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QUESTION EXPLANATIONS
For Test (Math Test - No Calculator)1 2 3 4 5 6 7 8 9 10
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1. (D) —
You can add 4 to both sides to get 12x = 7 and then divide both sides by 12 to get x = 7⁄12.
2. (D) —
The equation for line B is given in the form y = mx + b, where m = 3 is the slope. If the slope of line B is one-third the slope of line A, the slope of line A must be 9. Only answer (D) has a slope of 9.
3. (C) —
You can divide both sides of the given equation by 4 to get p⁄q = 5⁄4.
4. (A) —
To sum f (x) and g (x), simply collect like-terms and simplify:
5. (B) —
To find the solutions for x, you can set each factor to 0 and solve that factor for x. The solutions are 0, 3⁄2, 4, and 5⁄6. Multiplying these together gives you 0. If you realize that one of the solutions is 0, you dont need to find the other ones, since you know that multiplying anything by 0 will be 0.
If you picked (A), you may have forgotten 0 was a solution.
6. (C) —
Since f (x) = x + 4, you know that f (2x) = 2x + 4. Both of these functions have the same y-intercept (when x = 0, y = 4), so (A) and (B) are incorrect. Since f (2x) has a slope of 2 and f (x) has a slope of 1, (C) is correct.
7. (B) —
You can subtract the second equation from the first equation to get 0 = 2x + 6, which you can solve to get x = 3. You can then plug this value into either of the original equations and solve for y to get y = 3, meaning that the solution is (3, 3). Alternatively, you could try each answer individually to find which one satisfies both equations.
8. (D) —
To add two rational expressions, they must have a common denominator. Multiply the first term by 
to get }{(x+1)(x-2)}+\frac{3(x+1)}{(x+1)(x-2)} "\frac{2(x-2)}{(x+1)(x-2)}+\frac{3(x+1)}{(x+1)(x-2)}")
. You can now add these terms and simplify the numerator to get 
.
9. (D) —
Just by looking at the answer choices, you know that the graph above is of an absolute value function. In this case, the distance along wall B is acting as the x-axis, so you can see that the function is shifted to the right. This means that the constant in the absolute value signs must be subtracted from B, eliminating answer choices (A) and (C). Additionally, you can see that for every meter the ball travels to the right, it also goes 2 meters up along wall A, meaning the function is horizontally compressed by a factor of 2. This leaves you with the correct answer, A = |2(B 3)| = |2B 6|, or (D).
10. (A) —
Since there were at least 1,000 men, m ≥ 1,000. Since there were at least 3,000 women, w ≥ 3,000. Since there were twice as many women as men, w = 2m. Since the city had fewer than 15,000 inhabitants, w + m < 15,000. The only answer that combines all these statements is (A).
11. (A) —
You can multiply both the numerator and denominator by 4 + 2i to get }{(4-2i)(4+2i)}=\frac{40+20i}{16+8i-8i-4i^2}=\frac{40+20i}{16+4}=2+i "\frac{10(4+2i)}{(4-2i)(4+2i)}=\frac{40+20i}{16+8i-8i-4i^2}=\frac{40+20i}{16+4}=2+i")
.
12. (C) —
The parabola is opening downwards so the coefficient in front of the x2 will be negative, eliminating choices (A) and (B). You can find the vertex of the parabola when given the equation in standard form, y = ax2 + bx + c, by using the formula for the x-coordinate, where x = b⁄2a. It follows that the x-coordinate of the vertex for (C) is 3 and for (D) is 1⁄4. You can see on the graph that the vertex is located at x = 3, so (C) is correct.
13. (B) —
You can manipulate the equation as follows:
If you picked (A), you forgot to also take the square root of c.
14. (C) —
The correct answer is (C). Since the sine of (k a) is equal to the cosine of (k b), you know that the two angles must add to 90 degrees: k a + k b = 90. You also know that a + b = 90, since a and b are the two acute angles of a right-angled triangle, so you can solve for k:
Finally, k a b = 90 (a + b) = 90 90 = 0.
15. (C) —
To find the height difference between the baseball and the football, simply subtract f (t) from b
(t) to get:
16. (900) —
You can simply plug in n = 7 and evaluate to get s = 180(7 2) = 180 Χ 5 = 900.
17. (2) —
You know that 3a = 24, so you can divide both sides by 3 to get a = 8. Plug this into the first equation for x and solve:
18. (5) —
The given equation can be rearranged to get 
, which has a slope of 
, meaning that the laser beam will have a slope of 
, since it is perpendicular to this line. You can use the two given points to find the rise and the run of the laser beam and set up the equation 
. Cross-multiply to get 2a 4 = 3a 9, which you can solve to get a = 5.
19. (60) —
Since the radius of the circle is 1, as indicated on the diagram, the circles circumference is 2πr = 2π. Since arc AB has length π⁄3, you can set up the equation }{2\pi}=\frac{x}{360} "\frac{(\frac{\pi}{3})}{2\pi}=\frac{x}{360}")
, which you can solve to get x = 60.
20. (9) —
Since a + b = 7, you can square both sides to get a2 + 2ab + b2 = 49, or a2 + b2 = 49 2ab. You know that a2 + b2 = 31 also, so you can set the two right hand sides of these equations equal and solve for ab:
You can add 4 to both sides to get 12x = 7 and then divide both sides by 12 to get x = 7⁄12.
The equation for line B is given in the form y = mx + b, where m = 3 is the slope. If the slope of line B is one-third the slope of line A, the slope of line A must be 9. Only answer (D) has a slope of 9.
You can divide both sides of the given equation by 4 to get p⁄q = 5⁄4.
To sum f (x) and g (x), simply collect like-terms and simplify:
To find the solutions for x, you can set each factor to 0 and solve that factor for x. The solutions are 0, 3⁄2, 4, and 5⁄6. Multiplying these together gives you 0. If you realize that one of the solutions is 0, you dont need to find the other ones, since you know that multiplying anything by 0 will be 0.
If you picked (A), you may have forgotten 0 was a solution.
Since f (x) = x + 4, you know that f (2x) = 2x + 4. Both of these functions have the same y-intercept (when x = 0, y = 4), so (A) and (B) are incorrect. Since f (2x) has a slope of 2 and f (x) has a slope of 1, (C) is correct.
You can subtract the second equation from the first equation to get 0 = 2x + 6, which you can solve to get x = 3. You can then plug this value into either of the original equations and solve for y to get y = 3, meaning that the solution is (3, 3). Alternatively, you could try each answer individually to find which one satisfies both equations.
To add two rational expressions, they must have a common denominator. Multiply the first term by to get . You can now add these terms and simplify the numerator to get .
Just by looking at the answer choices, you know that the graph above is of an absolute value function. In this case, the distance along wall B is acting as the x-axis, so you can see that the function is shifted to the right. This means that the constant in the absolute value signs must be subtracted from B, eliminating answer choices (A) and (C). Additionally, you can see that for every meter the ball travels to the right, it also goes 2 meters up along wall A, meaning the function is horizontally compressed by a factor of 2. This leaves you with the correct answer, A = |2(B 3)| = |2B 6|, or (D).
Since there were at least 1,000 men, m ≥ 1,000. Since there were at least 3,000 women, w ≥ 3,000. Since there were twice as many women as men, w = 2m. Since the city had fewer than 15,000 inhabitants, w + m < 15,000. The only answer that combines all these statements is (A).
The parabola is opening downwards so the coefficient in front of the x2 will be negative, eliminating choices (A) and (B). You can find the vertex of the parabola when given the equation in standard form, y = ax2 + bx + c, by using the formula for the x-coordinate, where x = b⁄2a. It follows that the x-coordinate of the vertex for (C) is 3 and for (D) is 1⁄4. You can see on the graph that the vertex is located at x = 3, so (C) is correct.
You can manipulate the equation as follows:
If you picked (A), you forgot to also take the square root of c.
The correct answer is (C). Since the sine of (k a) is equal to the cosine of (k b), you know that the two angles must add to 90 degrees: k a + k b = 90. You also know that a + b = 90, since a and b are the two acute angles of a right-angled triangle, so you can solve for k:
Finally, k a b = 90 (a + b) = 90 90 = 0.
To find the height difference between the baseball and the football, simply subtract f (t) from b
(t) to get:
You can simply plug in n = 7 and evaluate to get s = 180(7 2) = 180 Χ 5 = 900.
You know that 3a = 24, so you can divide both sides by 3 to get a = 8. Plug this into the first equation for x and solve:
The given equation can be rearranged to get , which has a slope of , meaning that the laser beam will have a slope of , since it is perpendicular to this line. You can use the two given points to find the rise and the run of the laser beam and set up the equation . Cross-multiply to get 2a 4 = 3a 9, which you can solve to get a = 5.
Since the radius of the circle is 1, as indicated on the diagram, the circles circumference is 2πr = 2π. Since arc AB has length π⁄3, you can set up the equation , which you can solve to get x = 60.
Since a + b = 7, you can square both sides to get a2 + 2ab + b2 = 49, or a2 + b2 = 49 2ab. You know that a2 + b2 = 31 also, so you can set the two right hand sides of these equations equal and solve for ab: