is (A) — You can start by multiplying the second and third binomials together, remembering that i² = –1. Since they are a difference of squares, you get (i + 1)(25 – 25i²) = (i + 1)(25 + 25) = (i + 1)(50) = 50 + 50i.

If you got (E), you forgot that –25i² = –25(–1) = +25, so you thought the last two binomials multiplied to zero.

40.

The Correct Answer is (D) — Calculate the distance between points R and J using the distance formula, and solve for a:

$\begin{align*} 10 &= \sqrt{(a-6)^2\;+\;(a-(-2))^2\;+\;(a-0)^2}\\ 10&=\sqrt{a^2-12a+36 \;\;+\;\;a^2+4a+4\;\;+\;\;a^2}\\ 100&=3a^2 -8a+40\\ 0&=3a^2-8a-60\\ 0&=(3a+10)(a-6)\\ a = 6 \;\;&\text{OR}\;\;a=-\frac{10}{3} \end{align*}$

Since only a = 6 is an answer choice, the answer is (D).

41.

The Correct Answer is (E) — If |a| < |b|, then a + b will always have the same sign as b and a – b will always have the opposite sign as b. This means that $\frac{a+b}{a-b}$ will be negative, since the numerator and denominator will have opposite signs. Note that option (E) rules out the case b = 0 because |0| = 0 and so there is no way |a| could be less than this.

(A) is incorrect because this will make the expression equal to 0.
You can see that (B) is incorrect because if b = –1 and a = 2 then the expression is equal to ⅓ > 0.
(C) is incorrect because if a = –2 and b = 1 then the expression is again equal to ⅓ > 0.
(D) is incorrect because of either of the examples above.

42.

The Correct Answer is (E) — Given any three points, there exists a circle that passes through all of them, unless the points are collinear (meaning that they lie on the same line). The points (–1, –2), (3, 4) and (5, 7) all lie on the line $y=\frac32x-\frac12$ , so there is no circle that passes through all of them.

43.

The Correct Answer is (D) — The easiest way to solve this question is to plug all the answer options into your calculator. If x = –150° then 2x = –300°, and cos⁡(–300°) = ½.

The other answer options are incorrect because the cosine of two times them is not equal to ½.

44.

The Correct Answer is (D) — Remember that the $x^{th}$ root of an expression is equal to the expression raised to the power of $\frac{1}{x}$. You can rewrite the equation using exponent rules to arrive at the value of a:

$\begin{align*} 64^{\frac{1}{x}}\;\times\; 64^{\frac{1}{y}}&= 64^{\frac{1}{a}}\\ 64^{\frac{1}{x}+\frac{1}{y}}&= 64^{\frac{1}{a}}\\ \frac{1}{x}+\frac{1}{y} &= \frac{1}{a}\\ \frac{1}{\frac{1}{x}+\frac{1}{y}} &=a \end{align*}$

45.

The Correct Answer is (C) — To find the point or points where a graph crosses its oblique asymptote, set the function equal to the asymptote and solve for x (the asymptote is y = x + 5, the polynomial part of the function):

$\begin{align} x+5-\frac{x-3}{x^2-1} &= x+5 \\ \frac{x-3}{x^2-1} &= 0 \\ x-3 &= 0 \\ x = 3 \end{align}$

If you got (D) or (E), you were trying to set the entire function equal to zero.

46.

The Correct Answer is (B) — The easiest way to answer this question is to graph the given equations on your graphing calculator, remembering that 0 ≤ t ≤ π.

If you mistakenly include the values π ≤ t ≤ 2π, you will get a complete circle, or answer choice (A).

47.

The Correct Answer is (D) — The easiest way to solve this problem is to move the lines off the coordinate plane and treat it as a geometry problem. Because you know the slopes of the lines are 2 and –2, you can add in a horizontal and vertical line and label your diagram with these measurements:

Each of the smaller angles at the bottom has measure $\frac{\theta}{2}$. From the diagram, you know that $\frac{\theta}{2} = \tan ^{-1} \left(\frac12 \right ) = 26.565°$, so $\theta = 53.13°$.

48.

The Correct Answer is (C) — The four possible outcomes are rain and sun, rain and no sun, no rain and sun, and no rain and no sun. The probabilities for these four options must add up to 1, since there are no other options. The question tells you that the probability of rain and no sun is 0.3. It also tells you that the probability of no rain, which is equal to the probability of no rain and sun plus the probability of no rain and no sun, is 0.4. This means that three of the four options combined have a 0.7 probability of happening, meaning that the fourth option, rain and sun, has a probability of 0.3 of occurring.

49.

The Correct Answer is (A) — You can plug the given numbers into the equation to get 750 = 100e^r5. Divide both sides by 100 to get 7.5 = e^r5, and then take the natural logarithm of both sides: 5r = ln(7.5) ≈ 2.015. Divide by 5 to get r ≈ 0.403.

50.

The Correct Answer is (B) — If the radius of the sphere is 8, its diameter is 16. Since opposite vertices of the cube lie on the diameter of the sphere, the distance between opposite vertices is also 16. You can use the distance formula to find the side length of the cube; if a is the side length, then 16² = a² + a² + a² = 3a², so a = $\sqrt{\frac{256}{3}}$ = 9.2376. Since the area of any face of the cube is equal to a², the surface area of the entire cube is equal to 6a² = 6 × 9.2376² = 512.00.