Ultimate Challenge Set — Hardest & Most Confusing (30 Questions)

Directions

For each question, choose the best answer from the four choices given, or enter your answer in the response box provided for student-produced response questions. As soon as you choose or submit an answer, the explanation will appear so you can check your work immediately. You may use the on-screen calculator by clicking the "Calculator" button at any time.

1Question 1

Systems

Math

For what value of $k$ does the system $2x+ky=10$ and $4x+8y=20$ have infinitely many solutions?

The second equation is exactly twice the first when $k=4$: $2(2x+4y)=4x+8y=20$ matches $2(10)=20$. Since the equations become identical multiples of each other, there are infinitely many solutions when $k=4$.

2Question 2

Quadratics

Math

The length of a rectangle is 3 more than its width, and its area is 130. What is the width? (Only a positive value makes sense.)

Let width $=w$. Then $w(w+3)=130 \Rightarrow w^2+3w-130=0 \Rightarrow (w-10)(w+13)=0$. So $w=10$ or $w=-13$; a negative width is impossible, so $w=10$.

3Question 3

Algebra

Math

For which value of $x$ is the expression $\dfrac{x+5}{x^2-4x-12}$ undefined?

Ax=-5

Bx=-2

Cx=6

DBoth x=-2 and x=6

Factor the denominator: $x^2-4x-12=(x-6)(x+2)$. The expression is undefined when the denominator is 0, i.e., $x=6$ or $x=-2$.

4Question 4

Exponential

Math

An investment of \$1000 grows at 5% annual interest, compounded yearly, following $A=1000(1.05)^t$. Approximately how many years until the investment reaches \$1500?

A6.2 years

B7.1 years

C8.3 years

D9.6 years

Set $1000(1.05)^t=1500$, so $(1.05)^t=1.5$. Taking the log of both sides: $t=\dfrac{\ln 1.5}{\ln 1.05}\approx8.3$ years.

5Question 5

Sequences

Math

A sequence is defined by $a_1=4$ and $a_{n}=2a_{n-1}-3$ for $n>1$. What is $a_4$?

$a_2=2(4)-3=5$; $a_3=2(5)-3=7$; $a_4=2(7)-3=11$.

6Question 6

Trigonometry

Math

If $\sin\theta=\dfrac{5}{13}$ and $\theta$ is in the first quadrant, what is $\cos\theta$?

A5/13

B8/13

C12/13

D13/5

Using $\sin^2\theta+\cos^2\theta=1$: $\cos^2\theta=1-\left(\dfrac{5}{13}\right)^2=1-\dfrac{25}{169}=\dfrac{144}{169}$. Since $\theta$ is in Quadrant I, $\cos\theta>0$, so $\cos\theta=\dfrac{12}{13}$.

7Question 7

Algebra

Math

How many solutions does $|2x+1|=3x-4$ have?

DInfinitely many

Case 1: $2x+1=3x-4$ gives $x=5$; check: $|11|=11$ and $3(5)-4=11$ ✓. Case 2: $-(2x+1)=3x-4$ gives $-2x-1=3x-4$, so $5x=3$, $x=3/5$; check: RHS $=3(3/5)-4=-11/5<0$, but the left side $|2x+1|\ge0$, so this is extraneous. Only $x=5$ works: 1 solution.

8Question 8

Functions

Math

If $f(x)=3x-2$ and $f^{-1}$ is the inverse of $f$, what is $f^{-1}(10)$?

D28

Set $f(x)=10$: $3x-2=10 \Rightarrow 3x=12 \Rightarrow x=4$. So $f^{-1}(10)=4$.

9Question 9

Exponents

Math

If $5^x=40$, which is closest to the value of $x$?

A1.6

B2.3

C3.1

D3.7

Taking $\log$ of both sides: $x=\log_5 40=\dfrac{\ln 40}{\ln 5}\approx2.3$.

10Question 10

Geometry

Math

A sphere and a cylinder each have radius 3. The cylinder's height equals the sphere's diameter (6). What is the ratio of the sphere's volume to the cylinder's volume?

A1:1

B2:3

C3:4

D4:3

Sphere volume $=\dfrac43\pi(3)^3=36\pi$. Cylinder volume $=\pi(3)^2(6)=54\pi$. Ratio $=36\pi:54\pi=2:3$.

11Question 11

Word Problems

Math

Pump A fills a pool in 6 hours; Pump B fills it in 10 hours. Pump C alone can drain a full pool in 15 hours. If all three operate together (A and B filling, C draining), how many hours does it take to fill the empty pool? (Enter as a decimal or fraction.)

Combined rate $=\dfrac16+\dfrac1{10}-\dfrac1{15}$. Using a common denominator of 30: $\dfrac{5}{30}+\dfrac{3}{30}-\dfrac{2}{30}=\dfrac{6}{30}=\dfrac15$ pool/hr. Time $=1\div\dfrac15=5$ hours.

12Question 12

Coordinate Geometry

Math

Points $A(1,2)$ and $B(7,8)$ are endpoints of a diameter of a circle. What is the area of the circle, in terms of $\pi$?

A9π

B18π

C36π

D72π

The diameter is $AB=\sqrt{(7-1)^2+(8-2)^2}=\sqrt{36+36}=6\sqrt2$, so the radius is $3\sqrt2$. Area $=\pi r^2=\pi(3\sqrt2)^2=\pi(18)=18\pi$.

13Question 13

Combinatorics

Math

A committee of 3 people is chosen from a group of 8 people. How many different committees are possible?

A24

B56

C112

D336

Order doesn't matter, so use combinations: $\binom{8}{3}=\dfrac{8!}{3!5!}=56$.

14Question 14

Statistics

Math

A data set has a standard deviation of 4. If every value in the set is increased by 10, what is the new standard deviation?

B10

C14

D40

Adding a constant to every value shifts the data but does not change its spread, so the standard deviation remains 4.

15Question 15

Polynomials

Math

If the polynomial $x^3+kx^2-5x+2$ has a remainder of 3 when divided by $x-1$, what is the value of $k$?

By the Remainder Theorem, $p(1)=3$: $1+k-5+2=3 \Rightarrow k-2=3 \Rightarrow k=5$.

16Question 16

Functions

Math

A function is defined as $f(x)=x^2$ for $x<2$ and $f(x)=3x+1$ for $x\ge2$. What is $f(2)+f(-3)$?

A10

B14

C16

D19

Since $2\ge2$, $f(2)=3(2)+1=7$. Since $-3<2$, $f(-3)=(-3)^2=9$. Sum $=7+9=16$.

17Question 17

Systems

Math

How many points of intersection are there between the circle $x^2+y^2=25$ and the line $y=x+1$?

Substitute $y=x+1$ into the circle equation: $x^2+(x+1)^2=25 \Rightarrow 2x^2+2x-24=0 \Rightarrow x^2+x-12=0 \Rightarrow (x+4)(x-3)=0$. This gives two x-values ($x=-4$ and $x=3$), so there are 2 intersection points.

18Question 18

Percent

Math

An item is discounted 20%, then 8% sales tax is applied to the discounted price, resulting in a final price of \$64.80. What was the original price, in dollars?

Let original price $=p$. $p(0.80)(1.08)=64.80 \Rightarrow 0.864p=64.80 \Rightarrow p=75$.

19Question 19

Exponents

Math

What is the value of $\left(\dfrac{4}{9}\right)^{-1/2}$?

A2/3

B3/2

C4/9

D9/4

A negative exponent means take the reciprocal: $\left(\dfrac94\right)^{1/2}=\sqrt{\dfrac94}=\dfrac32$.

20Question 20

Geometry

Math

In the circle with center $O$, central angle $\angle AOB=120°$. What is the measure of the major arc $AB$ (the arc NOT containing the central angle's interior, i.e., going the long way from $A$ to $B$)?

A120°

B180°

C240°

D300°

A full circle is 360°. The minor arc $AB$ equals the central angle, 120°. The major arc is the rest of the circle: $360°-120°=240°$.

21Question 21

Inequalities

Math

What is the solution set of $(x-1)(x+2)(x-4)<0$?

Ax<-2 or 1

Bx<-2

C-24

Dx<-2 or x>4

The roots are $x=-2,1,4$, dividing the number line into 4 intervals. Testing each interval shows the product is negative for $-24$.

22Question 22

Systems

Math

If $3x-2y=7$ and $x+4y=9$, what is the value of $x$? (Enter as a decimal or fraction.)

From the second equation, $x=9-4y$. Substitute: $3(9-4y)-2y=7 \Rightarrow 27-12y-2y=7 \Rightarrow -14y=-20 \Rightarrow y=10/7$. Then $x=9-4(10/7)=9-40/7=23/7$.

23Question 23

Functions

Math

What is the domain of $f(x)=\sqrt{2x-8}$?

Ax≥0

Bx≥4

Cx≥8

Dx≤4

For the square root to be defined (real), the radicand must be $\ge0$: $2x-8\ge0 \Rightarrow x\ge4$.

24Question 24

Geometry

Math

Two similar triangles have corresponding sides $(x+2)$ and 6 in one triangle, and 8 and 12 in the other (with $(x+2)$ corresponding to 8, and 6 corresponding to 12). What is the value of $x$?

Set up the proportion $\dfrac{x+2}{8}=\dfrac{6}{12}$. Cross-multiplying: $12(x+2)=48 \Rightarrow x+2=4 \Rightarrow x=2$.

25Question 25

Sequences

Math

What is the sum of the first 5 terms of the geometric sequence with first term 2 and common ratio 3?

A121

B180

C242

D364

Sum $=a_1\dfrac{r^n-1}{r-1}=2\cdot\dfrac{3^5-1}{3-1}=2\cdot\dfrac{242}{2}=242$.

26Question 26

Statistics

Math

A data set is 12, 15, 18, 20, 25. If the value 50 is added to the set, which statement about the mean and median is true?

AThe mean increases more than the median.

BThe median increases more than the mean.

CThe mean and median increase by the same amount.

DThe median stays the same, but the mean increases.

The original mean is 18, and the new mean is $140/6\approx23.3$, an increase of about 5.3. The original median is 18; the new median (of 12,15,18,20,25,50) is $(18+20)/2=19$, an increase of only 1. Outliers pull the mean much more than the median.

27Question 27

Exponential

Math

Function $f(x)=10+5x$ and function $g(x)=2(2)^x$ are graphed. For large values of $x$, which statement is true?

Af(x) will always be greater than g(x).

Bg(x) will eventually exceed f(x) and continue growing faster.

Cf(x) and g(x) grow at the same rate.

Df(x) and g(x) never intersect.

Exponential functions eventually outgrow linear functions no matter the starting values, because exponential growth compounds while linear growth adds a constant amount each step.

28Question 28

Percent

Math

A \$250 item is marked down 6%, and then the discounted price is increased by a 3% processing fee. To the nearest cent, what is the final price, in dollars?

After the 6% markdown: $250\times0.94=235$. After the 3% fee: $235\times1.03=242.05$.

29Question 29

Geometry

Math

A line is tangent to a circle at point $P$, where $O$ is the center. If $OP=9$ and the tangent line segment from the external point $T$ to $P$ has length 12, what is the distance $OT$?

A13

B14

C15

D21

A radius drawn to the point of tangency is perpendicular to the tangent line, so triangle $OPT$ is a right triangle with legs 9 and 12. $OT=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.

30Question 30

Quadratics

Math

What is the sum of all solutions to $(x-3)^2=4(x-3)$?

C10

D13

Let $u=x-3$. The equation becomes $u^2=4u \Rightarrow u^2-4u=0 \Rightarrow u(u-4)=0$, so $u=0$ or $u=4$, giving $x=3$ or $x=7$. Sum $=3+7=10$. (A common error is dividing both sides by $(x-3)$, which loses the solution $x=3$.)

Answer Key — Ultimate Challenge Set — Hardest & Most Confusing (30 Questions)

#	Answer	#	Answer
1	B	2	10
3	D	4	C
5	11	6	C
7	B	8	B
9	B	10	B
11	5	12	B
13	B	14	A
15	5	16	C
17	C	18	75
19	B	20	C
21	C	22	23/7
23	B	24	2
25	C	26	A
27	B	28	242.05
29	C	30	C

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Answer Key — Ultimate Challenge Set — Hardest & Most Confusing (30 Questions)

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