Full Length CLEP College Mathematics Practice Test

The correct answer is $2\ x^2\ -\ 4\ x\ +\ 4$
The correct answer is
In order to figure out what the equation of the graph is, fist find the vertex. From the graph we can determine that the vertex is at $(1,\ 2)$.
We can use vertex form to solve for the equation of this graph. Recall vertex form, $y=a\ (x\ -\ h)^2\ +\ k$, where $h$ is the $x$ coordinate of the vertex,
and $k$ is the $y$ coordinate of the vertex. Plugging in our values, you get $y=a\ (x\ -\ 1)^2\ +\ 2$
To solve for $a$, we need to pick $a$ point on the graph and plug it into the equation.
Let’s pick $(-\ 1,\ 10)$. $10=a\ (-\ 1\ -\ 1)^2\ +\ 2\ →\ 10=a\ (-\ 2)^2\ +\ 2\ →\ 10=4\ a\ +\ 2$
$8=4\ a\ →\ a=2$ Now the equation is : $y=2\ (x\ -\ 1)^2\ +\ 2$
Let’s expand this, $y=2\ (x^2\ -\ 2\ x\ +\ 1)\ +\ 2\ →\ y=2\ x^2\ -\ 4\ x\ +\ 2\ +\ 2\ →\ y=2\ x^2\ -\ 4\ x\ +\ 4$
The equation in Choice C is the same.

The correct answer is $7\ ≤\ x\ ≤\ 13$
$|x\ -\ 10|\ ≤\ 3\ →\ -\ 3\ ≤\ x\ -\ 10\ ≤\ 3\ →\ -\ 3\ +\ 10\ ≤\ x\ -\ 10\ +\ 10\ ≤\ 3\ +\ 10\ →\ 7\ ≤\ x\ ≤\ 13$

The correct answer is {$5,\ 11$}
The union of $A$ and $B$ is: $A∪B=${$1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 11,\ 15$}
The intersection of $(A∪B)$ and $C$ is: $(A∪B)∩C=${$5,\ 11$}

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 $7$.

The correct answer is $60$ feet
The relationship among all sides of special right triangle
$30^\circ\ -\ 60^\circ\ -\ 90^\circ$ is provided in this triangle:
In this triangle, the opposite side of $30^\circ$ angle is half of the hypotenuse.
Draw the shape of this question:
The ladder is the hypotenuse. Therefore, the ladder is $60$ ft.

The correct answer is $9$
The union of $A$ and $B$ is: $A∪B=${$1,\ 2,\ 4,\ 8,\ 12,\ 16,\ 24,\ 32,\ 48$}. There are $9$  elements in $A∪B$. 

The correct answer is $15$ cm
Use Pythagorean Theorem: $a^2\ +\ b^2=c^2$.
$9^2\ +\ 12^2 = c^2 ⇒ 81\ +\ 144=c^2 ⇒ 225=c^2\ ⇒c=15$

The correct answer is $10\%$
The percent of girls playing tennis is: $40\%\ ×\ 25\% = 0.40\ ×\ 0.25 = 0.10 = 10\%$

The correct answer is $16\ π$
Area of the circle is less than $16\ π$. Use the formula of areas of circles.
Area $= π\ r^2 ⇒ 64\ π\ >\ π\ r^2\ ⇒ 64\ >\ r^2⇒ r\ <\ 8$
Radius of the circle is less than $8$. Let’s put $8$ for the radius. Now, use the circumference formula: Circumference $=2\ π\ r=2\ π\ (8)=16\ π$
Since the radius of the circle is less than $8$. Then, the circumference of the circle must be less than $16\ π$. Only choice $A$ is less than $16\ π$.

The correct answer is $x=-\ 2,\ y=3$
$$\begin{cases}x\ +\ 4\ y=10\\5\ x\ +\ 10\ y=20\end{cases}$$ → Multiply the top equation by $-\ 5$ then,
$$\begin{cases}-\ 5\ x\ -\ 20\ y=-\ 50\\5\ x\ +\ 10\ y=20\end{cases}$$ → Add two equations. $-\ 10\ y=-\ 30\ →\ y=3$ ,
plug in the value of $y$ into the first equation: $x\ +\ 4\ y=10\ →\ x\ +\ 4\ (3)=10\ →\ x\ +\ 12=10$
Subtract $12$ from both sides of the equation. Then: $x\ +\ 12=10\ →\ x=-\ 2$

The correct answer is $300\%$
Write the equation and solve for $B: 0.60\ A = 0.20\ B$, divide both sides by $0.20$, then:
$\frac{0.60}{0.20}$ $A=B$, therefore: $B=3\ A$, and $B$ is $3$ times of $A$ or it’s $300\%$ of $A$.

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 $\frac{12800\ -\ 16000}{16000}\ ×\ 100\%=20\%$

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 $= (27)\ (9)\ (3) = 729$

The correct answer is $90$
To find the number of possible outfit combinations, multiply number of options for each factor: $6\ ×\ 3\ ×\ 5=90$

The correct answer is $$ 1,800$
Use simple interest formula: $I=$prt $(I =$ interest,p $=$ principal,$r =$ rate,$t =$ time$)$
$I=(8,000)\ (0.045)\ (5)=1,800$

The correct answer is $(9,\ 3)$
First, find the equation of the line. All lines through the origin are of the form $y=m\ x$, so the equation is $y=\frac{1}{3}\ x$.
Of the given choices, only choice $C\ (9,\ 3)$, satisfies this equation:
$y=\frac{1}{3}\ x\ →\ 3=\frac{1}{3}\ (9)=3$

The correct answer is $125\%$
Use percent formula: part $=\frac{percent}{100}\ ×$ whole
$25= \frac{percent}{100}\ ×\ 20 \ ⇒\ 25= \frac{percent\ ×\ 20}{100}\ ⇒\ 25=\frac{percent\ ×\ 2}{10}$, multiply both sides by $10$.
$250=$percent $×\ 2$, divide both sides by $2$.
125=percent

The correct answer is $46$
The area of the trapezoid is: Area$=\frac{1}{2}\ h\ (b_1\ +\ b_2 )=\frac{1}{2}\ (x)\ (13\ +\ 8)=126\ →\ 10.5\ x\ =126\ →\ x=12$.
$y=\sqrt{5^2\ +\ 12^2 }=\sqrt{25\ +\ 144}=\sqrt{169}=13$
The perimeter of the trapezoid is: $12\ +\ 13\ +\ 8\ +\ 13=46$

The correct answer is $475$ km
Add the first $5$ numbers. $40\ +\ 45\ +\ 50\ +\ 35\ +\ 55 = 225$
To find the distance traveled in the next $5$ hours, multiply the average by number of hours.
Distance $=$ Average $×$ Rate $= 50\ ×\ 5 = 250$. Add both numbers. $250\ +\ 225 = 475$

The correct answer is $8$ hours and $24$ minutes
Use distance formula: Distance $=$ Rate $×$ time $⇒ 420 = 50\ ×\ T$,
divide both sides by $50.\frac{ 420} { 50} = T ⇒ T = 8.4$ hours.
Change hours to minutes for the decimal part. $0.4$ hours $= 0.4\ ×\ 60 = 24$ minutes.

The correct answer is $(-\ 1,\ -\ 2)$
The equation of a line is in the form of $y=m\ x\ +\ b$, where $m$ is the slope of the line and $b$ is the $y\ -$intercept of the line. Two points $(4,\ 3)$ and $(3,\ 2)$ are on line $A$. Therefore, the slope of the line $A$ is: slope of line $A=\frac{y_2\ -\ y_1}{x_2\ -\ x_1} = \frac{2\ -\ 3}{3\ -\ 4}=\frac{-\ 1}{-\ 1}=1$
The slope of line $A$ is $1$. Thus, the formula of the line $A$ is: $y=m\ x\ +\ b=x\ +\ b$, choose a point and plug in the values of $x$ and $y$ in the equation to solve for $b$. Let’s choose point $(4,\ 3)$. Then:
$y=x\ +\ b\ →\ 3=4\ +\ b\ →\ b=3 \ -\ 4=-\ 1$
The equation of line $A$ is: $y=x\ -\ 1$. Now, let’s review the choices provided:
$A.$        $(-\ 1,\ 2) y=x\ -\ 1\ →\ 2=-\ 1\ -\ 1=-\ 2$                     This is not true.
$B.$        $(5,\ 7) y=x\ -\ 1\ →\ 7=5\ -\ 1=4$                                This is not true.
$C.$        $(3,\ 4) y=x\ -\ 1\ →\ 4=3 \ -\ 1=2$                               This is not true.
$D.$        $(-\ 1,\ -\ 2) y=x\ -\ 1\ →\ -\ 2=-\ 1\ -\ 1=-\ 2$              This is true!

The correct answer is $30$
Let $x$ be the number. Write the equation and solve for $x$.
$\frac{2}{3}\ ×\ 18= \frac{2}{5} .\ x\ ⇒\frac{2\ ×\ 18}{3}=\frac{2\ x}{5}$ , use cross multiplication to solve for $x$.
$5\ ×\ 36=2\ x\ ×\ 3\ ⇒180=6\ x\ ⇒\ x=30$

The correct answer is $0.88$ $D$
To find the discount, multiply the number by $(100\%\ –$ rate of discount$)$.
Therefore, for the first discount we get: $(D)\ (100\% \ –\ 20\%) = (D)\ (0.80) = 0.80\ D$
For increase of $10\%: (0.85\ D)\ (100\%\ +\ 10\%) = (0.85\ D)\ (1.10) = 0.88\ D = 88\%$ of $D$

The correct answer is $30\%$
Use the formula for Percent of Change $\frac{New \ Value\ -\ Old\ Value}{Old\ Value}\ ×\ 100\%$
$\frac{28\ -\ 40}{40}\ ×\ 100\%= –\ 30\%$ (Negative sign here means that the new price is less than old price).

The correct answer is $15$
Some of prime numbers are: $2,\ 3,\ 5,\ 7,\ 11,\ 13$. Find the product of two consecutive prime numbers: $2\ ×\ 3 = 6$ (not in the options), $3\ ×\ 5 = 15$ (bingo!), $5\ ×\ 7 = 35$ (not in the options)
$7\ ×\ 11 = 77$ (not in the options)

The correct answer is $\frac{2}{7},\ \frac{3}{8},\ \frac{5}{11},\ \frac{3}{4}$
Let’s compare each fraction: $\frac{2}{7}\ <\ \frac{3}{8}\ <\ \frac{5}{11}\ <\ \frac{3}{4}$ Only choice C provides the right order.

The correct answer is $50$ miles
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: $a^2\ +\ b^2=c^2$
$40^2\ +\ 30^2=c^2 \ ⇒\ 1600\ +\ 900= c^2 ⇒ 2500=c^2 \ ⇒ \ c=50$

The correct answer is $12$
The ratio of boy to girls is $4:7$. Therefore, there are $4$ boys out of $11$ students. To find the answer,
first divide the total number of students by $11$, then multiply the result by $4$.
$44\ ÷\ 11=4\ ⇒\ 4\ ×\ 4=16$. There are $16$ boys and $28\ (44\ –\ 16)$ girls. So,
$12$ more boys should be enrolled to make the ratio $1:1$

The correct answer is $15\%$
The question is this: $530.40$ is what percent of $624$? Use percent formula: part  $=$ $\frac{ percent}{100}\ ×$ whole. $530.40=$$\frac{ percent}{100}\ ×\ 624 ⇒ 530.40= \frac{percent \ ×\ 624}{100}\ ⇒\ 53040 =$ percent $×\ 624\ ⇒$ percent $= \frac{53040}{624}=85$
$530.40$ is $85\%$ of $624$. Therefore, the discount is: $100\%\ –\ 85\%=15\%$

The correct answer is $15$
If the score of Mia was $60$, therefore the score of Ava is $30$. Since, the score of Emma was half as that of Ava, therefore, the score of Emma is $15$.

The correct answer is $\frac{17}{18}$
If $17$ balls are removed from the bag at random, there will be one ball in the bag.
The probability of choosing a brown ball is $1$ out of $18$. Therefore,
the probability of not choosing a brown ball is $17$ out of $18$ and the probability of having not a brown ball after removing $17$ balls is the same.

The correct answer is $36$
Let $x$ be the smallest number. Then, these are the numbers: $x,\ x\ +\ 1,\ x\ +\ 2,\ x\ +\ 3,\ x \ +\ 4$
average $=\frac{sum\ of\ terms}{number\ of\ terms} ⇒ 38=\frac{x\ +\ (x\ +\ 1)\ +\ (x\ +\ 2)\ +\ (x\ +\ 3)\ +\ (x\ +\ 4)}{5}\ ⇒\ 38= \frac{5\ x\ +\ 10}{5}\ ⇒\ 190=5\ x\ +\ 10\ ⇒\ 180=5\ x\ ⇒ x=36$

The correct answer is $18$
The area of the floor is: $6$ cm $×\ 24$ cm $= 144$ cm$^2$
The number of tiles needed $= 144\ ÷\ 8 = 18$

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

The correct answer is $600$ ml
$4\%$ of the volume of the solution is alcohol. Let $x$ be the volume of the solution.
Then: $4\%$ of $x=24$ ml $⇒ 0.04\ x=24\ ⇒\ x\ =24\ ÷\ 0.04=600$

The correct answer is $61.28$
average $=\frac{sum\ of\ terms}{number\ of\ terms}$
The sum of the weight of all girls is: $18\ ×\ 60=1080$ kg
The sum of the weight of all boys is: $32\ ×\ 62=1984$ kg
The sum of the weight of all students is: $1080\ +\ 1984=3064$ kg
average$=\frac{3064}{50}=61.28$

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 $x$.

The correct answer is $12$
Given the two equations, substitute the numerical value of a into the second equation to solve for . $a=\sqrt{3},\ 4\ a=\sqrt{4\ x}$
Substituting the numerical value for a into the equation with $x$ is as follows.
$4(\sqrt{3})=\sqrt{4x},$ From here, distribute the $4$. $4\sqrt{3}=\sqrt{4x}$
Now square both side of the equation.$(4\sqrt{3})^2=(\sqrt{4x})^2$
Remember to square both terms within the parentheses. Also, recall that squaring a square root sign cancels them out. $4^2 \sqrt{3}^2=4\ x,\ 16\ (3)=4\ x,\ 48=4\ x,\ x=12$

The correct answer is $5$ cm
Formula for the Surface area of a cylinder is: $S\ A=2\ π\ r^2\ +\ 2\ π\ r\ h\ →\ 150\ π=2\ π\ r^2\ +\ 2\ π\ r\ (10)\ →r^2\ +\ 10\ r\ -\ 75=0\ →\ (r\ +\ 15)\ (r\ -\ 5)=0\ →\ r=5$ or
$r=-\ 15$ (unacceptable)

The correct answer is $I\ >\ 2000\ x\ +\ 24000$
Let $x$ be the number of years. Therefore, $$ 2,000$ per year equals $2000\ x$.
starting from $$24,000$ annual salary means you should add that amount to $2000\ x$.
Income more than that is: $I\ >\ 2000\ x\ +\ 24000$

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 $g(x)$, which has three distinct zeros, must have three $x\ -$intercepts.
Only the graph in choice $C$ has three $x\ -$intercepts.

The correct answer is $225$
$0.6\ x=(0.3)\ ×\ 20\ →\ x=10\ →\ (x\ +\ 5)^2=(15)^2=225$

The correct answer is $y=x$
The slop of line $A$ is: $m=\frac{y_2\ -\ y_1}{x_2\ -\ x_1 }=\frac{3\ -\ 2}{4\ -\ 3}=1$
Parallel lines have the same slope and only choice $D\ (y=x)$ has slope of $1$.

The correct answer is $x$ is divided by $3$
Replace $z$ by $\frac{z}{3}$ and simplify.
$x_1=\frac{8\ y\ +\frac{r}{r\ +\ 1}}{\frac{6}{\frac{z}{3}}}$$=\frac{8\ y\ +\ \frac{r}{r\ +\ 1}}{\frac{3\ ×\ 6}{z}}$$=\frac{8\ y+\frac{r}{r\ +\ 1}}{3\ ×\ \frac{6}{z}}$$=\frac{1}{3}$$×\frac{8\ y\ + \frac{r}{r\ +\ 1}}{\frac{6}{z}}=\frac{x}{3}$
When $z$ is divided by $3,\ x$ is also divided by $3$.

The correct answer is $50$ miles
Use the information provided in the question to draw the shape.
Use Pythagorean Theorem: $a^2\ +\ b^ 2 = c^ 2$
$40^2\ +\ 30^2 = c^ 2\ ⇒\ 1600\ +\ 900 = c^2\ ⇒\ 2500 = c^ 2\ ⇒\ c = 50$

The correct answer is$6$
The four$-$term polynomial expression can be factored completely, by grouping, as follows: $(x^3\ -\ 6\ x^2 )\ +\ (3\ x\ -\ 18)=0$
$x^2\ (x\ -\ 6)\ +\ 3\ (x\ -\ 6)=0$ $(x\ -\ 6)\ (x^2\ +\ 3)=0$
By the zero$-$product property, set each factor of the polynomial equal to $0$
and solve each resulting equation for $x$. This gives $x=6$ or $x=±\ i\sqrt{3}$, respectively.
Because the equation the question asks for the real value of $x$ that satisfies the equation, The correct answer is $6$.

The correct answer is$p^3$
To solve for $f\ (3\ g(P))$, first, find $3\ g(p)$. $g(x)=\log_3$ $x\ →\ g(p)=\log_3\ p\ →\ 3\ g(p)=3\ \log_3\ p=\log_3\ p^3$
Now, find $f\ (3\ g(p)): f\ (x)=3^x\ →\ f\ (\log_3\ p^3 )=3^{\log_3\ p^3}$
Logarithms and exponentials with the same base cancel each other.
This is true because logarithms and exponentials are inverse operations. Then: $f\ (\log_3\ p^3 )=3^(\log_3\ p^3 )=p^3$

The correct 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 $c=0.35\ (60\ h)$

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 5th score: Sum of $5$ score: $(5)\ (92)=460$
To get her sum from $360$ to $460$, Mary needs to score $460\ -\ 360=100$.

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.

The correct answer is $105$
$y = 4\ a\ b\ +\ 3\ b^3$. Plug in the values of a and b in the equation: $a = 2$ and $b = 3$
$y = 4\ (2)\ (3)\ +\ 3\ (3)^3 = 24\ +\ 3\ (27) = 24\ +\ 81 = 105$

The correct answer is $–\ x^2\ –\ 3\ x\ –\ 6$
$(g\ –\ f)\ (x) = g(x)\ –\ f(x) = (–\ x^2\ –\ 1\ –\ 2\ x)\ –\ (5\ +\ x)$
$–\ x^2\ –\ 1\ –\ 2\ x\ –\ 5\ –\ x = –\ x^2\ –\ 3\ x\ –\ 6$

The correct answer is $\frac{100\ x}{y}$
Let the number be $A$. Then: $x=y\%\ ×\ A$. Solve for $A.\ x=\frac{y}{100}\ ×\ A$
Multiply both sides by $\frac{100}{y}: x\ ×\ \frac{100}{y}=\frac{y}{100}\ ×\ \frac{100}{y}\ ×\ A\ →\ A=\frac{100\ x}{y}$

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
$\frac{m}{6}=\frac{12}{m}\ →\ m^2=72\ →\ m=±\sqrt{72}=±\sqrt{36×2}=±\sqrt{36}\ ×\ \sqrt{2}=±6\ \sqrt{2}$

The correct answer is $6$
It is given that $g(5)=4$. Therefore, to find the value of f$(g(5))$, then $f(g(5))=f(4)=6$

The correct answer is $(–\ 1,\ 3)$
Plug in each pair of number in the equation:
A.              $(2,\ 1):$ $2\ (2)\ +\ 4\ (1) = 8$                Nope!
B.              $(–\ 1,\ 3):$ $2\ (–\ 1)\ +\ 4\ (3) = 10$      Bingo!
C.              $(–\ 2,\ 2):$ $2\ (–\ 2)\ +\ 4\ (2) = 4$        Nope!
D.              $(2,\ 2):$ $2\ (2)\ +\ 4\ (2) = 12$               Nope!

The correct answer is $29$
Here we can substitute $8$ for $x$ in the equation. Thus, $y\ -\ 3=2\ (8\ +\ 5),\ y\ -\ 3=26$
Adding $3$ to both side of the equation: $y=26\ +\ 3 ,\ y=29$

The correct answer is I and III only
Let’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 all 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.

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 $1$. $a\ >\ 1$

The correct answer is $0$
$g(x)=-\ 2,$ then $f(g(x) )= f\ (-\ 2)=2\ (-\ 2)^3\ +\ 5\ (-\ 2)^2\ +\ 2\ (-\ 2)= -\ 16 \ +\ 20\ -\ 4=0$