 ## Full Length THEA Mathematics Practice Test

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THEA Mathematics
Practice Test 1

(Calculator)   50 questions
Total time for this section: (Students have 5 hours to complete all three sections of the THEA Test)
You may use a non-programmable calculator for this test.

1-  If $$8\ x\ -\ 8\ = 24$$ , what is the value of $$6\ x\ - \ 4$$?
(A) $$15$$
(B) $$20$$
(C) $$25$$
(D) $$30$$
2- If $$x\ +\ y=0,4\ x\ -\ 2 \ y \ =24$$ which of the following ordered pairs$$(x,y)$$ satisfies both equations?
(A) $$(4, 3)$$
(B) $$( 5, 4 )$$
(C) $$( 4 , -\ 4 )$$
(D) $$( 4 , -\ 6 )$$
3- If $$f(x)= 3 \ x \ + 4\ (\ x \ +\ 1 ) \ +\ 2$$ then $$f(4 \ x \ ) = ?$$
(A) $$28 \ x \ + \ 6$$
(B) $$16 \ x \ - \ 6$$
(C) $$25 \ x \ + \ 4$$
(D) $$12 \ x \ + \ 3$$
4-

A line in the $$x \ y$$-plane passes through origin and has a slope of  $$\frac{1}{3}$$. Which of the following points lies on the line?

(A) $$(2,1)$$
(B) $$(4,1)$$
(C) $$(9,3)$$
(D) $$(6,3)$$
5- Which of the following is equivalent to $$(3 \ n^2 \ + \ 2 \ n \ + \ 6\ )\ -\ (2\ n^2\ -\ 4$$)?
(A) $$n \ + \ 4 \ n^2$$
(B) $$n^2 \ - \ 3$$
(C) $$n^2 \ + \ 2 \ n\ + \ 10$$
(D) $$n \ + \ 2$$
6- If $$( a \ x \ + \ 4) \ (b\ x \ + \ 3)=10\ x^2\ \ +\ c\ x \ +\ 12$$ for all values of  $$x$$ and $$a\ + \ b = 7$$ , what are the two possible values for $$c$$ ?
(A) $$22 \ , \ 21$$
(B) $$20 \ , \ 22$$
(C) $$23 \ , \ 26$$
(D) $$24 \ , \ 23$$
7- If $$x ≠ - \ 4$$ and $$x ≠ 5$$ , which of the following is equivalent $$\frac{1}{{\frac{1}{x\ -\ 5}}\ +\ {\frac{1}{x\ +\ 4}}}$$ ?
(A) $$\frac {( \ x \ - \ 5 \ ) \ ( \ x \ + \ 4 \ )}{( \ x \ - \ 5 \ ) \ + \ ( \ x \ + \ 4 \ )}$$
(B) $$\frac{( \ x \ + \ 4 \ ) \ + \ ( \ x \ - \ 5 \ ) }{( \ x \ + \ 4 \ )( \ x \ - \ 5 \ )}$$
(C) $$\frac{( \ \ x \ + \ 4 \ ) \ ( \ x \ - \ 5 \ )}{( \ x \ + \ 4 \ ) \ - \ ( \ x \ + \ 5 \ )}$$
(D) $$\frac{( \ x \ + \ 4 \ ) \ + \ ( \ x \ - \ 5 \ )}{( \ x \ + \ 4 \ ) \ - \ ( \ x \ - \ 5 \ )}$$
8-

In the $$x \ y$$ -plane, if $$(0 \ , \ 0)$$ is a solution to the system of inequalities above, which of the following relationships between $$a$$  and $$b$$ must be true?
$$y \ < \ a\ - \ x \ , \ y\ > \ x \ + \ b$$

(A) $$a \ < \ b$$
(B) $$a \ > \ b$$
(C) $$a \ = \ b$$
(D) $$a \ = \ b \ + \ a$$
9- Which of the following points lies on the line that goes through the points $$(2 \ , \ 4)$$ and $$(4 \ , \ 5)$$ ?
(A) $$9 \ , \ 9$$
(B) $$9 \ , \ 6$$
(C) $$6 \ , \ 9$$
(D) $$6 \ , \ 6$$
10- Calculate $$f(5)$$ for the following function $$f$$
$$f(x)= \ x^2 \ - \ 3 \ x$$
(A) $$5$$
(B) $$10$$
(C) $$15$$
(D) $$20$$
11- John buys a pepper plant that is $$6$$ inches tall. With regular watering the plant grows $$4$$ inches a year. Writing John’s plant’s height as a function of time, what does the $$y-$$ intercept represen
(A) The $$y-$$intercept represents the rate of grows of the plant which is $$5$$ inches
(B) The $$y-$$intercept represents the starting height of $$6$$ inches
(C) The $$y-$$intercept represents the rate of growth of plant which is $$3$$ inches per year
(D) There is no $$y-$$intercept
12- If $$\frac{4}{x}=\frac{12}{x \ - \ 8}$$what is the value of $$\frac{x}{2}$$ ?
(A) $$1$$
(B) $$3$$
(C) $$-2$$
(D) $$2$$
13- If $$4\ n \ - \ 3 \ ≥ \ 1$$ , what is the least possible value of $$4 \ n\ + \ 3$$ ?
(A) $$3$$
(B) $$4$$
(C) $$7$$
(D) $$9$$
14- What is the ratio of the minimum value to the maximum value of the following function?
$$f(x)=- \ 3 \ + \ 1$$         $$- \ 2 \ ≤\ x\ ≤ \ 3$$
(A) $$\frac{7}{8}$$
(B) $$-\frac{8}{7}$$
(C) $$-\frac{ 7}{8}$$
(D) $$\frac{8}{7}$$
15- The equation $$x^2= 4 \ x \ - \ 3$$  has how many distinct real solutions?
(A) $$0$$
(B) $$1$$
(C) $$2$$
(D) $$4$$
16- The table above shows the distribution of age and gender for $$30$$ employees in a company. If one employee is selected at random, what is the probability that the employee selected be either a female under age $$45$$ or a male age $$45$$ or older?
 Gender Under $$45$$ $$45$$ or older total Male $$12$$ $$6$$ $$18$$ Female $$5$$ $$7$$ $$12$$ Total $$17$$ $$13$$ $$30$$
(A) $$\frac{5}{6}$$
(B) $$\frac{5}{30}$$
(C) $$\frac{6}{30}$$
(D) $$\frac{11}{30}$$
17- In the triangle below, if the measure of angle A  is $$37$$ degrees, then what is the value of $$y$$ ? (figure is NOT drawn to scale) (A) $$62$$
(B) $$70$$
(C) $$78$$
(D) $$86$$
18- If $$y \ = n \ x \ + \ 2$$ , where is a constant, and when  $$x = 6 \ , \ y = 14$$ , what is the value of  when  $$x=10$$?
(A) $$10$$
(B) $$12$$
(C) $$18$$
(D) $$22$$
19- Which of the following numbers is NOT a solution of the inequality $$2 \ x \ - \ 5 \ ≥ \ 3 \ x \ - \ 1$$ ?
(A) $$-\ 2$$
(B) $$-\ 4$$
(C) $$-\ 5$$
(D) $$-\ 8$$
20-  If the following equations are true, what is the value of $$x$$
$$a=\sqrt{3} \\ 4\ a =\sqrt{4\ x}$$
(A) $$2$$
(B) $$3$$
(C) $$6$$
(D) $$12$$
21- If $$\sqrt{4 \ m\ - \ 3}=m$$ what is (are) the value(s) of $$m$$?
(A) $$0$$
(B) $$1$$
(C) $$1\ , 3$$
(D) $$- \ 1\ , 3$$
22- If $$\frac{6}{5}\ y$$ = $$\frac{3}{2}$$, what is the value of $$y$$?
(A) $$\frac{5}{6}$$
(B) $$\frac{5}{4}$$
(C) $$\frac{4}{5}$$
(D) $$\frac{3}{2}$$
23- Jack walks $$30$$ meters in $$15$$ seconds. If he walks at this same rate, which of the following is the distance he will walk in $$6$$ minutes?
(A) $$720$$ m
(B) $$360$$ m
(C) $$180$$ m
(D) $$100$$ m
24- If the function $$g$$$$(x)$$ has three distinct zeros, which of the following could represent the graph of $$g$$$$(x)$$?
(A) (B) (C) (D) 25- If the positive integer $$x$$ leaves a remainder of $$2$$ when divided by $$8$$ , what will the remainder be when $$x \ + \ 9$$ is divided by $$8$$?
(A) $$3$$
(B) $$2$$
(C) $$1$$
(D) $$0$$
26- The cost of using a car is $$0.35$$  per minutes. Which of the following equations represents the total cost $$c$$ , in dollars, for $$h$$ hours of using the car?
(A) $$c= \frac{60 \ h}{035}$$
(B) $$c=\frac{0.35}{60 \ h}$$
(C) $$c=0.35 \ ( \ 60 \ h)$$
(D) $$c=60 \ h \ + \ 0.35$$
27- Mary’s average score after $$4$$ tests is $$90$$ . What score on the $$5^{th}$$ test would bring Mary’s average up to exactly $$92$$  ?
(A) $$102$$
(B) $$100$$
(C) $$98$$
(D) $$94$$
28- The equation $$x^2= 5 \ x \ - \ 4$$ has how many distinct real solutions?
(A) $$0$$
(B) $$1$$
(C) $$2$$
(D) $$3$$
29- A library has $$840$$ books that include Mathematics, Physics, Chemistry, English and History.
What is the product of the number of Mathematics and number of English books in the library? (A) $$31,752$$
(B) $$26,460$$
(C) $$21,168$$
(D) $$17,640$$
30- What are the values of angle $$α$$ and  $$β$$ in the graph? (A) $$90˚$$, $$54˚$$
(B) $$120˚$$, $$36˚$$
(C) $$120˚$$, $$45˚$$
(D) $$108˚$$, $$54˚$$
31- The librarians decided to move some of the books in the Mathematics section to Chemistry section. How many books are in the Chemistry section if now $$γ=\frac {2}{5}\ α$$ ? (A) $$80$$
(B) $$120$$
(C) $$150$$
(D) $$180$$
32- In the $$x \ y$$  -plane, the line determined by the points $$( 6 \ , \ m )$$ and  $$( m \ , \ 12)$$  passes through the origin. Which of the following could be the value of $$m$$ ?
(A) $$\sqrt{6}$$
(B) $$12$$
(C) $$6 \sqrt{2}$$
(D) $$9$$
33- A function $$g(3)=5$$ and  $$g(5)=4$$ . A function $$f(5)=2$$ and $$f(4)=6$$ . What is the value of $$f(g(5))$$?
(A) $$5$$
(B) $$6$$
(C) $$7$$
(D) $$8$$
34- What is the area of the following equilateral triangle if the side AB =$$8$$ cm ? (A) $$16\sqrt{3 }$$ cm$$^2$$
(B) $$8\sqrt{3 }$$ cm$$^2$$
(C) $$\sqrt{3 }$$ cm$$^2$$
(D) $$8$$ cm$$^2$$
35- A function  $$g(x)$$ satisfies  $$g(4)= 5$$ and $$g(5)=8$$ . A function $$f(x)$$  satisfies $$f(5)= 18$$ and $$f(8)=32$$ . What is the value of $$f(g(5))$$?
(A) $$12$$
(B) $$22$$
(C) $$32$$
(D) $$42$$
36- In the system of equations above, $$c$$  and $$d$$  are constants. For which of the following values of $$c$$  and $$d$$ does the system of equations have exactly two real solutions?
$$y=c \ x^2 \ + \ d \ ,\ y=5$$
(A) $$c=- \ 2 \ , \ d=6$$
(B) $$c= \ 1 \ , \ d=7$$
(C) $$c= -\ 3 \ , \ d=4$$
(D) $$c= \ 5 \ , \ d=5$$
37- From the figure below, which of the following must be true? (figure not drawn to scale) (A) $$y = z$$
(B) $$y = 5 \ x$$
(C) $$y \ ≥ \ x$$
(D) $$y \ + \ 4 \ x=z$$
38- Point A lies on the line with equation $$y \ - \ 3=2\ ( \ x \ + \ 5)$$. If the $$x$$-coordinate of A is $$8$$, what is the $$y$$-coordinate of A ?
(A) $$14$$
(B) $$16$$
(C) $$22$$
(D) $$29$$
39- If $$|a| \ < \ 1$$ , then which of the following is true? $$(b \ > \ 0 )$$?
I.  $$\ – \ b \ < \ b a < \ b$$
II.$$\ - \ a \ < \ a \ ^2 \ < \ a$$        if     $$a \ < \ 0$$
III.$$\ - \ 5 \ < 2 \ a \ - \ 3 \ < \ - \ 1$$
(A) I only
(B) III only
(C) I and III only
(D) I , II and III only
40- In the equation above, if $$c$$ is negative and $$d$$ is positive, which of the following must be true?
$$\frac { c \ - \ d }c=\ a$$
(A) $$a \ < \ 1$$
(B) $$a=0$$
(C) $$a \ > \ 1$$
(D) $$a \ < \ - \ 1$$
41- If $$m$$ is a positive integer and $$\sqrt { 2 \ m \ + \ 48 }=m$$ , what is the value of  $$m$$ ?
(A) $$4$$
(B) $$8$$
(C) $$12$$
(D) $$16$$
42- $$f(a)=| \ 11 \ - \ a^ 2 \ |$$ , where $$x$$ is a positive integer. If $$f(a)=20$$ , what is the value of $$a$$ that satisfies the equation above?
(A) $$3$$
(B) $$4$$
(C) $$5$$
(D) $$6$$
43- The price of a car was $$20,000$$ in 2014, $$16,000$$ in 2015 and $$12,800$$ in 2016. What is the rate of depreciation of the price of car per year?
(A) $$15\%$$
(B) $$20\%$$
(C) $$25\%$$
(D) $$30\%$$
44- 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 $$27$$ cm , what is the volume of the box?
(A) $$81$$ cm$$^3$$
(B) $$162$$ cm$$^3$$
(C) $$243$$ cm$$^3$$
(D) $$729$$ cm$$^3$$
45- How many liters of a solution that is $$30 \%$$  salt must be added to $$3$$  liters of a solution that is $$75 \%$$  salt to obtain a $$39 \%$$  salt solution?
(A) $$24$$
(B) $$18$$
(C) $$12$$
(D) $$8$$
46- A boat sails $$40$$ miles south and then $$30$$  miles east. How far is the boat from its start point?
(A) $$45$$ miles
(B) $$50$$ miles
(C) $$60$$ miles
(D) $$70$$ miles
47- The sum of four numbers is $$600$$ . One of the numbers, $$x$$  is $$50 \%$$ more than the sum of the other three numbers. What is the value of $$x$$  ?
(A) $$240$$
(B) $$360$$
(C) $$420$$
(D) $$540$$
48-  The profit in dollars from a carwash is given by the function $$P(x)=\frac{40\ a \ - \ 500 }{ a } \ + \ b$$ , where $$a$$ is the number of cars washed and $$b$$ is a constant. If  $$50$$ cars were washed today for a total profit of $$600$$ , what is the value of $$b$$ ?
(A) $$450$$
(B) $$540$$
(C) $$750$$
(D) $$570$$
49- A rope weighs $$600$$ grams per meter of length. What is the weight in kilograms of $$12.2$$ meters of this rope? ( $$1$$  kilograms = $$1000$$ grams)
(A) $$0.0732$$
(B) $$0.732$$
(C) $$7.32$$
(D) $$73.20$$
50-

In the following figure, ABCD is a rectangle, and E and F are points on AD and DC, respectively and DE $$=4$$ and DF $$=3$$. The area of ∆BED is $$16$$, and the area of  ∆BDF is $$18$$. What is the perimeter of the rectangle? (A) $$40$$
(B) $$75$$
(C) $$90$$
(D) $$95$$
 1- Choice B is correct The correct answer is $$20$$Add $$8$$ both sides of the equation $$8\ x \ - \ 8=24$$ gives $$8 \ x =24 \ + \ 8=32$$.Dividing each side of the equation $$8 \ x=32$$ by $$8$$ gives $$x=4$$. Substituting $$4$$ for $$x$$ in the expression $$6\ x \ - \ 4$$ gives $$6\ (4)\ - \ 4=20$$. 2- Choice C is correct The correct answer is $$( 4 , -\ 4 )$$ Method 1: Plug in the values of $$x$$ and $$y$$ provided in the options into both equationA.$$(4,3)$$ $$x \ + \ y=0→4 \ + \ 3≠0$$B.$$(5,4)$$ $$x \ + \ y=0→5 \ + 4 ≠0$$C.$$(4,- \ 4)$$ $$x \ + \ y=0→4 \ + \ ( -\ 4)=0$$D.$$(4,-\ 6)$$ $$x \ + \ y=0→4 \ + \ (-\ 6)=0$$Only option C is correct.Method 2: Multiplying each side of $$x \ + \ y=0$$ by $$2$$ gives $$2 \ x \ + \ 2 \ y=0$$. Then, adding the corresponding side of $$2 \ x \ + \ 2 \ y=0$$ and $$4 \ x \ - \ 2 \ y=24$$ gives $$6 \ x=24$$ .Dividing each side of $$6 \ x=24$$ by $$6$$ gives $$x=4$$ .Finally, substituting $$4$$ for $$x$$ in $$x \ + \ =0$$ , or $$y= - \ 4$$. Therefore, the solution to the given system of equations is $$( 4,-\ 4)$$ 3- Choice A is correct The correct answer is  $$28 \ x \ + \ 6$$If $$f(x)= 3 \ x \ + 4\ (\ x \ +\ 1 ) \ +\ 2$$ , then find $$f(4 \ x)$$  by substituting   $$4 \ x$$ for  every $$x$$ in the function.This gives:$$f(4\ x) = 3 \ (4\ x \ ) \ + \ 4 \ (4 \ x \ + \ 1 \ ) \ + \ 2$$,It simplifies to: $$f(4 \ x \ )=3 \ (4 \ x \ ) \ + \ 4 \ (4 \ x \ + \ 1 ) \ + \ 2=12 \ x \ + \ 16 \ x \ + \ 4 \ + \ 2=28 \ x \ + \ 6$$ 4- Choice C is correct The correct answer is $$(9,3)$$First, find the equation of the line.All lines through the origin are of the form $$y=m$$so the equation is $$y=\frac {1}{3}\ x$$ .Of the given choices, only choice C $$(9 \ , \ 3)$$ , satisfies this equation:$$y=\frac{1}{3} \ x→\ 3 =\frac {1}{3} \ (9)=3$$ 5- Choice C is correct The correct answer is $$n^2 \ + \ 2 \ n \ + \ 10$$$$(3 \ n ^2 \ + \ 2 \ n \ + \ 6 ) \ - \ (2 \ n^2 \ - \ 4 )$$Add like terms together:$$3 \ n^2 \ - \ 2 \ n^2=n^2$$$$2 \ n$$ doesn’t have like terms.$$6 \ - \ ( - \ 4)=10$$Combine these terms into one expression to find the answer:$$n^2 \ + \ 2 \ n \ + \ 10$$ 6- Choice C is correct The correct answer is $$23$$ , $$26$$You can find the possible values of $$a$$ and $$b$$ in $$(a \ x \ + \ 4) \ (b \ x \ + \ 3)$$ by using the given equation $$a \ + \ b=7$$and finding another equation that relates the variables $$a$$ and $$b$$.Since $$(a \ x \ + \ 4)\ (b \ x \ + \ 3)=10 \ x^2 \ + \ c \ x \ + \ 12$$,expand the left side of the equation to obtain $$a \ b \ x^2 \ + \ 4 \ b \ x \ + \ 3 \ a \ x \ + \ 12=10 \ x^2 \ + \ c \ x \ + \ 12$$Since $$a \ b$$ is the coefficient of $$x^2$$ on the left side of the equation and $$10$$ is the coefficient of $$x^2$$ on the right side of the equation,it must be true that $$a \ b=10$$The coefficient of $$x$$ on the left side is $$4 \ b \ + \ 3 \ a$$ and the coefficient of $$x$$ in the right side is $$c$$.Then: $$4 \ b \ + \ 3 \ a=c$$$$a \ + \ b=7$$, then: $$a=7 \ - \ b$$Now, plug in the value of a in the equation $$a \ b=10$$.Then: $$a \ b=10→(7 \ - \ b) \ b=10→7 \ b \ - \ b^2=10$$Add $$\ - \ 7 \ b \ + \ b^2$$ both sides.Then: $$b^2 \ - \ 7 \ b \ + \ 10=0$$Solve for $$b$$ using the factoring method.$$b^2 \ - \ 7 \ b \ + \ 10=0→(b \ - \ 5)\ (b \ - \ 2)=0$$Thus, either $$b=2$$ and $$a = 5$$, or $$b = 5$$ and $$a = 2$$.If $$b = 2$$ and $$a = 5$$,then $$4 \ b \ + \ 3 \ a=c→4\ ( \ 2 \ ) \ + \ 3 \ ( \ 5\ )=c→c=23$$ 7- Choice A is correct The correct answer is $$\frac{( \ x \ - \ 5 \ ) \ ( \ x \ + \ 4 \ )}{( \ x \ - \ 5 \ ) \ + \ ( \ x \ + \ 4 \ )}$$ To rewrite $$\frac{1}{{\frac{1}{x\ -\ 5}}\ +\ {\frac{1}{x\ +\ 4}}}$$,first simplify $${\frac{1}{x\ - \ 5}}\ + \ {\frac{1}{x\ +\ 4}}$$. $${\frac{1}{(x \ - \ 5)}+\frac{1}{( x \ + \ 4)}=\frac{1 \ (x \ + \ 4)}{(x \ - \ 5)(x \ + \ 4)}+\frac{1\ (x \ - \ 5 )}{(x \ + \ 4)\ (x \ - \ 5)}}=\frac {(x \ + \ 4) \ + \ (x \ - \ 5)}{(x \ + \ 4)\ (x \ - \ 5)}$$Then$$\frac{1}{{\frac{1}{x\ -\ 5}}\ +\ {\frac{1}{x\ +\ 4}}}$$=$$\frac {1}{\frac{( \ x \ + \ 4 \ ) \ + \ ( \ x \ - \ 5 \ ) }{( \ x \ + \ 4 \ )\ ( \ x \ - \ 5 \ )}}$$=$$\frac{( \ x \ - \ 5 \ ) \ ( \ x \ + \ 4 \ )}{( \ x \ - \ 5 \ ) \ + \ ( \ x \ + \ 4 \ )}$$ . (Remember,$$\frac{1}{\frac{1}{x }}=x$$)This result is equivalent to the expression in choice A. 8- Choice B is correct The correct answer is $$a \ > \ b$$Since $$(0 \ , \ 0)$$ is a solution to the system of inequalities, substituting $$0$$ for $$x$$ and $$0$$ for $$y$$  in the given system must result in two true inequalities.After this substitution, $$y \ < \ a \ - \ x$$ becomes $$0 \ < \ a$$ , and  $$y \ > \ x \ + \ b$$ becomes  $$0 \ > \ b$$.Hence, $$a$$  is positive and $$b$$ is negative.Therefore, $$a \ > \ b$$. 9- Choice D is correct The correct answer is $$6$$ , $$6$$First find the slope of the line using the slope formula.$$m=\frac {y_2 \ - \ y_1}{x_2 \ - \ x_1}$$Substituting in the known information.$$(\ x_1 \ , \ y_1 )=(2 \ ,4 )$$$$( \ x_2 \ ,\ y_2 \ )=( \ 4 \ , \ 5 \ )$$$$m=\frac {5 \ - \ 4}{4 \ - \ 2}=\frac{1}{2}$$Now the slope to find the equation of the line passing through these points.$$y=m \ x \ + \ b$$Choose one of the points and plug in the values of $$x$$ and $$y$$ in the equation to solve for $$b$$. Let’s choose point $$(\ 4 \ , \ 5)$$. Then:$$\ y=m \ x \ + \ b→5=\frac{1}{2} \ (4) \ + \ b→5=2 \ + \ b→b=5 \ - \ 2=3$$The equation of the line is: $$y=\frac{1}{2} \ x \ + \ 3$$Now, plug in the points provided in the choices into the equation of the line.$$(\ 9\ ,\ 9\ )$$ $$y=\frac {1}{2} \ + \ 3→9=\frac{1}{2}(\ 9 ) \ + \ 3→9=7.5$$ This is NOT true.$$(\ 9\ ,\ 6\ )$$ $$y=\frac{1}{2} x \ + \ 3→6=\frac{1}{2}\ (\ 9) \ + \ 3→6=7.5$$ This is NOT true.$$(\ 6\ , \ 9\ )$$ $$y=\frac{1}{2} x \ + \ 3→9=\frac{1}{2}\ ( 6) \ + \ 3→9=6$$ This is NOT true.$$(\ 6\ , \ 6\ )$$ $$y=\frac{1}{2} x \ + \ 3→6=\frac{1}{2}( \ 6) \ + \ 3→6=6$$ This is true!Therefore, the only point from the choices that lies on the line is $$(6 , 6)$$ . 10- Choice B is correct The correct answer is $$10$$The input value is $$5$$. Then: $$x = 5$$$$f(x) = x^2\ - \ 3 \ x → f(5) = 5^2 \ - \ 3 \ (5) = 25 \ - \ 15 = 10$$ 11- Choice B is correct The correct answer is  The $$y-$$intercept represents the starting height of $$6$$ inchesTo solve this problem, first recall the equation of $$a$$ line: $$y=m \ x \ + \ b$$Where $$m=$$ slope$$y=y-$$interceptRemember that slope is the rate of change that occurs in a function and that the $$y-$$intercept is the $$y$$ value corresponding to $$x=0$$.Since the height of John’s plant is $$6$$ inches tall when he gets it. Time (or $$x$$) is zero. The plant grows $$4$$ inches per year.Therefore, the rate of change of the plant’s height is $$4$$. The $$y-$$intercept represents the starting height of the plant which is $$6$$ inches. 12- Choice C is correct The correct answer is $$- \ 2$$Multiplying each side of $$\frac{4}{x}=\frac{12}{x \ - \ 8}$$ by $$x \ (x \ - \ 8)$$ gives $$4 \ ( \ x \ - \ 8 \ )=12 \ ( \ x \ )$$,distributing the $$4$$ over the values within the parentheses yields $$x \ - \ 8=3 \ x$$ or $$x=- \ 4$$. Therefore, the value of  $$\frac{x}{2}$$=$$\frac{(- \ 4 )}{2}=- \ 2$$. 13- Choice C is correct The correct answer is $$7$$Adding $$6$$ to each side of the inequality $$4 \ n \ - \ 3 \ ≥ \ 1$$ yields the inequality $$4 \ n \ + \ 3 \ ≥ \ 7$$. Therefore, the least possible value of $$4 \ n \ + \ 3$$ is $$7$$. 14- Choice B is correct The correct answer is $$-\frac{8}{7}$$Since$$f(x)$$ is linear function with a negative slop, then when $$x=-\ 2$$ , $$f(x)$$ is maximum and when $$x=3$$ , $$f(x)$$ is minimum.Then the ratio of the minimum value to the maximum value of the function is:$$\frac{f(3)}{f(-2)}$$=$$\frac{- \ 3 \ ( \ 3 \ ) \ + \ 1 \ }{- \ 3 \ ( \ - \ 2) \ + \ 1}$$=$$\frac {- \ 8}{7}$$=$$-\frac {8}{7}$$ 15- Choice D is correct The correct answer is $$2$$ Method 1: There can be $$0$$, $$1$$, or $$2$$ solutions to a quadratic equation.In standard form, a quadratic equation is written as: $$a \ x^2 \ + \ b \ x \ + \ c=0$$For the quadratic equation, the expression $$b^2 \ - \ 4 \ a \ c$$ is called discriminant. If discriminant is positive, there are $$2$$ distinct solutions for the quadratic equation.If discriminant is $$0$$, there is one solution for the quadratic equation and if it is negative the equation does not have any solutions. To find number of solutions for $$x^2=4 \ x \ - \ 3$$, first, rewrite it $$a \ s \ x^2 \ - \ 4 \ x \ + \ 3=0$$.Find the value of the discriminant. $$b^2\ -\ 4 \ a \ c=( - \ 4)^2 \ - \ 4 \ (1) \ (3)=16 \ - \ 12=4$$Since the discriminant is positive, the quadratic equation has two distinct solutions. 16- Choice D is correct The correct answer is  $$\frac{11}{30}$$Of the $$30$$ employees, there are $$5$$ females under age $$45$$ and $$6$$ males age $$45$$ or older. Therefore, the probability that the person selected will be either a female under age $$45$$ or a male age $$45$$ or older is:$$\frac{5}{30} \ + \frac {6}{30}=\frac{11}{30}$$ 17- Choice D is correct The correct answer is $$86$$In the figure angle A is labeled $$(3 \ x \ - \ 2)$$ and it measures $$37$$. Thus, $$3 \ x \ - \ 2=37$$ and $$3 \ x=39$$ or $$x=13$$. That means that angle B, which is labeled $$(5 \ x)$$, must measure $$5 \ × \ 13=65$$.Since the three angles of a triangle must add up to $$180$$ , $$37 \ + \ 65 \ + \ y \ - \ 8=180$$ , then:$$y \ + \ 94=108→y=180 \ - \ 94=86$$ 18- Choice D is correct The correct answer is $$22$$Substituting $$6$$ for $$x$$ and $$14$$ for $$y$$ in $$y = n \ + \ 2$$ gives $$14=( \ n \ )\ ( \ 6 \ )+2$$ ,which gives $$n=2$$. Hence, $$y=2 \ x \ + \ 2$$. Therefore, when$$= 10$$ , the value of $$y$$ is$$y=( \ 2 \ ) \ ( \ 10 \ ) \ + \ 2 = 22$$. 19- Choice A is correct The correct answer is $$-\ 2$$Subtracting $$2 \ x$$ and adding $$5$$ to both sides of $$2 \ x \ - \ 5 \ ≥ \ 3 \ x \ - \ 1$$ gives $$- \ 4 \ ≥ \ x$$. Therefore, $$x$$ is a solution to $$2 \ x \ - \ 5 \ ≥ \ 3 \ x \ - \ 1$$ if and only if $$x$$ is less than or equal to $$- \ 4$$ and$$x$$ is NOT a solution to $$2 \ x \ - \ 5 \ ≥ \ 3 \ x \ - 1$$ if and only if $$x$$ is greater than $$- \ 4$$.Of the choices given, only $$- \ 2$$ is greater than $$- \ 4$$ and,therefore, cannot be a value of $$x$$. 20- Choice D is correct The correct answer is $$12$$Given the two equations,substitute the numerical value of $$a$$ into the second equation to solve for $$x\ . \ a=\sqrt{3}$$, $$4\ a=\sqrt{4 \ x}$$Substituting the numerical value for ainto the equation with $$x$$ is as follows.$$4 \ (\sqrt{3})=\sqrt{4 \ x}$$,From here, distribute the $$4$$ . $$4\sqrt{3}=\sqrt { \ 4 \ x}$$Now square both side of the equation.$$(4\sqrt{3})^2=(\sqrt{4 \ x})^2$$Remember to square both terms within the parentheses.Also, recall that squaring a square root sign cancels them out.$$4^2 \sqrt{3}^2=4 \ x$$ ,$$16 \ (3)=4 \ x$$ ,$$48=4 \ x$$ ,$$x=12$$ 21- Choice C is correct The correct answer is $$1 \ , \ 3$$First square both sides of the equation to get $$4 \ m-3=m^2$$Subtracting both sides by $$4 \ m \ - \ 3$$ gives us the equation $$\ m^2 \ - \ 4 \ m \ + \ 3=0$$Here you can solve the quadratic equation by factoring to get $$(\ m \ - \ 1) \ ( \ m \ - \ 3 )=0$$For the expression $$( \ m \ - \ 1) \ ( \ m \ - \ 3 )$$ to equal zero, $$m=1$$ or $$m=3$$ 22- Choice B is correct The correct answer is $$\frac{5}{4}$$To solve the equation for $$y$$ , multiply both sides of the equation by the reciprocal of  $$\frac{6}{5}$$ , which is  $$\frac{5}{6}$$ , this gives $$\frac{(5)}{(6)}×\frac {6}{5} y= \frac {3}{2} \ × \frac {(5)}{(6)}$$, which simplifies to $$y=\frac{15}{12}=\frac{5}{4}$$. 23- Choice A is correct The correct answer is $$720$$ mBecause Jack walks $$30$$ meters in $$15$$ seconds, and $$6$$ minutes is equal to $$360$$ seconds,use the proportion to solve. $$\frac{30\ meters}{15\ s ec}=\frac{x \ meters}{360 \ sec}$$The proportion can be simplified to $$\frac{30}{15}=\frac{x}{360}$$ then each side of the equation can be multiplied by $$360$$ , giving $$\frac{(360)(30)}{15}=x=720$$. Therefore, $$720$$ meters is the distance Jack will walk in $$6$$ minutes. 24- Choice C is correct A zero of a function corresponds to an $$x-$$intercept of the graph of the function in the $$x \ y-$$plane. Therefore, the graph of the function $$g ()$$, which has three distinct zeros, must have three $$x-$$intercepts.Only the graph in choice C has three $$x-$$intercepts. 25- Choice A is correct The correct answer is $$3$$The fastest way to find the answer is to pick numbers.Pick a number for that has a remainder of $$2$$ when divided by $$8$$, such as $$10$$. Increase the number you picked by $$9$$. In this case $$10 \ + \ 9 \ =19$$ .Now divide $$19$$ by $$8$$, which gives you remainder $$3$$. Therefore, the answer is $$3$$. 26- Choice C is correct The answer is  $$c=0.35 \ ( \ 60 \ h)$$$$0.35$$ per minute to use car.This per-minute rate can be converted to the hourly rate using the conversion $$1$$ hour = $$60$$ minutes, as shown below.$$\frac{0.35}{minute} \ × \frac {60 minutes}{1 hours}=\frac{ \ ( \ 0.35 \ × \ 60 \ )}{ hour}$$Thus, the car costs $$( \ 0.35 \ × \ 60 \ )$$ per hour.Therefore, the cost $$c$$ , in dollars, for h hours of use is $$c=( \ 0.35 \ × \ 60) \ h$$,Which is equivalent to $$c=0.35 \ (60 \ h)$$ 27- Choice B is correct The correct answer is $$100$$The best way to deal with changing averages is to use the sum.Use the old average to figure out the total of the first $$4$$ scores:Sum of first $$4$$ scores: $$(4) \ (90) = 360$$Use the new average to figure out the total she needs after the $$5^{th}$$ score:Sum of score: $$(5) \ (92) = 460$$ To get her sum from $$360$$ to $$460$$ ,Mary needs to score $$460 \ - \ 360=100$$. 28- Choice C is correct The correct answer is $$2$$To solve a quadratic equation, put it in the $${a \ x}^2 \ + \ b \ x \ + \ c=0$$ form, factor the left side,and set each factor equal to $$0$$ separately to get the two solutions.To solve $${x}^2=5 \ x \ - \ 4$$ , first, rewrite it as $${x}^2 \ - \ 5 \ x \ + \ 4=0$$.Then factor the left side: $${x}^2 \ - \ 5 \ x \ + \ 4=0$$ , $$(x \ - \ 4) \ (x \ - \ 1)=0$$$$x=1$$ Or $$x=4$$ , There are two solutions for the equation. 29- Choice A is correct The correct answer is $$31,752$$$$30\%$$ of the books are Mathematics books and $$15\%$$ of the books are English books. Thus, number of Mathematics books: $$0.3 \ × \ 840=252$$Number of English books: $$0.15 \ × \ 840=126$$The product of number of Mathematics and number of English books: $$252 \ × \ 126=31,752$$ 30- Choice D is correct The correct answer is $$108^\circ$$, $$54^\circ$$All central angles in a circle sum up to $$360$$ degrees.Thus, the angle $$α$$  is: $$0.3 \ × \ 360=108^\circ$$ 31- Choice B is correct The correct answer is $$120$$According to the graph, $$50\%$$ of the books are in the Mathematics and Chemistry sections. Therefore, there are $$420$$ books in these two sections.$$0.50 \ × \ 840 = 420$$$$γ \ + \ α=420$$, and $$γ=\frac{2}{5} \ α$$Replace$$γ$$ by $$\frac{2}{5} \ α$$ in the first equation.$$γ \ + \ α=420 → \frac{2}{5} α \ + \ α=420 → \frac{7}{5} α=420$$ → multiply both sides by $$\frac{5}{7}$$$$\frac{(5)}{(7)} \frac{7}{5} \ α = 420 \ × \frac{(5)}{(7)} → α = \frac{420 \ × \ 5}{7}=300$$$$α=300 → γ = \frac{2}{5}\ α → γ = \frac {2}{5} \ × \ 300=120$$There are $$120$$ books in the Chemistry section. 32- Choice C is correct The correct answer is $$6 \sqrt{2}$$The line passes through the origin, $$(6 \ , \ m)$$ and $$(m \ , \ 12)$$.Any two of these points can be used to find the slope of the line. Since the line passes through $$(0 \ , \ 0)$$ and $$(6 \ , \ m)$$,the slope of the line is equal to$$\frac{m \ - \ 0}{6 \ - \ 0}$$=$$\frac{m}{6}$$.Similarly, since the line passes through $$(0 \ , \ 0)$$ and $$(m \ , \ 12)$$ , the slope of the line is equal to  $$\frac{12 \ - \ 0}{m \ - \ 0}$$=$$\frac{12}{m}$$.since each expression gives the slope of the same line, it must be true that $$\frac{m}{6}=\frac{12}{m}$$Using cross multiplication gives$$\frac{m}{6}=\frac{12}{m}→m^2=72→m=± \sqrt{72}=± \sqrt{36 \ × \ 2}=± \sqrt{36} \ × \sqrt{2}=± \ 6 \sqrt{2}$$ 33- Choice B is correct The correct answer is $$6$$It is given that $$g(5)=4$$. Therefore, to find the value of $$f(g(5))$$, then $$f(g(5))=f(4)=6$$ 34- Choice A is correct The correct answer is  $$16\sqrt{3 }$$ cm$$^2$$Area of the triangle is: $$\frac{1}{2}$$ AD$$×$$BC and AD is perpendicular to BC.Triangle ADC is a $$30^° \ - \ 60^° \ - \ 90^°$$ right triangle. The relationship among all sides of right triangle $$30^° \ - \ 60^° \ - \ 90^°$$ is provided in the following triangle: In this triangle, the opposite side of $$30^°$$ angle is half of the hypotenuse.And the opposite side of $$60^°$$ is opposite of  $$30^° \ × \sqrt{3}$$CD = $$4$$ , then AD = $$4 \ ×\sqrt{3}$$Area of the triangle ABC is :$$\frac {1}{2}$$ AD×BC =$$\frac {1}{2} 4\sqrt {3} \ × \ 8=16\sqrt{3}$$ 35- Choice C is correct The correct answer is $$32$$It is given that $$g(5)=8$$. Therefore, to find the value of $$f(g(5) )$$, substitute $$8$$ for $$g(5)$$.$$f(g(5) )=f(8)=32$$. 36- Choice A is correct The correct answer is  $$c=- \ 2\ ,\ d=6$$Substituting $$5$$ for yin $$y=c \ x^2 \ + \ d$$ gives $$5=c \ x^2 \ + \ d$$ which can be rewritten as $$5 \ - \ d=c \ x^2$$ .Since $$y = 5$$ is one of the equations in the given system, any solution $$x$$ of $$5 \ - \ d=c \ x^2$$ corresponds to the solution $$(x \ , \ 5)$$ of the given system. Since the square of a real number is always nonnegative, and a positive number has two square roots,the equation $$5 \ - \ d=c \ x^2$$ will have two solutions for $$x$$ if and only if $$(1) \ c \ > \ 0$$ and $$d \ < \ 5$$ or $$(2) \ c \ < \ 0$$ and $$d \ > \ 5$$. Of the values for cand dgiven in the choices, only $$c=- \ 2$$, $$d=6$$ satisfy one of these pairs of conditions.Alternatively, if $$c=- \ 2$$ and $$d=6$$ , then the second equation would be$$y=- \ 2 \ x^2 \ + \ 6$$The equation above has two real answer. 37- Choice D is correct The correct answer is  $$y \ + \ 4 \ x=z$$$$x$$ and $$z$$ are colinear. $$y$$ and $$5 \ x$$ are colinear. Therefore,$$x \ + \ z=y \ + \ 5 \ x$$ ,subtractxfrombothsides,then,  $$z=y \ + \ 4 \ x$$ 38- Choice D is correct The correct answer is $$29$$Here we can substitute $$8$$ for $$x$$ in the equation.Thus, $$y \ - \ 3=2 \ (8 \ + \ 5)$$ , $$- \ 3=26$$Adding $$3$$ to both side of the equation:$$y=26 \ + \ 3$$ , $$y=29$$ 39- Choice C is correct The correct answer is I and III onlyLet’s review the options:I. $$|a| \ < \ 1→- \ 1 \ < \ a \ < \ 1$$Multiply all sides by $$b$$. Since, $$b \ > \ 0→- \ b \ < \ b \ a \ < \ b$$II. Since , $$- \ 1 \ < \ a \ < \ 1$$ , and $$a \ < \ 0→- \ a \ > \ a^2 \ > \ a$$  (plug in $$\frac{- \ 1}{2}$$, and check!)III. $$- \ 1 \ < \ a \ < \ 1$$ ,multiply ll sides by $$2$$ , then: $$- \ 2 \ < \ 2 \ a \ < \ 2$$ , subtract $$3$$ from all sides,then:$$- \ 2 \ - \ 3 \ < \ 2 a \ - \ 3 \ < \ 2 \ - \ 3→- \ 5 \ < \ 2 \ a \ - \ 3 \ < \ - \ 1$$I and III are correct. 40- Choice C is correct The correct answer is  $$a \ > \ 1$$The equation can be rewritten as$$c \ - \ d=a \ c$$ →(divide both sides by $$c$$ ) $$1 \ - \frac {d}{c}=a$$ ,since $$c \ < 0$$ and $$d \ > \ 0$$ , the value of $$- \frac{d}{c}$$ is positive. Therefore, $$1$$ plus a positive number is positive. $$a$$  must be greater than $$1$$. $$a \ > \ 1$$ 41- Choice B is correct The correct answer is $$8$$Squaring both sides of the equation gives $$2 \ m \ + \ 48=m^2$$Subtracting both sides by $$2 \ m \ + \ 48$$ gives us the equation $$m^2 \ - \ 2 \ m \ - \ 48=0$$Here you can solve the quadratic by factoring to get $$(m \ - \ 8) \ (m \ + \ 6)=0$$For the expression $$(m \ - \ 8) \ (m \ + \ 6)$$ to equal zero, $$m=8$$  or  $$m=- \ 6$$Since $$m$$ is a positive integer, $$8$$ is the answer. 42- Choice A is correct The correct answer is $$3$$Since we are dealing with an absolute value, $$f(a)=20$$ means that either $$11 \ - \ a^2=20$$  or $$11 \ - \ a^2=- \ 20$$Let’s start with the positive value $$(20)$$ and see what we get.  If $$11 \ - \ a^2=20$$ , then $$a^2=9$$Taking the square root, we get $$a=3$$ or $$- \ 3$$On the other hand, if $$11 \ - \ a^2=- \ 20$$,  then $$a=\sqrt {- \ 31}$$Notice that the question states that $$a$$ is a positive integer, therefore the answer is $$3$$. 43- Choice B is correct The correct answer is $$20\%$$ Use this formula: Percent of Change$$\frac{New Value \ - \ Old Value}{Old Value}×100\%$$$$\frac{16000 \ - \ 20000}{20000} \ × \ 100\%=20\%$$ and $$\frac{12800 \ - \ 16000}{16000} \ × \ 100 \%=20\%$$ 44- Choice D is correct The correct answer is $$729$$ cm$$^3$$If the length of the box is $$27$$ , then the width of the box is one third of it, $$9$$ , and the height of the box is $$3$$ (one third of the width). The volume of the box is:$$V = lwh = (27) \ (9) \ (3) = 729$$ 45- Choice C is correct The correct answer is $$12$$Let $$x$$ represent the number of liters of the $$30\%$$ solution. The amount of salt in the $$30\%$$ solution $$(0.30\ x)$$ plus the amount of salt in the $$75\%$$ solution $$(0.75) \ × ( \ 5 )$$ must be equal to the amount of salt in the $$39\%$$ mixture $$(0.39 \ × \ (x \ + \ 5))$$.Write the equation and solve for $$x$$. $$0.30 \ x \ + \ 0.75 \ ( 3) = 0.39 \ (x \ + \ 3) →0.30 \ x \ + \ 2.25 = 0.39 \ x \ + \ 1.17 → 0.39 \ x \ - \ 0.30 \ x = 2.25 \ - \ 1.17→0.09 \ x = 1.08 → x = \frac{1.08}{0.09}=12$$ 46- Choice B is correct The correct answer is $$50$$ melisUse the information provided in the question to draw the shape.Use Pythagorean Theorem: $$a^2 \ + \ b^2=c^2$$$$40^2 \ + \ 30^2$$= $$c^2$$⇒ $$1600 \ + \ 900$$ = $$c^2$$ ⇒ $$2500$$ = $$c^2$$ ⇒ $$c=50$$ 47- Choice B is correct The correct answer is $$360$$One of the four numbers is $$x$$ ;let the other three numbers be $$y$$ , $$z$$ and $$w$$.Since the sum of four numbers is $$600$$ , the equation $$x \ + \ y \ + \ z \ + \ w = 600$$ is true.The statement that $$x$$ is $$50\%$$ more than the sum of the other three numbers can be represented as $$x = 1.5 \ (y \ + \ w)$$ or $$\frac{x}{1.5} = y \ + \ z \ + \ w → \frac{2x}{3} = y \ + \ z \ + \ w$$Substituting the value $$y \ + \ z \ + \ w$$ in the equation $$x \ + \ y \ + \ z \ + \ w=600$$gives $$x \ + \frac {2 \ x}{3} = 600 → \frac{5 \ x}{3} = 600 → 5 \ x = 1,800 → x = \frac{1,800}{5} = 360$$ 48- Choice D is correct The correct answer is $$570$$This is a simple matter of substituting values for variables.We are given that the $$50$$ cars were washed today, therefore we can substitute that for $$a$$.Giving us the expression $$\frac{40\ (50)\ -\ 500}{50} \ + \ b$$We are also given that the profit was $$600$$, which we can substitute for $$f(a)$$.Which gives us the equation $$600= \frac{40\ (50)\ -\ 500}{50} \ + \ b$$Simplifying the fraction gives us the equation $$600=30 \ + \ b$$And subtracting both sides of the equation by $$30$$ gives us $$b=570$$, which is the answer. 49- Choice C is correct The correct answer is $$7.32$$The weight of $$12.2$$ meters of this rope is: $$12.2 \ × \ 600$$ g = $$7,320$$ g$$1$$ kg $$= 1,000$$ g , therefore , $$7,320$$ g $$÷ \ 1000 = 7.32$$ kg 50- Choice A is correct The correct answer is $$40$$The area of ∆BED is $$16$$, then: $$\frac{4 × \ AB}{2}$$=$$16→4 \ × \ AB=32→AB=8$$The area of ∆BDF is $$18$$, then: $$\frac{3×BC } {2}=18→3 \ × \ BC=36→BC=12$$The perimeter of the rectangle is = $$2 \ × \ (8 \ + \ 12)=40$$

### Practice Test 1

Simulate test day with an official practice test. Then, score your test. The answers come with explanations so you can learn from your mistakes.

### Practice Test 2

Simulate test day with an official practice test. Then, score your test. The answers come with explanations so you can learn from your mistakes.

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