## Full Length ACT Mathematics Practice Test

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ACT Mathematics
Practice Test 4

60 questions

Total time for this section: 60 Minutes

You can use a scientific calculator on this test.

1- If the interior angles of a quadrilateral are in the ratio $$1:2:3:4$$, what is the measure of the smallest angle?
(A) $$36^\circ$$
(B) $$72^\circ$$
(C) $$108^\circ$$
(D) $$144^\circ$$
(E) $$168^\circ$$
2- What is the value of x in the following system of equations?
$$2 \ x \ + \ 3 \ y=12$$
$$4 \ x \ - \ 4 \ y= \ - \ 16$$
(A) $$0$$
(B) $$1$$
(C) $$2$$
(D) $$- \ 2$$
(E) $$- \ 1$$
3- If sin A $$= \frac{1}{5}$$ in a right triangle and the angle A is an acute angle, then what is cos A?
(A) $$\frac{\sqrt{6}}{5}$$
(B) $$\frac{\sqrt{6}}{10}$$
(C) $$\frac{2 \ \sqrt{6}}{5}$$
(D) $$\frac{5 }{\sqrt{6}}$$
(E) $$\frac{5 }{2 \ \sqrt{6}}$$
4- The ratio of boys to girls in a school is $$3:4$$. If there are $$560$$ students in a school, how many boys are in the school.
(A) $$350$$
(B) $$410$$
(C) $$120$$
(D) $$420$$
(E) $$510$$
5- If $$f(x) = 2 \ x \ – \ 1$$ and $$g(x) = x^2 \ + \ 2 \ x$$, then find $$(\frac{f}{g})(x)$$.
(A) $$\frac{2 \ x \ − \ 1}{x^2 \ + \ 2 \ x}$$
(B) $$\frac{2 \ x \ + \ 1}{x^2 \ - \ 2 \ x}$$
(C) $$\frac{x^2 \ - \ 2 \ x}{2 \ x \ + \ 1}$$
(D) $$\frac{x^2 \ + \ 2 \ x}{2 \ x \ - \ 1}$$
(E) $$\frac{1}{2 \ x \ - \ 1}$$
6- If $$(x \ - \ 2)^2 \ + \ 1 \ > \ 3 \ x \ - \ 1$$, then $$x$$ can equal which of the following?
(A) $$1$$
(B) $$6$$
(C) $$8$$
(D) $$3$$
(E) $$4$$
7- In the standard $$(x, y)$$ coordinate system plane, what is the area of the circle with the following equation?
$$(x \ + \ 2)^2 \ + \ (y \ - \ 4)^2=25$$
(A) $$31\ π$$
(B) $$36\ π$$
(C) $$5\ π$$
(D) $$25\ π$$
(E) $$16 \ π$$
8- In the standard $$(x, y)$$ coordinate plane, which of the following lines contains the points
$$(3, - \ 5)$$ and $$(8, 15)$$?
(A) $$y=\frac{1}{4} \ x \ + \ 13$$
(B) $$y= \ - \ \frac{1}{4} \ x \ + \ 17$$
(C) $$y= \ - \ 4 \ x \ + \ 7$$
(D) $$y=4 \ x \ − \ 17$$
(E) $$y= 2 \ x \ - \ 11$$
9- If the area of a circle is $$49$$ square meters, what is its diameter?
(A) $$\frac{8 \sqrt{π}}{π}$$
(B) $$\frac{\pi}{8 \sqrt{π}}$$
(C) $$\frac{\pi}{7 \sqrt{π}}$$
(D) $$\frac{7\sqrt{π}}{π}$$
(E) $$\frac{1}{7\sqrt{π}}$$
10- A bank is offering $$2.5\%$$ simple interest on a savings account. If you deposit $$15,000$$, how much interest will you earn in two years?
(A) $$800$$
(B) $$700$$
(C) $$750$$
(D) $$850$$
(E) $$900$$
11- What is the solution of the following inequality?
$$|x \ - \ 5| \ ≥ \ 4$$
(A) $$x\ ≥9$$
(B) $$x≤ 1$$
(C) $$x\ ≥9 \ ∪ \ x≤ 1$$
(D) $$x\ ≥9 \ ∪ \ x≤- \ 1$$
(E) $$x\ ≥ \ - \ 9 \ ∪ \ x≤- \ 1$$
12- The length of a rectangle is $$\frac{3}{5}$$ times its width. If the width is $$15$$, what is the perimeter of this rectangle?
(A) $$42$$
(B) $$45$$
(C) $$48$$
(D) $$53$$
(E) $$64$$
13- If $$120\%$$ of a number is $$90$$, then what is $$80\%$$ of that number?
(A) $$70$$
(B) $$60$$
(C) $$80$$
(D) $$65$$
(E) $$75$$
14- In two successive years, the population of a town is increased by $$12\%$$ and $$25\%$$. What percent of the population is increased after two years?
(A) $$40\%$$
(B) $$30\%$$
(C) $$50\%$$
(D) $$60\%$$
(E) $$20\%$$
15- If the ratio of home fans to visiting fans in a crowd is $$3:2$$ and all $$25,000$$ seats in a stadium are filled, how many visiting fans are in attendance?
16- A card is drawn at random from a standard $$57–$$card deck, what is the probability that the card is of Hearts? (The deck includes $$12$$ of each suit clubs, diamonds, hearts, and spades)
(A) $$\frac{1}{19}$$
(B) $$\frac{19}{4}$$
(C) $$\frac{1}{4}$$
(D) $$\frac{1}{9}$$
(E) $$\frac{4}{19}$$
17- Simplify:
$$4 \ x^2 \ y^3 \ + \ 5 \ x^3 \ y^5 \ – \ (9 \ x^2 \ y^3 \ – \ 4 \ x^3 \ y^5)$$
(A) $$7\ x^3\ y^5\ + \ 5 \ x^2\ y^3$$
(B) $$9\ x^3\ y^5\ -\ 5 \ x^2\ y^3$$
(C) $$- \ 9\ x^3\ y^5\ + \ 7 \ x^2\ y^3$$
(D) $$- \ 5\ x^3\ y^5$$
(E) $$9\ x^3\ y^5\ +\ 9 \ x^2\ y^3$$
18- In the figure below, line A is parallel to line B. What is the value of angle $$x$$?
(A) $$112$$ degree
(B) $$180$$ degree
(C) $$142$$ degree
(D) $$153$$ degree
(E) $$124$$ degree
19- If tan⁡$$x=\frac{6}{8}$$, then sin ⁡$$x=$$
(A) $$\frac{3}{10}$$
(B) $$\frac{3}{5}$$
(C) $$\frac{6}{5}$$
(D) $$\frac{1}{5}$$
(E) It cannot be determined from the information given.
20- An angle is equal to one fourth of its supplement. What is the measure of that angle?
(A) $$30$$
(B) $$42$$
(C) $$49$$
(D) $$21$$
(E) $$36$$
21- If $$x \ +$$ sin$$^2 \ a \ +$$ cos$$^2 \ a=5$$, then $$x =$$?
(A) $$1$$
(B) $$2$$
(C) $$3$$
(D) $$4$$
(E) $$5$$
22- Sophia purchased a sofa for $$474.24$$. The sofa is regularly priced at $$624$$. What was the percent discount Sophia received on the sofa?
(A) $$18\%$$
(B) $$12\%$$
(C) $$24\%$$
(D) $$31\%$$
(E) $$36\%$$
23- Which of the following expressions is equal to $$\sqrt{\frac{x^2}{3} \ + \ \frac{x^2}{12}}$$?
(A) $$\frac{\sqrt{3} }{2 \ \sqrt{5}} \ x$$
(B) $$\frac{\sqrt{5} }{2 \ \sqrt{3}} \ x$$
(C) $$\frac{\sqrt{5} }{2 } \ x$$
(D) $$\frac{\sqrt{5} }{ \sqrt{3}} \ x$$
(E) $$\frac{2 \ \sqrt{5} }{ \sqrt{3}} \ x$$
24- Last week $$18,000$$ fans attended a football match. This week three times as many bought tickets, but one sixth of them cancelled their tickets. How many are attending this week?
(A) $$54,000$$
(B) $$35,000$$
(C) $$45,000$$
(D) $$81,000$$
(E) $$9,000$$
25- The average of six consecutive numbers is $$30$$. What is the smallest number?
(A) $$22$$
(B) $$21.5$$
(C) $$24$$
(D) $$25.5$$
(E) $$27.5$$
26- What is the slope of a line that is perpendicular to the line?
$$4 \ x \ - \ 2 \ y=16$$?
(A) $$\frac{1}{2}$$
(B) $$2$$
(C) $$- \ 2$$
(D) $$− \ \frac{1}{2}$$
(E) $$1$$
27- What is the value of the expression $$5 \ (x \ - \ 2 \ y) \ + \ (2 \ - \ x)^2$$ when $$x=2$$ and $$y= \ - \ 2$$?
(A) $$25$$
(B) $$20$$
(C) $$35$$
(D) $$15$$
(E) $$30$$
28- Convert $$320,000$$ to scientific notation.
(A) $$32\ ×\ 10^5$$
(B) $$32\ ×\ 10^6$$
(C) $$3.2\ ×\ 10^6$$
(D) $$3.2\ ×\ 10^4$$
(E) $$3.2\ ×\ 10^5$$
29- If $$\sqrt{8 \ x}=\sqrt{y}$$, then $$x=$$
(A) $$\frac{y}{9}$$
(B) $$\frac{9}{y}$$
(C) $$\frac{y}{8}$$
(D) $$\frac{8}{y}$$
(E) $$\frac{1}{y}$$
30- In following rectangle which statement is true?
(A) Length of AD equal to length BC
(B) Length of AB equal to length BC
(C) Length of AB equal to length AD
(D) The measure of all the angles equals $$180^°$$.
(E) The answer cannot be found from the information given.
31- The surface area of a cylinder is $$48 \ π$$ cm$$^2$$. If its height is $$10$$ cm, what is the radius of the cylinder?
(A) $$2$$ cm
(B) $$4$$ cm
(C) $$5$$ cm
(D) $$7$$ cm
(E) $$1$$ cm
32- Let $$r$$ and $$p$$ be constants. If $$x^2 \ + \ 6 \ x \ + \ r$$ factors into $$(x \ + \ 2) \ (x \ + \ p)$$, the values of $$r$$ and $$p$$ respectively are?
(A) $$4, 6$$
(B) $$6, 3$$
(C) $$4, 8$$
(D) $$8, 4$$
(E) The answer cannot be found from the information given.
33- In a coordinate plane, triangle ABC has coordinates: $$(− \ 1, 6), (− \ 2,5)$$, and $$(5,8)$$. If triangle ABC is reflected over the $$y-$$axis, what are the coordinates of the new image?
(A) $$(− \ 1, − \ 6), (− \ 2, − \ 5), (5, − \ 8)$$
(B) $$(− \ 1,6), (− \ 2,5), (5,8)$$
(C) $$(1,6), (2,5), (5,8)$$
(D) $$(− \ 1, − \ 6), (− \ 2, − \ 5), (− \ 5, − \ 8)$$
(E) $$(1,6), (2,5), (− \ 5,8)$$

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34- A cruise line ship left Port A and traveled $$80$$ miles due west and then $$150$$ miles due north. At this point, what is the shortest distance from the cruise to port A?
(A) $$120$$ miles
(B) $$80$$ miles
(C) $$150$$ miles
(D) $$240$$ miles
(E) $$170$$ miles
35- If the ratio of $$2 \ a$$ to $$3 \ b$$ is $$\frac{1}{15}$$, what is the ratio of $$a$$ to $$b$$?
(A) $$\frac{1}{10}$$
(B) $$\frac{1}{15}$$
(C) $$\frac{1}{25}$$
(D) $$10$$
(E) $$15$$
36- Two-kilograms apple and four-kilograms orange cost $$25.4$$. If one-kilogram apple costs $$4.2$$ how much does one-kilogram orange cost?
(A) $$6$$
(B) $$5.5$$
(C) $$5$$
(D) $$4.5$$
(E) $$4$$
37- Tickets to a movie cost $$12.50$$ for adults and $$7.50$$ for students. A group of $$12$$ friends purchased tickets for $$125$$. How many adults tickets did they buy?
(A) $$3$$
(B) $$4$$
(C) $$5$$
(D) $$6$$
(E) $$7$$
38- If A $$=\begin{bmatrix}- \ 1 & 3 \\1 & - \ 3 \end{bmatrix}$$ and B $$=\begin{bmatrix}2 & 1 \\- \ 2 & 2 \end{bmatrix}$$ then $$2$$ A $$-$$ B $$=$$?
(A) $$\begin{bmatrix}4&5\\ 4&-\ 6\end{bmatrix}$$
(B) $$\begin{bmatrix}4&6\\ - \ 4&-\ 6\end{bmatrix}$$
(C) $$\begin{bmatrix}-\ 4&3\\ 2&-\ 6\end{bmatrix}$$
(D) $$\begin{bmatrix}-\ 4&- \ 8\\ 2&-\ 6\end{bmatrix}$$
(E) $$\begin{bmatrix}-\ 4&5\\ 4&-\ 8\end{bmatrix}$$
39- $$(x^5)^{\frac{3}{7}}$$ equal to?
(A) $$x ^ {\frac{7}{17}}$$
(B) $$x ^ {\frac{7}{15}}$$
(C) $$x ^ {\frac{15}{7}}$$
(D) $$x ^ {\frac{17}{7}}$$
(E) $$x ^ {\frac{1}{7}}$$
40- The width of a box is one third of its length. The height of the box is one third of its width. If the length of the box is $$45$$ cm, what is the volume of the box?
(A) $$3,375$$
(B) $$375$$
(C) $$2,125$$
(D) $$2,625$$
(E) $$3,415$$
41- What is the amplitude of the graph of the equation $$y \ - \ 1=2$$ cos $$3 \ x$$?
(half the distance between the graph’s minimum and maximum $$y-$$values in standard $$(x, y)$$ coordinate plane is the amplitude of a graph.)
(A) $$1$$
(B) $$- \ 1$$
(C) $$- \ 2$$
(D) $$2$$
(E) $$0$$
42- If one angle of a right triangle measures $$60^\circ$$, what is the cos of the other acute angle?
(A) $$\frac{1}{2}$$
(B) $$\frac{\sqrt{2}}{2}$$
(C) $$\frac{\sqrt{2}}{3}$$
(D) $$\frac{\sqrt{3}}{2}$$
(E) $$1$$
43- What is the difference in area between a $$8$$ cm by $$4$$ cm rectangle and a circle with diameter of $$10$$ cm? $$(π=3)$$
(A) $$32$$
(B) $$33$$
(C) $$51$$
(D) $$42$$
(E) $$43$$
44- The average weight of $$15$$ girls in a class is $$60$$ kg and the average weight of $$37$$ boys in the same class is $$64$$ kg. What is the average weight of all the $$52$$ students in that class?
(A) $$62.84$$ kg
(B) $$64.34$$ kg
(C) $$58.90$$ kg
(D) $$65.32$$ kg
(E) $$71.23$$ kg
45- Between which two of the months shown was there a twenty percent decreased in the number of pants sold?
(A) January and February
(B) February and March
(C) March and April
(D) April and May
(E) May and June
46- During the six-month period shown, what is the median number of shirts and mean number of shoes per month?
(A) $$156.5, 30$$
(B) $$178.5, 29$$
(C) $$147.5, 30$$
(D) $$123.5, 31$$
(E) $$135.5, 32$$
47- How many shoes need to be added in April until the ratio of pants to shoes in April equals to five-seventeenth of this ratio in May?
(A) $$30$$
(B) $$40$$
(C) $$50$$
(D) $$60$$
(E) $$70$$
48- What is the value of $$x$$ in the following equation?
$$5^{ \ x}=3125$$
(A) $$4$$
(B) $$5$$
(C) $$6$$
(D) $$7$$
(E) $$3$$
49- If $$x=5$$, what is the value of $$y$$ in the following equation?
$$3 \ y =\frac{2 \ x^3}{5} \ + \ 4$$
(A) $$17$$
(B) $$21$$
(C) $$32$$
(D) $$18$$
(E) $$12$$
50- In the following figure, ABCD is a rectangle. If $$a=\sqrt{3}$$, and $$b=3 \ a$$, find the area of the shaded region. (the shaded region is a trapezoid)
(A) $$\frac{15 \ \sqrt{4}}{2}$$
(B) $$\frac{15 \ \sqrt{3}}{2}$$
(C) $$15 \ \sqrt{3}$$
(D) $$\sqrt{3}$$
(E) $$\frac{\sqrt{3}}{15}$$
51- Which of the following is one solution of this equation?
$$x^2 \ + \ 2 \ x \ - \ 5=0$$
(A) $$\sqrt{2}\ -\ 1$$
(B) $$\sqrt{6}\ +\ 1$$
(C) $$\sqrt{12}$$
(D) $$\sqrt{2}\ +\ 1$$
(E) $$\sqrt{6}\ -\ 1$$
52- A football team had $$23,000$$ to spend on supplies. The team spent $$16,000$$ on new balls. New sport shoes cost $$110$$ each. Which of the following inequalities represent how many new shoes the team can purchase?
(A) $$110 \ x \ - \ 16,000 \ ≤ \ 23,000$$
(B) $$110 \ x \ + \ 16,000 \ ≤ \ 23,000$$
(C) $$110 \ - \ 16,000 \ x \ ≤ \ 23,000$$
(D) $$110 \ + \ 7,000 \ x \ \geq \ 24,000$$
(E) $$110 \ - \ 19,000 \ \ \geq \ 24,000 \ x$$
53- If $$f(x)=3 \ x^2\ + \ 5$$ and $$g(x)=- \ \frac{1}{x}$$, what is the value of $$f(g(x))$$?
(A) $$\frac{3}{x^2} \ - \ 5$$
(B) $$\frac{3}{x^2} \ + \ 5$$
(C) $$\frac{2}{x^3} \ + \ 5$$
(D) $$\frac{3}{x^2}$$
(E) $$\frac{3}{x^2} \ - \ 2$$
54- The length of a rectangle is $$4$$ meters greater than $$3$$ times its width.  The perimeter of the rectangle is $$40$$ meters.  What is the area of the rectangle?
(A) $$45$$ m$$^2$$
(B) $$52$$ m$$^2$$
(C) $$64$$ m$$^2$$
(D) $$69$$ m$$^2$$
(E) $$73$$ m$$^2$$
55- What is the sum of prime numbers between $$20$$ and $$40$$?
(A) $$120$$
(B) $$60$$
(C) $$92$$
(D) $$86$$
(E) $$112$$
56- Simplify $$\frac{4 \ - \ 3 \ i}{- \ 4 \ i}$$?
(A) $$\frac{3}{4} \ -\ i$$
(B) $$\frac{1}{4} \ -\ i$$
(C) $$i$$
(D) $$\frac{3}{4} \ +\ i$$
(E) $$\frac{1}{4} \ +\ i$$
57- What are the zeroes of the function $$f(x)=x^3 \ + \ 5 \ x^2 \ + \ 6 \ x$$?
(A) $$0, -\ 2 , 3$$
(B) $$0, 2 , 3$$
(C) $$1, 2 , 3$$
(D) $$1, 2$$
(E) $$0, -\ 2 , -\ 3$$
58- In the following figure, what is the perimeter of $$\triangle$$ABC if the area of $$\triangle$$ADC is $$12$$?
(A) $$32.5$$
(B) $$15$$
(C) $$21$$
(D) $$24$$
(E) The answer cannot be determined from the information given
59- A swimming pool holds $$5,000$$ cubic feet of water. The swimming pool is $$20$$ feet long and $$20$$ feet wide. How deep is the swimming pool?
(A) $$10$$ feet
(B) $$8$$ feet
(C) $$8.5$$ feet
(D) $$12.5$$ feet
(E) $$15.5$$ feet
60- If $$y=(- \ 5 \ x^4)^2$$, which of the following expressions is equal to $$y$$?
(A) $$25 \ x^6$$
(B) $$25 \ x^8$$
(C) $$5 \ x^8$$
(D) $$- \ 25 \ x^8$$
(E) $$25 \ x^4$$
 1- Choice A is correct The correct answer is $$36^\circ$$ The sum of all angles in a quadrilateral is $$360$$ degrees.Let $$x$$ be the smallest angle in the quadrilateral.Then the angles are: $$x,\ 2\ x\ ,\ 3\ x\ ,\ 4\ x$$$$x\ +\ 2\ x\ +\ 3\ x\ +\ 4 \ x=360→10\ x=360→x=36$$ The angles in the quadrilateral are: $$36^\circ, \ 72^\circ,\ 108^\circ$$, and $$144^\circ$$ 2- Choice A is correct The correct answer is $$0$$ Solving Systems of Equations by EliminationMultiply the first equation by $$- \ 2$$, then add it to the second equation.\cfrac{\begin{align} - \ 2 \ (2 \ x \ + \ 3 \ y \ = \ 12) \\ 4 \ x \ - \ 4 \ y \ = - \ 16 \end{align}}{}\cfrac{ \begin{align} - \ 4 \ x \ - \ 6 \ y \ = \ - \ 24 \\ 4 \ x \ - \ 4 \ y \ = - \ 16 \end{align} }{\begin{align} - \ 10\ y \ = - \ 40 \\ ⇒ y \ = \ 4 \end{align}}Plug in the value of $$y$$ into one of the equations and solve for $$x$$.$$2 \ x \ + \ 3 \ (4)= 12 ⇒$$$$2 \ x \ + \ 12= 12 ⇒$$$$2 \ x=0⇒ x= 0$$ 3- Choice C is correct The correct answer is $$\frac{2 \ \sqrt{6}}{5}$$sin $$A=\frac{1}{5}⇒$$Since sin$$θ=\frac{opposite}{hypotenuse}$$, we have the following right triangle. Then:$$c=\sqrt{5^2\ -\ 1^2 }= \sqrt{25\ -\ 1}=\sqrt{24}=2 \ \sqrt{6}$$cos $$A=\frac{2 \ \sqrt{6}}{5}$$ 4- Choice D is correct The correct answer is $$420$$The ratio of boy to girls is $$3:4$$.Therefore, there are $$3$$ boys out of $$4$$ students.To find the answer, first divide the total number of students by $$4$$, then multiply the result by $$3$$. $$560 \ ÷\ 4 = 140 ⇒ 140 \ ×\ 3 = 420$$ 5- Choice A is correct The correct answer is $$\frac{2 \ x \ − \ 1}{x^2 \ + \ 2 \ x}$$$$(\frac{f}{g})(x) = \frac{f(x)}{g(x)} = \frac{2 \ x \ − \ 1}{x^2 \ + \ 2 \ x}$$ 6- Choice C is correct The correct answer is$$8$$Plug in the value of each option in the inequality.A. $$1$$    $$(1\ -\ 2)^2\ +\ 1>3\ (1) \ -\ 1→2>2$$        No!B. $$6$$    $$(6\ -\ 2)^2\ +\ 1>3\ (6)\ -\ 1→17>17$$    No!C. $$8$$    $$(8\ -\ 2)^2\ +\ 1>3\ (8)\ -\ 1→37>23$$    Bingo!D. $$3$$    $$(3\ -\ 2)^2\ +\ 1>3\ (3)\ -\ 1→2>8$$        No!E $$4$$     $$(4\ -\ 2)^2\ +\ 1>3\ (4)\ -\ 1→5>11$$      No! 7- Choice D is correct The correct answer is $$25\ π$$The equation of a circle in standard form is: $$(x\ -\ h)^2\ +\ (y\ -\ k)^2=r^2$$, where $$r$$ is the radius of the circle. In this circle the radius is $$5$$.$$r^2=25→r=5$$$$(x\ +\ 2)^2\ +\ (y\ -\ 4)^2=25$$Area of a circle: $$A=π\ r^2=π\ (5)^2=25\ π$$ 8- Choice D is correct The correct answer is $$y=4 \ x \ − \ 17$$The equation of a line is: $$y=m \ x \ + \ b$$, where m is the slope and b is the $$y-$$intercept.First find the slope: $$m=\frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1} }=\frac{15 \ -\ (- \ 5)}{8 \ - \ 3}=\frac{20}{5}=4$$Then, we have: $$y=4 \ x \ + \ b$$Choose one point and plug in the values of $$x$$ and $$y$$ in the equation to solve for $$b$$.Let’s choose the point $$(3, - \ 5)$$$$y=4 \ x \ + \ b→ \ - \ 5=4 \ (3) \ + \ b→ \ - \ 5=12 \ + \ b→b= \ - \ 17$$The equation of the line is: $$y=4 \ x \ - \ 17$$ 9- Choice D is correct The correct answer is $$\frac{7\sqrt{π}}{π}$$Formula for the area of a circle is: $$A=π\ r^2$$ Using $$49$$ for the area of the circle we have:$$49=π\ r^2$$ Let’s solve for the radius $$(r)$$. $$\frac{49}{π}=r^2→r=\sqrt{\frac{49}{π}}=\frac{7}{\sqrt{π}}=\frac{7}{\sqrt{π}}\ ×\ \frac{\sqrt{π}}{\sqrt{π}}=\frac{7\sqrt{π}}{π}$$ 10- Choice C is correct The correct answer is $$750$$Use simple interest formula:$$I=prt$$($$I =$$ interest, $$p =$$ principal, $$r =$$ rate, $$t =$$ time)$$I=(15000) \ (0.025) \ (2)=750$$ 11- Choice C is correct The correct answer is $$x\ ≥9 \ ∪ \ x≤ 1$$$$x\ -\ 5\ ≥\ 4→x\ ≥4\ +\ 5→x≥9$$Or$$x\ -\ 5≤-\ 4→x≤-\ 4\ +\ 5→x≤ 1$$Then, solution is: $$x\ ≥9 \ ∪ \ x≤ 1$$ 12- Choice C is correct The correct answer is $$48$$ Length of the rectangle is: $$\frac{3}{5}\ ×\ 15=9$$ perimeter of rectangle is: $$2\ ×\ (9\ +\ 15)=48$$ 13- Choice B is correct The correct answer is $$60$$First, find the number.Let $$x$$ be the number.Write the equation and solve for $$x$$. $$120 \%$$ of a number is $$90$$, then:$$1.2\ ×\ x=90 ⇒ x=90 \ ÷\ 1.2=75$$$$80 \%$$ of $$75$$ is: $$0.7 \ ×\ 75 = 60$$ 14- Choice A is correct The correct answer is $$40\%$$ The population is increased by $$12\%$$ and $$25\%$$.$$12\%$$ increase changes the population to $$112\%$$ of original population.For the second increase, multiply the result by $$125\%$$.$$(1.12) \ ×\ (1.25) = 1.40= 140\%$$$$40$$ percent of the population is increased after two years. 14- Choice A is correct The correct answer is $$40\%$$ The population is increased by $$12\%$$ and $$25\%$$.$$12\%$$ increase changes the population to $$112\%$$ of original population.For the second increase, multiply the result by $$125\%$$.$$(1.12) \ ×\ (1.25) = 1.40= 140\%$$$$40$$ percent of the population is increased after two years. 16- Choice E is correct The correct answer is $$\frac{4}{19}$$The probability of choosing a Hearts is $$\frac{12}{57} = \frac{4}{19}$$ 17- Choice B is correct The correct answer is $$9\ x^3\ y^5\ -\ 5 \ x^2\ y^3$$$$4\ x^2\ \ y^3\ + \ 5\ x^3\ y^5\ –\ 9\ x^2\ y^3\ +\ 4\ x^3\ y^5\ =$$$$4\ x^2\ y^3\ -\ 9\ x^2\ y^3\ +\ 5\ x^3\ y^5\ + \ 4 \ x^3 \ y^5\ =$$$$9\ x^3\ y^5\ -\ 5 \ x^2\ y^3$$ 18- Choice C is correct The correct answer is $$142$$ degree The angle $$x$$ and $$35$$ are complementary angles.Therefore: $$x\ +\ 38=180$$ $$180^\circ\ - \ 38^\circ=142^\circ$$ 19- Choice B is correct The correct answer is $$\frac{3}{5}$$tan ⁡$$=\frac{opposite}{djacent}$$, and tan⁡ $$x=\frac{6}{8}$$, therefore, the opposite side of the angle $$x$$ is $$6$$ and the adjacent side is $$8$$.Let’s draw the triangle.Using Pythagorean theorem, we have:$$a^2\ +\ b^2=c^2→6^2\ +\ 8^2=c^2→36\ +\ 64=c^2→c=10$$sin ⁡$$=\frac{opposite}{hypotnuse}=\frac{6}{10} = \frac{3}{5}$$ 20- Choice E is correct The correct answer is $$36$$The sum of supplement angles is $$180$$. Let $$x$$ be that angle.Therefore, $$x \ +\ 4\ x\ = 180$$ $$5\ x\ = 180$$, divide both sides by $$5$$: $$x = 36$$ 21- Choice D is correct The correct answer is $$4$$sin$$^2 \ a\ +$$ cos$$^2 \ a=1$$, then: $$x\ +\ 1=5, x=4$$ 22- Choice C is correct The correct answer is $$24\%$$The question is this: $$474.24$$ is what percent of $$624$$?Use percent formula: part $$= \frac{percent}{100} \ ×$$ whole$$474.24 = \frac{percent}{100} \ ×\ 624 ⇒ 474.24= \frac{percent \ ×\ 624}{100} ⇒47424 =$$ percent $$×\ 624 ⇒$$ percent $$= \frac{47424}{624} = 83.39$$$$47424$$ is $$76 \%$$ of $$624$$. Therefore, the discount is: $$100\% \ –\ 76\% = 24\%$$ 23- Choice B is correct The correct answer is $$\frac{\sqrt{5} }{2 \ \sqrt{3}} \ x$$Simplify the expression.$$\sqrt{\frac{x^2}{3}\ +\ \frac{x^2}{12}}=\sqrt{\frac{4 \ x^2}{12}\ +\ \frac{x^2}{12}}=\sqrt{\frac{5x^2}{12}}=\sqrt{\frac{5}{12} \ x^2 }=\sqrt{\frac{5}{12}} \ ×\ \sqrt{x^2} = \frac{\sqrt{5}}{\sqrt{12}}\ ×\ \sqrt{x^2}= \frac{\sqrt{5}}{2 \ \sqrt{3}}\ ×\ \sqrt{x^2}=\frac{\sqrt{5} }{2 \ \sqrt{3}} \ x$$ 24- Choice C is correct The correct answer is $$45,000$$ Three times of $$18,000$$ is $$54,000$$. One sixth of them cancelled their tickets.One sixth of $$54,000$$ equals $$9,000\ (\frac{1}{6} \ ×\ 54000 = 9000)$$. $$45,000 \ (54000 \ –\ 9000 = 45000)$$ fans are attending this week 25- Choice E is correct The correct answer is $$27.5$$Let $$x$$ be the smallest number.Then, these are the numbers:$$x, x\ + \ 1, x \ + \ 2, x \ + \ 3, x \ + \ 4, x \ + \ 5$$average $$= \frac{sum \ of \ terms}{number \ of \ terms} ⇒$$$$30 = \frac{x \ + \ (x \ + \ 1) \ + \ (x \ + \ 2) \ + \ (x \ + \ 3) \ + \ (x \ + \ 4)\ + \ (x \ + \ 5)}{6}⇒$$$$30=\frac{6 \ x \ + \ 15}{6} ⇒$$ $$180 = 6 \ x \ + \ 15 ⇒$$$$165 = 6 \ x ⇒ x=27.5$$ 26- Choice D is correct The correct answer is $$− \ \frac{1}{2}$$The equation of a line in slope intercept form is: $$y=m \ x \ + \ b$$Solve for $$y$$.$$4 \ x \ - \ 2 \ y=16 ⇒$$$$- \ 2 \ y=16 \ - \ 4 \ x ⇒$$$$y=(16 \ - \ 4 \ x) \ ÷ \ (- \ 2) ⇒$$$$y=2 \ x \ - \ 8$$The slope is $$2$$.The slope of the line perpendicular to this line is:$$m_{1} \ × \ m{2} = \ - \ 1 ⇒$$$$2 \ × \ m_{2} = \ - \ 1 ⇒ m_{2} = \ - \ \frac{1}{2}$$ 27- Choice E is correct The correct answer is $$30$$Plug in the value of $$x$$ and $$y$$. $$x=2$$ and $$y=-\ 2$$$$5\ (x\ -\ 2\ y)\ +\ (2 \ - \ x)^2=5\ (2\ -\ 2(-\ 2))\ +\ (2\ -\ 2)^2=5\ (2\ +\ 4)\ +\ (0)^2 = 30\ +\ 0=30$$ 28- Choice E is correct The correct answer is $$3.2\ ×\ 10^5$$$$320000=3.2\ ×\ 10^5$$ 29- Choice C is correct The correct answer is $$\frac{y}{8}$$Solve for $$x$$.$$\sqrt{8 \ x}=\sqrt{y}$$Square both sides of the equation:$$(\sqrt{8 \ x})^2=(\sqrt{y})^2$$  ,  $$8\ x=y$$  ,   $$x=\frac{y}{8}$$ 30- Choice A is correct The correct answer is Length of AD equal to length BCIn any rectangle opposite sides are equal. 31- Choice A is correct The correct answer is $$2$$ cmFormula for the Surface area of a cylinder is:S A $$=2 \ π \ r^2 \ + \ 2 \ π \ r \ h→$$$$48 \ π=2 \ π \ r^2 \ + \ 2 \ π \ r \ (10)→$$$$r^2 \ + \ 10 \ r \ - \ 24=0$$Factorize and solve for $$r$$.$$(r \ + \ 12)(r \ - \ 2)=0→r=2$$ or $$r= \ - \ 12$$ (unacceptable) 32- Choice D is correct The correct answer is $$8, 4$$$$(x\ +\ 2)\ (x\ +\ p)=x^2\ +\ (2\ +\ p)x\ +\ 2\ p→2\ +\ p=6→p=4$$ and$$r=2\ p=8$$ 33- Choice E is correct The correct answer is $$(1,6), (2,5), (− \ 5,8)$$Since the triangle ABC is reflected over the $$y-$$axis, then all values of $$y$$’s of the points don’t change and the sign of all $$x$$’s change. (remember that when a point is reflected over the $$y-$$axis, the value of $$y$$ does not change and when a point is reflected over the $$x-$$axis, the value of $$x$$ does not change).Therefore:$$(− \ 1,6)$$ changes to $$(1, 6)$$$$(− \ 2, 5)$$ changes to $$(2, 5)$$$$(5, 8)$$ changes to $$(− \ 5, 8)$$ 34- Choice E is correct The correct answer is $$170$$ milesUse the information provided in the question to draw the shape.Use Pythagorean Theorem: $$a^2 \ + \ b^2 = c^2$$$$80^2 \ + \ 150^2 = c^2 ⇒$$$$6400 \ + \ 22500 = c^2 ⇒$$$$28900 = c^2 ⇒ c = 170$$ 35- Choice A is correct The correct answer is $$\frac{1}{10}$$Write the ratio of $$2 \ a$$ to $$3 \ b$$.$$\frac{2 \ a}{3 \ b}=\frac{1}{15}$$Use cross multiplication and then simplify.$$2 \ a \ × \ 15=3 \ b \ ×\ 1→30 \ a=3 \ b→a=\frac{3 \ b}{30}=\frac{b}{10}$$Now, find the ratio of $$a$$ to $$b$$.$$\frac{a}{b}=\frac{\frac{b}{10}}{b}→\frac{b}{10} \ ÷\ b=\frac{b}{10} \ × \ \frac{1}{b}=\frac{b}{10 \ b}=\frac{1}{10}$$ 36- Choice D is correct The correct answer is $$4.5$$Let $$x$$ be the cost of one-kilogram orange, then: $$4\ x\ + \ (2 \ × \ 4.2)=25.4→$$$$4\ x\ +\ 8.4=25.4→4\ x=25.4\ -\ 8.4→4\ x=18→x=\frac{18}{4}=4.5$$ 37- Choice E is correct The correct answer is $$7$$Let $$x$$ be the number of adult tickets and $$y$$ be the number of student tickets. Then:$$x \ + \ y =12, \ 12.50 \ x \ + \ 7.50y=125$$Use elimination method to solve this system of equation.Multiply the first equation by $$- \ 7.5$$ and add it to the second equation.$$- \ 7.5 \ (x \ + \ y=12), \ - \ 7.5 \ x \ - \ 7.5 \ y= \ - \ 90, 12.50 \ x \ + \ 7.50y=125$$$$5 \ x=35, \ x=7$$There are $$7$$ adult tickets and $$5$$ student tickets. 38- Choice E is correct The correct answer is $$\begin{bmatrix}-\ 4&5\\ 4&-\ 8\end{bmatrix}$$First, find $$2$$ A.A$$=\begin{bmatrix}-\ 1 & 3\\ 1 & -\ 3\end{bmatrix}$$$$2$$A$$=2\ ×\ \begin{bmatrix}-\ 1&3\\ 1 & -\ 3\end{bmatrix}=\begin{bmatrix} -\ 2& 6\\ 2 & -\ 6\end {bmatrix}$$Now, solve for $$2$$ A $$-$$ B.$$2$$ A$$-$$B$$=\begin{bmatrix} -\ 2&6 \\ 2& -\ 6\end {bmatrix}\ -\ \begin{bmatrix} 2&1\\ -\ 2 & 2\end{bmatrix}=\begin{bmatrix}-\ 2 \ -\ 2 & 6 \ -\ 1\\ 2\ -\ (-\ 2) & -\ 6\ -\ 2\end{bmatrix}=\begin{bmatrix}-\ 4&5\\ 4&-\ 8\end{bmatrix}$$ 39- Choice C is correct The correct answer is $$x ^ {\frac{15}{7}}$$$$(x^5)^{\frac{3}{7}} = x^{5 \ ×\ \frac{3}{7}} =x ^ {\frac{15}{7}}$$ 40- Choice A is correct The corrcet answer is $$3,375$$If the length of the box is $$45$$, then the width of the box is one third of it, $$15$$, and the height of the box is $$5$$ (one third of the width).The volume of the box is:$$V = lwh = (45)\ (15)\ (5) = 3,375$$ 41- Choice D is correct The correct answer is $$2$$The amplitude in the graph of the equation $$y=a$$ cos $$b\ x$$ is $$a$$. ($$a$$ and $$b$$ are constant)In the equation $$y\ -\ 1=2$$ cos $$3\ x$$, the amplitude is $$2$$. 42- Choice D is correct The correct answer is $$\frac{\sqrt{3}}{2}$$The relationship among all sides of right triangle $$30^\circ \ - \ 60^\circ \ - \ 90^\circ$$ is provided in the following triangle: cos of $$30^\circ$$ equals to:$$\frac{adjacent}{hypotenuse}=\frac{x \ \sqrt{3}}{2\ x}=\frac{\sqrt{3}}{2}$$ 43- Choice E is correct The correct answer is $$43$$The area of rectangle is: $$8 \ × \ 4=32$$ cm$$^2$$The area of circle is: $$π \ r^2=π \ × \ ( \frac{10}{2})^2=3 \ × \ 25=75$$ cm$$^2$$Difference of areas is: $$75 \ - \ 32=43$$ 44- Choice A is correct The correct answer is $$62.84$$ kgaverage$$= \frac{sum \ of\ terms }{number \ of\ terms}$$The sum of the weight of all girls is: $$18 \ ×\ 60 = 900$$ kgThe sum of the weight of all boys is: $$37 \ ×\ 64 = 2368$$ kgThe sum of the weight of all students is: $$900 \ +\ 2368 = 3268$$ kgaverage $$= \frac{3268}{52} = 62.84$$ 45- Choice A is correct The correct answer is January and FebruaryFirst find the number of pants sold in each month.A. January: $$110$$, February: $$88$$, March: $$90$$, April: $$70$$, May: $$85$$, June: $$65$$Check each option provided.January and February, $$(\frac{110 \ - \ 88}{110}) \ × \ 100=\frac{22}{110} \ × \ 100=20\%$$B. February and March, there is an increase from February to March.C. March and April$$(\frac{90 \ - \ 70}{90}) \ × \ 100=\frac{20}{90} \ × \ 100=22.22\%$$D. April and May: there is an increase from April to MayMay and June$$(\frac{85 \ - \ 65}{85}) \ × \ 100=\frac{20}{85} \ × \ 100=23.53\%$$ 46- Choice C is correct The correct answer is $$147.5, 30$$Let’s order number of shirts sold per month:$$130,140,145,150,160,170$$median is: $$\frac{145 \ + \ 150}{2}=147.5$$Let’s list the number of shoes sold per month:$$20,25,25,35,35,40$$mean is: $$\frac{20 \ + \ 25 \ + \ 25 \ + \ 35 \ + \ 35 \ + \ 40}{6}=\frac{180}{6}=30$$ 47- Choice C is correct The correct answer is $$50$$Let $$x$$ be the number of shoes need to be added in April. Then:$$\frac{70}{20 \ + \ x}=(\frac{5}{17}) \ (\frac{85}{25}) →\frac{70}{20 \ + \ x}=\frac{425}{425}=1→$$$$70=20 \ + \ x→x=50$$ 48- Choice B is correct The correct answer is $$5$$$$3125=5^5→5^{ \ x}=5^5→x=5$$ 49- Choice D is correct The correct answer is $$18$$Plug in the value of $$x$$ in the equation and solve for $$y$$.$$3 \ y=\frac{2 \ x^3}{5} \ + \ 4→$$$$3 \ y =\frac{2(5)^3}{5} \ + \ 4→$$$$3 \ y=\frac{2 \ (125)}{5} \ + \ 4→$$$$3 \ y= 50 \ + \ 4=54$$$$3 \ y = 54→y=18$$ 50- Choice B is correct The correct answer is $$\frac{15 \ \sqrt{3}}{2}$$Based on triangle similarity theorem:$$\frac{a}{a\ +\ b}=\frac{c}{4}→c=\frac{4\ a}{a\ +\ b}=\frac{4\sqrt3}{4\sqrt3}=1→$$ area of shaded region is:$$(\frac{c\ +\ 4}{2})\ (b)=\frac{5}{2}\ ×\ 3\sqrt{3}=\frac{15 \ \sqrt{3}}{2}$$ 51- Choice E is correct The correct answer is $$\sqrt{6}\ - \ 1$$$$x_{1,2} =\frac{ -\ b \ ±\ \sqrt{b^2\ -\ 4\ a\ c }}{2\ a}$$ $$a\ x^2\ +\ b\ x\ +\ c = 0$$$$x^2 \ +\ 2\ x\ –\ 5 = 0 ⇒$$ then:$$a = 1, b = 2$$ and $$c = \ – \ 5$$ $$x =\frac{ -\ 2 \ +\ \sqrt{2^2 \ -\ 4 .1 .-5}}{2 .1} = \sqrt{6} \ –\ 1$$$$x = \frac{-\ 2 \ -\ \sqrt{2^2 \ -\ 4 .1 .-\ 5 }}{2 .1} = –\ 1 \ –\ \sqrt{6}$$ 52- Choice B is correct The correct answer is $$110 \ x \ + \ 16,000 \ ≤ \ 23,000$$Let $$x$$ be the number of shoes the team can purchase.Therefore, the team can purchase $$110 \ x$$.The team had $$23,000$$ and spent $$16000$$.Now the team can spend on new shoes $$7000$$ at most.Now, write the inequality: $$110 \ x \ + \ 16,000 \ ≤ \ 23,000$$ 53- Choice B is correct The correct answer is $$\frac{3}{x^2} \ + \ 5$$$$f(g(x))=3 \ × \ (- \ \frac{1}{x})^2 \ + \ 5=\frac{3}{x^2} \ + \ 5$$ 54- Choice C is correct The correct answer is $$64$$ m$$^2$$Let $$L$$ be the length of the rectangular and $$W$$ be the with of the rectangular. Then,$$L=3 \ W \ + \ 4$$The perimeter of the rectangle is $$40$$ meters.Therefore: $$2 \ L \ + \ 2 \ W=40$$$$L \ + \ W=20$$Replace the value of $$L$$ from the first equation into the second equation and solve for $$W$$:$$(3 \ W \ + \ 4) \ + \ W=20→4 \ W \ + \ 4=20→4 \ W=16→W=4$$The width of the rectangle is $$4$$ meters and its length is:$$L=3 \ W \ + \ 4=3 \ (4) \ + \ 4=16$$The area of the rectangle is:length $$×$$ width $$= 4 \ × \ 16 = 64$$ 55- Choice A is correct The correct answer is $$120$$Here is the list of all prime numbers between $$20$$ and $$40$$:$$23, 29, 31, 37$$The sum of all prime numbers between $$20$$ and $$40$$ is:$$23 \ + \ 29 \ + \ 31 \ + \ 37 = 120$$ 56- Choice D is correct The correct answer is $$i\ +\frac{ 3}{4}$$To simplify the fraction, multiply both numerator and denominator by $$i$$. $$\frac{4\ -\ 3\ i}{-\ 4\ i}\ ×\ \frac{i}{i}=\frac{4\ i\ -\ 3\ i^2} { -\ 4\ i^2}$$$$i^2\ -\ 1$$, Then: $$\frac{4\ i\ -\ 3 \ i^2}{-\ 4\ i^2 }=\frac{4\ i\ -\ 3(-\ 1)}{-\ 4(-\ 1)}=\frac{4\ i\ +\ 3}{4}=\frac{4\ i}{4}\ +\ \frac{3}{4}=i\ +\frac{ 3}{4}$$ 57- Choice E is correct The correct answer is $$0, -\ 2 , -\ 3$$Frist factor the function:$$f(x)=x^3\ +\ 5\ x^2\ +\ 6\ x\ =x\ (x\ +\ 2)\ (x\ +\ 3)$$To find the zeros, $$f(x)$$ should be zero.$$f(x)=x \ (x\ +\ 2)\ (x\ +\ 3)=0$$Therefore, the zeros are:$$x=0$$$$(x\ +\ 2)=0 ⇒ x= -\ 2$$$$(x\ +\ 3)=0 ⇒ x= -\ 3$$ 58- Choice D is correct The correct answer is $$24$$Let $$x$$ be the length of AB, then:$$12=\frac{x×3}{2}→x=8$$The length of AC $$=\sqrt{6^2\ +\ 8^2 }=\sqrt{100}=10$$The perimeter of ∆ABC$$=6\ +\ 8\ +\ 10=24$$ 59- Choice D is correct The correct answer is $$12.5$$ feetUse formula of rectangle prism volume.$$V =$$ (length) (width) (height) $$⇒ 5000 = (20) \ (20)$$ (height) $$⇒$$ height $$= 5000 \ ÷ \ 400 = 12.5$$ 60- Choice B is correct The correct answer is $$25 \ x^8$$$$y=(-\ 5 \ x^4)^2=(-\ 5^2) \ (x^4)^2= 25 \ x^8$$

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