Grades 11 / 12 Tests and Answer Keys

Find the exact value of x satisfying ${\log }_3\sqrt{x}={\log }_{\sqrt{x}}3$ and .

  (2 pts) 1.

Given that $x$ and $y$ are distinct nonzero real numbers such that

$$x+\frac{2}{x}=y+\frac{2}{y}$$

what is the value of $xy$?

1. A.

   $\frac{1}{4}$

2. B.

   $\frac{1}{2}$

3. C.

   $1$

4. D.

   $2$

5. E.

   $4$

  (3 pts) 2.

What is the square root of the largest perfect square that divides $12!$ ?

  (3 pts) 3.

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$?

1. A.

   1

2. B.

   $\frac{\sqrt{6}}{2}$

3. C.

   $\sqrt{3}$

4. D.

   $2$

5. E.

   $3\sqrt{2}$

  (3 pts) 4.

The region in the first quadrant bounded by the line $3x+2y=7$ and the coordinate axes is rotated about the $x$-axis. Approximately, what is the volume of the resulting solid?

1. A.

   8 units³

2. B.

   20 units³

3. C.

   30 units³

4. D.

   90 units³

5. E.

   120 units³

  (3 pts) 5.

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

1. A.

   15

2. B.

   30

3. C.

   40

4. D.

   60

5. E.

   70

  (3 pts) 6.

Let $A_1$ be the area of a square with a side length of 1. For $n\ge 2$, $A_n={\displaystyle \frac{3}{4}}{A}_{n-1}$. Find the value of

$$\frac{A_1}{A_3}\cdot \frac{A_5}{A_7}\cdot \frac{A_9}{A_{1}1}\cdot \mathrm{\cdots }\cdot \frac{A_{3}3}{A_{3}5}\cdot \frac{A_{3}5}{A_{3}9}$$

Round your answer to two decimal places.

  (3 pts) 7.

Find the exact value of x satisfying ${\log }_3\sqrt{x}={\log }_{\sqrt{x}}3$ and .

  (2 pts) 1. $\frac{1}{9}$

Given that $x$ and $y$ are distinct nonzero real numbers such that

$$x+\frac{2}{x}=y+\frac{2}{y}$$

what is the value of $xy$?

1. A.

   $\frac{1}{4}$

2. B.

   $\frac{1}{2}$

3. C.

   $1$

4. D.

   $2$

5. E.

   $4$

  (3 pts) 2. D $2$

What is the square root of the largest perfect square that divides $12!$ ?

  (3 pts) 3. $2^5\cdot 3^2\cdot 5=1440$

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$?

1. A.

   1

2. B.

   $\frac{\sqrt{6}}{2}$

3. C.

   $\sqrt{3}$

4. D.

   $2$

5. E.

   $3\sqrt{2}$

  (3 pts) 4. B ${\displaystyle \frac{\sqrt{6}}{2}}$

The region in the first quadrant bounded by the line $3x+2y=7$ and the coordinate axes is rotated about the $x$-axis. Approximately, what is the volume of the resulting solid?

1. A.

   8 units³

2. B.

   20 units³

3. C.

   30 units³

4. D.

   90 units³

5. E.

   120 units³

  (3 pts) 5. C 30 units³

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

1. A.

   15

2. B.

   30

3. C.

   40

4. D.

   60

5. E.

   70

  (3 pts) 6. E 70

Let $A_1$ be the area of a square with a side length of 1. For $n\ge 2$, $A_n={\displaystyle \frac{3}{4}}{A}_{n-1}$. Find the value of

$$\frac{A_1}{A_3}\cdot \frac{A_5}{A_7}\cdot \frac{A_9}{A_{1}1}\cdot \mathrm{\cdots }\cdot \frac{A_{3}3}{A_{3}5}\cdot \frac{A_{3}5}{A_{3}9}$$

Round your answer to two decimal places.

  (3 pts) 7. ${\left({\displaystyle \frac{4}{3}}\right)}^{20}\approx 315.34$

Find the exact value of x satisfying ${\log }_3\sqrt{x}={\log }_{\sqrt{x}}3$ and .

  (2 pts) 1. $\frac{1}{9}$

Given that $x$ and $y$ are distinct nonzero real numbers such that

$$x+\frac{2}{x}=y+\frac{2}{y}$$

what is the value of $xy$?

1. A.

   $\frac{1}{4}$

2. B.

   $\frac{1}{2}$

3. C.

   $1$

4. D.

   $2$

5. E.

   $4$

  (3 pts) 2. D $2$

What is the square root of the largest perfect square that divides $12!$ ?

  (3 pts) 3. $2^5\cdot 3^2\cdot 5=1440$

A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$?

1. A.

   1

2. B.

   $\frac{\sqrt{6}}{2}$

3. C.

   $\sqrt{3}$

4. D.

   $2$

5. E.

   $3\sqrt{2}$

  (3 pts) 4. B ${\displaystyle \frac{\sqrt{6}}{2}}$

The region in the first quadrant bounded by the line $3x+2y=7$ and the coordinate axes is rotated about the $x$-axis. Approximately, what is the volume of the resulting solid?

1. A.

   8 units³

2. B.

   20 units³

3. C.

   30 units³

4. D.

   90 units³

5. E.

   120 units³

  (3 pts) 5. C 30 units³

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

1. A.

   15

2. B.

   30

3. C.

   40

4. D.

   60

5. E.

   70

  (3 pts) 6. E 70

Let $A_1$ be the area of a square with a side length of 1. For $n\ge 2$, $A_n={\displaystyle \frac{3}{4}}{A}_{n-1}$. Find the value of

$$\frac{A_1}{A_3}\cdot \frac{A_5}{A_7}\cdot \frac{A_9}{A_{1}1}\cdot \mathrm{\cdots }\cdot \frac{A_{3}3}{A_{3}5}\cdot \frac{A_{3}5}{A_{3}9}$$

Round your answer to two decimal places.

  (3 pts) 7. ${\left({\displaystyle \frac{4}{3}}\right)}^{20}\approx 315.34$

Consider two identical beakers, beaker A and beaker B. Beaker B is full of liquid and beaker A is empty. A scientist pours one third of the liquid from beaker B into beaker A. She then pours one fourth of the liquid in beaker A back into beaker B. Finally, she pours half of the liquid in beaker B back into beaker A. After this process, what fraction of the liquid is in beaker A?

(a) $5/8$ (b) $2/3$ (c) $5/6$ (d) $7/8$ (e) none of these

  (2 pts) 1.

Suppose $r,s>0$. A disk of radius $r$ cm is cut into nine pieces and a second disk of radius $(r+s)$ cm is cut into sixteen pieces. If each of the twenty-five pieces has equal area, then $r/s$ is

(a) $1/3$ (b) $3/4$ (c) $4/3$ (d) $3$ (e) none of these

  (3 pts) 2.

For how many integers $n$ does the equation $x^2+nx+10=0$ have integer solutions?

(a) $0$ (b) $1$ (c) $2$ (d) $3$ (e) $4$

  (3 pts) 3.

A square $ABCD$ with side $1$ is given in the plan. How many points $P$ in the plane of the square satisfy

$$PA+PC+PB+PD=2\sqrt{2}\mathrm{?}$$

(a) $0$ (b) $1$ (c) $2$ (d) $3$ (e) infinitely many

  (3 pts) 4.

Evan’s living room has twice as much area as Victoria’s, and three times as much area as Kyle’s. Kyle mops floors half as fast as Victoria and one third as fast as Evan. If they all start mopping their floor at the same time, who will finish mopping first?

  (3 pts) 5.

Let $p(x)=3x^4+a{x}^2+b$ where $a,b$ are real numbers. Suppose that $p(x)$ has exactly three distinct real roots $r$,$s$ and $t$ with . Which one of the following statements must be true?

(a) $r+t=0$ (b) $s+t=0$ (c) $rt\ge 0$ (d) $r+s\ge 0$ (e) none of these

  (3 pts) 6.

What is the number of integers $m$, with $1\le m\le 300$, for which $m^m$ is a perfect cube?

(a) $100$ (b) $101$ (c) $103$ (d) $104$ (e) $106$

  (3 pts) 7.

Consider two identical beakers, beaker A and beaker B. Beaker B is full of liquid and beaker A is empty. A scientist pours one third of the liquid from beaker B into beaker A. She then pours one fourth of the liquid in beaker A back into beaker B. Finally, she pours half of the liquid in beaker B back into beaker A. After this process, what fraction of the liquid is in beaker A?

(a) $5/8$ (b) $2/3$ (c) $5/6$ (d) $7/8$ (e) none of these

  (2 pts) 1. A

Suppose $r,s>0$. A disk of radius $r$ cm is cut into nine pieces and a second disk of radius $(r+s)$ cm is cut into sixteen pieces. If each of the twenty-five pieces has equal area, then $r/s$ is

(a) $1/3$ (b) $3/4$ (c) $4/3$ (d) $3$ (e) none of these

  (3 pts) 2. D

For how many integers $n$ does the equation $x^2+nx+10=0$ have integer solutions?

(a) $0$ (b) $1$ (c) $2$ (d) $3$ (e) $4$

  (3 pts) 3. E

A square $ABCD$ with side $1$ is given in the plan. How many points $P$ in the plane of the square satisfy

$$PA+PC+PB+PD=2\sqrt{2}\mathrm{?}$$

(a) $0$ (b) $1$ (c) $2$ (d) $3$ (e) infinitely many

  (3 pts) 4. B

Evan’s living room has twice as much area as Victoria’s, and three times as much area as Kyle’s. Kyle mops floors half as fast as Victoria and one third as fast as Evan. If they all start mopping their floor at the same time, who will finish mopping first?

  (3 pts) 5. B

Let $p(x)=3x^4+a{x}^2+b$ where $a,b$ are real numbers. Suppose that $p(x)$ has exactly three distinct real roots $r$,$s$ and $t$ with . Which one of the following statements must be true?

(a) $r+t=0$ (b) $s+t=0$ (c) $rt\ge 0$ (d) $r+s\ge 0$ (e) none of these

  (3 pts) 6. A

What is the number of integers $m$, with $1\le m\le 300$, for which $m^m$ is a perfect cube?

(a) $100$ (b) $101$ (c) $103$ (d) $104$ (e) $106$

  (3 pts) 7. D

Consider two identical beakers, beaker A and beaker B. Beaker B is full of liquid and beaker A is empty. A scientist pours one third of the liquid from beaker B into beaker A. She then pours one fourth of the liquid in beaker A back into beaker B. Finally, she pours half of the liquid in beaker B back into beaker A. After this process, what fraction of the liquid is in beaker A?

(a) $5/8$ (b) $2/3$ (c) $5/6$ (d) $7/8$ (e) none of these

  (2 pts) 1. A

Suppose $r,s>0$. A disk of radius $r$ cm is cut into nine pieces and a second disk of radius $(r+s)$ cm is cut into sixteen pieces. If each of the twenty-five pieces has equal area, then $r/s$ is

(a) $1/3$ (b) $3/4$ (c) $4/3$ (d) $3$ (e) none of these

  (3 pts) 2. D

For how many integers $n$ does the equation $x^2+nx+10=0$ have integer solutions?

(a) $0$ (b) $1$ (c) $2$ (d) $3$ (e) $4$

  (3 pts) 3. E

A square $ABCD$ with side $1$ is given in the plan. How many points $P$ in the plane of the square satisfy

$$PA+PC+PB+PD=2\sqrt{2}\mathrm{?}$$

(a) $0$ (b) $1$ (c) $2$ (d) $3$ (e) infinitely many

  (3 pts) 4. B

Evan’s living room has twice as much area as Victoria’s, and three times as much area as Kyle’s. Kyle mops floors half as fast as Victoria and one third as fast as Evan. If they all start mopping their floor at the same time, who will finish mopping first?

  (3 pts) 5. B

Let $p(x)=3x^4+a{x}^2+b$ where $a,b$ are real numbers. Suppose that $p(x)$ has exactly three distinct real roots $r$,$s$ and $t$ with . Which one of the following statements must be true?

(a) $r+t=0$ (b) $s+t=0$ (c) $rt\ge 0$ (d) $r+s\ge 0$ (e) none of these

  (3 pts) 6. A

What is the number of integers $m$, with $1\le m\le 300$, for which $m^m$ is a perfect cube?

(a) $100$ (b) $101$ (c) $103$ (d) $104$ (e) $106$

  (3 pts) 7. D

In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once?

  (2 pts) 1.

Suppose there are many socks of 4 different colors in a box: red, black, blue and yellow. Socks are randomly picked from the box one by one. What is the minimum number of socks that need to be picked from the box before 3 pairs of socks can be guaranteed? The pairs do not need to match each other, but socks within a pair must match.

  (3 pts) 2.

Two different positive numbers $x$ and $y$ each differ from their reciprocals by 1. What is the sum of $x$ and $y$ ?

  (3 pts) 3.

David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?

  (3 pts) 4.

Compute the sum of all roots of $(4x+5)(x-6)+(x-3)(4x+5)=0$

  (3 pts) 5.

What is the largest prime number that divides 2023 ?

  (3 pts) 6.

The largest circle in the figure has radius one. Seven circles are arranged in the largest circle such that the innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. Use $\pi =3.14$ and round your answer to three decimal places.

  (3 pts) 7.

In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once?

  (3 pts) 1. $28$

Suppose there are many socks of 4 different colors in a box: red, black, blue and yellow. Socks are randomly picked from the box one by one. What is the minimum number of socks that need to be picked from the box before 3 pairs of socks can be guaranteed? The pairs do not need to match each other, but socks within a pair must match.

  (2 pts) 2. 9

Two different positive numbers $x$ and $y$ each differ from their reciprocals by 1. What is the sum of $x$ and $y$ ?

  (3 pts) 3. $\sqrt{5}$ or $2.236$

David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?

  (3 pts) 4. $210$

Compute the sum of all roots of $(4x+5)(x-6)+(x-3)(4x+5)=0$

  (3 pts) 5. ${\displaystyle \frac{13}{4}}$ or $3.25$

What is the largest prime number that divides 2023 ?

  (3 pts) 6. $17$

The largest circle in the figure has radius one. Seven circles are arranged in the largest circle such that the innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. Use $\pi =3.14$ and round your answer to three decimal places.

  (3 pts) 7. $0.698$

Let $x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$. Which of the following is true?

1. a.

2. b.

   $x=2$

3. c.

   $x>2$

  (2 pts) 1.

If $(x-1)$ is a factor of the polynomial $P(x)=x^7-4x^4+2x^3+3x^2+ax-8$, find $a$.

  (3 pts) 2.

Suppose $a$ and $b$ are integers with $2a-3b=2$. Find ${\displaystyle \frac{9^a}{{27}^b}}$.

  (3 pts) 3.

Sally and Bob play the following game: Four fair coins are flipped. If exactly one of them comes up heads Bob wins. If exactly 1 of them comes up tails Bob wins. In all other cases Sally wins. What is the probability that Sally wins?

  (3 pts) 4.

In the picture below assume that A, B, and C are collinear, $\mathrm{\Delta }ABD$ is equilateral, and that AB= BC = 1. What is CD?

  (3 pts) 5.

Find $x$ if $x$ is positive and ${\displaystyle \frac{1}{\frac{1}{x+1}+\frac{1}{x+4}}}=x$.

  (3 pts) 6.

In the picture below $\mathrm{\square }AGHB$ and $\mathrm{\square }ACDF$ are rectangles and $C$, $G$, and $E$ are collinear. If $AF=AB=2$ and $BC=1$, what is $EG$?

  (3 pts) 7.

Let $x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$. Which of the following is true?

1. a.

2. b.

   $x=2$

3. c.

   $x>2$

  (2 pts) 1. a

If $(x-1)$ is a factor of the polynomial $P(x)=x^7-4x^4+2x^3+3x^2+ax-8$, find $a$.

  (3 pts) 2. 6

Suppose $a$ and $b$ are integers with $2a-3b=2$. Find ${\displaystyle \frac{9^a}{{27}^b}}$.

  (3 pts) 3. 9

Sally and Bob play the following game: Four fair coins are flipped. If exactly one of them comes up heads Bob wins. If exactly 1 of them comes up tails Bob wins. In all other cases Sally wins. What is the probability that Sally wins?

  (3 pts) 4. 50% or 1/2

In the picture below assume that A, B, and C are collinear, $\mathrm{\Delta }ABD$ is equilateral, and that AB= BC = 1. What is CD?

  (3 pts) 5. $\sqrt{3}$

Find $x$ if $x$ is positive and ${\displaystyle \frac{1}{\frac{1}{x+1}+\frac{1}{x+4}}}=x$.

  (3 pts) 6. 2

In the picture below $\mathrm{\square }AGHB$ and $\mathrm{\square }ACDF$ are rectangles and $C$, $G$, and $E$ are collinear. If $AF=AB=2$ and $BC=1$, what is $EG$?

  (3 pts) 7. $2\sqrt{5}$

Let $x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$. Which of the following is true?

1. a.

2. b.

   $x=2$

3. c.

   $x>2$

  (2 pts) 1. a

If $(x-1)$ is a factor of the polynomial $P(x)=x^7-4x^4+2x^3+3x^2+ax-8$, find $a$.

  (3 pts) 2. 6

Suppose $a$ and $b$ are integers with $2a-3b=2$. Find ${\displaystyle \frac{9^a}{{27}^b}}$.

  (3 pts) 3. 9

Sally and Bob play the following game: Four fair coins are flipped. If exactly one of them comes up heads Bob wins. If exactly 1 of them comes up tails Bob wins. In all other cases Sally wins. What is the probability that Sally wins?

  (3 pts) 4. 50% or 1/2

In the picture below assume that A, B, and C are collinear, $\mathrm{\Delta }ABD$ is equilateral, and that AB= BC = 1. What is CD?

  (3 pts) 5. $\sqrt{3}$

Find $x$ if $x$ is positive and ${\displaystyle \frac{1}{\frac{1}{x+1}+\frac{1}{x+4}}}=x$.

  (3 pts) 6. 2

In the picture below $\mathrm{\square }AGHB$ and $\mathrm{\square }ACDF$ are rectangles and $C$, $G$, and $E$ are collinear. If $AF=AB=2$ and $BC=1$, what is $EG$?

  (3 pts) 7. $2\sqrt{5}$

A rectangular garden is 75 feet long and 32 feet wide. Find the area of the garden. Express your answer in square feet.

  (20 pts) 1.

The wheels of a car are exactly 2 feet in diameter. As the car travels, the wheels of the car rotate at the rate of 8 revolutions per second. How fast is the car traveling? Express your answer in feet per second, and round your answer to two decimal places. You may assume that $\pi =3.14159$.

  (20 pts) 2.

The volume of a certain sphere is exactly 75 cubic inches. Find the radius of the sphere. Express your answer in inches, and round your answer to two decimal places. You may assume that $\pi =3.14159$.

  (20 pts) 3.

A deck of 52 cards has exactly 12 “face cards,” i.e. cards with a king, queen, or jack. Suppose you randomly select three cards from the deck. Find the probability that none of the selected cards is a face card. Round your answer to three decimal places.

  (20 pts) 4.

Mrs. Lopez plans to invest a total of $50,000 in two different types of bonds: bonds of type A and bonds of type B. Bonds of type A return a dividend of 4% per year, and bonds of type B return a dividend of 7% per year. For example, if Mrs. Lopez invests $40,000 in bonds of type A and $10,000 in bonds of type B, then the bonds will return a total dividend of $\text{\$40,000}\times 0.04+\text{\$10,000}\times 0.07=\text{\$2,300}$ per year. How much money should Mrs. Lopez invest in each type of bond if the bonds are to return a total dividend of $3,000 per year? Round your answers to the nearest cent.

  (20 pts) 5.

Mrs. Redfeather is designing a box. The top, the bottom, and the sides of the box are to be rectangles, but these rectangles are not necessarily of the same size. The dimensions of the box are to be $x$ inches by $10-2x$ inches by $7-2x$ inches. Here $x$ is a number which Mrs. Redfeather has not yet determined. Find the value of $x$ which will result in the box with the largest possible volume. Round your answer to two decimal places. Hint: All three dimensions must be positive real numbers.

  (20 pts) 6.

Let distinct real numbers $a$, $b$, and $c$ be given. Suppose that a certain polynomial $p(x)$ has real coefficients, and suppose that when $p(x)$ is divided by $x-a$, $x-b$, and $x-c$, it leaves remainders $a$, $b$, and $c$, respectively. What is the remainder when $p(x)$ is divided by $(x-a)(x-b)(x-c)$?

  (20 pts) 7.

What is the smallest positive integer $n$ for which ${136}^n$ will have more than $1000$ digits?

  (20 pts) 8.

Suppose that $p(x)$ is a polynomial of degree 2 such that $p(1)=1$, $p(2)=\frac{1}{2}$, and $p(3)=\frac{1}{3}$. Find $p(4)$.

  (20 pts) 9.

Consider a group of 6 people. Each of the six people knows exactly one piece of information, and all 6 pieces of information are different. Every time person “A” calls person “B” on the telephone, “A” tells “B” everything he or she knows, while “B” tells “A” nothing. What is the minimum number of telephone calls between pairs of people needed for everyone to know everything?

  (20 pts) 10.

A rectangular garden is 75 feet long and 32 feet wide. Find the area of the garden. Express your answer in square feet.

  (20 pts) 1. 2400 sq ft

The wheels of a car are exactly 2 feet in diameter. As the car travels, the wheels of the car rotate at the rate of 8 revolutions per second. How fast is the car traveling? Express your answer in feet per second, and round your answer to two decimal places. You may assume that $\pi =3.14159$.

  (20 pts) 2. 50.27 ft/sec

The volume of a certain sphere is exactly 75 cubic inches. Find the radius of the sphere. Express your answer in inches, and round your answer to two decimal places. You may assume that $\pi =3.14159$.

  (20 pts) 3. 2.62 in

A deck of 52 cards has exactly 12 “face cards,” i.e. cards with a king, queen, or jack. Suppose you randomly select three cards from the deck. Find the probability that none of the selected cards is a face card. Round your answer to three decimal places.

  (20 pts) 4. 0.447

Mrs. Lopez plans to invest a total of $50,000 in two different types of bonds: bonds of type A and bonds of type B. Bonds of type A return a dividend of 4% per year, and bonds of type B return a dividend of 7% per year. For example, if Mrs. Lopez invests $40,000 in bonds of type A and $10,000 in bonds of type B, then the bonds will return a total dividend of $\text{\$40,000}\times 0.04+\text{\$10,000}\times 0.07=\text{\$2,300}$ per year. How much money should Mrs. Lopez invest in each type of bond if the bonds are to return a total dividend of $3,000 per year? Round your answers to the nearest cent.

  (20 pts) 5. A: $16,666.67, B: $33,333.33

Mrs. Redfeather is designing a box. The top, the bottom, and the sides of the box are to be rectangles, but these rectangles are not necessarily of the same size. The dimensions of the box are to be $x$ inches by $10-2x$ inches by $7-2x$ inches. Here $x$ is a number which Mrs. Redfeather has not yet determined. Find the value of $x$ which will result in the box with the largest possible volume. Round your answer to two decimal places. Hint: All three dimensions must be positive real numbers.

  (20 pts) 6. 1.35 in

Let distinct real numbers $a$, $b$, and $c$ be given. Suppose that a certain polynomial $p(x)$ has real coefficients, and suppose that when $p(x)$ is divided by $x-a$, $x-b$, and $x-c$, it leaves remainders $a$, $b$, and $c$, respectively. What is the remainder when $p(x)$ is divided by $(x-a)(x-b)(x-c)$?

  (20 pts) 7. $x$

What is the smallest positive integer $n$ for which ${136}^n$ will have more than $1000$ digits?

  (20 pts) 8. $n=469$

Suppose that $p(x)$ is a polynomial of degree 2 such that $p(1)=1$, $p(2)=\frac{1}{2}$, and $p(3)=\frac{1}{3}$. Find $p(4)$.

  (20 pts) 9. $\frac{1}{2}$

Consider a group of 6 people. Each of the six people knows exactly one piece of information, and all 6 pieces of information are different. Every time person “A” calls person “B” on the telephone, “A” tells “B” everything he or she knows, while “B” tells “A” nothing. What is the minimum number of telephone calls between pairs of people needed for everyone to know everything?

  (20 pts) 10. 10 calls

A rectangular garden is 75 feet long and 32 feet wide. Find the area of the garden. Express your answer in square feet.

  (20 pts) 1. 2400 sq ft

The wheels of a car are exactly 2 feet in diameter. As the car travels, the wheels of the car rotate at the rate of 8 revolutions per second. How fast is the car traveling? Express your answer in feet per second, and round your answer to two decimal places. You may assume that $\pi =3.14159$.

  (20 pts) 2. 50.27 ft/sec

Solution: The circumference is $2\pi r=2\pi$ feet. The speed is

$$\frac{8\text{ rev}}{\text{sec}}\times \frac{2\pi \text{ ft}}{\text{rev}}=\frac{16\pi \text{ ft}}{\text{sec}}\approx 50.27\frac{\text{ft}}{\text{sec}}$$

The volume of a certain sphere is exactly 75 cubic inches. Find the radius of the sphere. Express your answer in inches, and round your answer to two decimal places. You may assume that $\pi =3.14159$.

  (20 pts) 3. 2.62 in

Solution: If $r$ is the radius, then the volume is $V=\frac{4}{3}\pi r^3$. Solving, this means

$$r=\sqrt[3]{\frac{3(75)}{4\pi }}\approx 2.62\text{ in}$$

A deck of 52 cards has exactly 12 “face cards,” i.e. cards with a king, queen, or jack. Suppose you randomly select three cards from the deck. Find the probability that none of the selected cards is a face card. Round your answer to three decimal places.

  (20 pts) 4. 0.447

Solution: Imagine that you select the cards in succession. So you randomly select one card. Then you select another card from the remaining 51 cards. Then you select a third card from the remaining 50 cards. We let $P(\mathrm{event})$ denote the probability of the event.

	$P(\text{first card is not a face card})$	$={\displaystyle \frac{40}{52}}$
	$P(\text{second card is not a face card, given that the first wasn’t})$	$={\displaystyle \frac{39}{51}}$
	$P(\text{third card is not, given that the first two weren’t})$	$={\displaystyle \frac{38}{50}}$
	$P(\text{none are face cards})$	$={\displaystyle \frac{40}{52}}\cdot {\displaystyle \frac{39}{51}}\cdot {\displaystyle \frac{38}{50}}\approx 0.447$

Mrs. Lopez plans to invest a total of $50,000 in two different types of bonds: bonds of type A and bonds of type B. Bonds of type A return a dividend of 4% per year, and bonds of type B return a dividend of 7% per year. For example, if Mrs. Lopez invests $40,000 in bonds of type A and $10,000 in bonds of type B, then the bonds will return a total dividend of $\text{\$40,000}\times 0.04+\text{\$10,000}\times 0.07=\text{\$2,300}$ per year. How much money should Mrs. Lopez invest in each type of bond if the bonds are to return a total dividend of $3,000 per year? Round your answers to the nearest cent.

  (20 pts) 5. A: $16,666.67, B: $33,333.33

Solution: Let $A$ be the amount of money, in dollars, that Mrs. Lopez invests in bonds of type A. Let $B$ be the amount of money, in dollars, that Mrs. Lopez invests in bonds of type B. Then

$$\{\begin{array}{cc}A+B=50,000\hfill & \\ 0.04A+0.07B=3000\hfill & \end{array}$$

But then $B=50,000-A$, so

	$0.04A+0.07(50,000-A)$	$=3000$
	$0.07(50,000)+0.04A-0.07A$	$=3000$
	$0.07(50,000)-3000$	$=0.03A$
	${\displaystyle \frac{0.07(50,000)-3000}{0.03}}$	$=A\approx 16,666.67,\text{ and}$
	$B$	$=50,000-A\approx 33,333.33$

Mrs. Redfeather is designing a box. The top, the bottom, and the sides of the box are to be rectangles, but these rectangles are not necessarily of the same size. The dimensions of the box are to be $x$ inches by $10-2x$ inches by $7-2x$ inches. Here $x$ is a number which Mrs. Redfeather has not yet determined. Find the value of $x$ which will result in the box with the largest possible volume. Round your answer to two decimal places. Hint: All three dimensions must be positive real numbers.

  (20 pts) 6. 1.35 in

Solution: All three dimensions must be positive real numbers. So $x>0$, $10-2x>0$, and $7-2x>0$, which means that $5>x$ and $\frac{7}{2}>x$. Thus . The volume of the box will be $V(x)=x(10-2x)(7-2x)$. Graphing this function on a calculator, we can zoom in to see that the largest $V$ occurs when $x\approx 1.35$ in.

Let distinct real numbers $a$, $b$, and $c$ be given. Suppose that a certain polynomial $p(x)$ has real coefficients, and suppose that when $p(x)$ is divided by $x-a$, $x-b$, and $x-c$, it leaves remainders $a$, $b$, and $c$, respectively. What is the remainder when $p(x)$ is divided by $(x-a)(x-b)(x-c)$?

  (20 pts) 7. $x$

Solution: By the division algorithm, there are polynomials $q_a(x)$, $q_b(x)$, and $q_c(x)$ such that

	$p(x)$	$=(x-a)q_a(x)+a$
	$p(x)$	$=(x-b)q_b(x)+b$
	$p(x)$	$=(x-c)q_c(x)+c.$

Thus,

$$p(a)=a,p(b)=b,\text{and}p(c)=c.$$

We may apply the division algorithm again to find that

$$p(x)=(x-a)(x-b)(x-c)q(x)+r(x).$$

Here $q(x)$ and $r(x)$ are polynomials, and $r(x)$ has degree at most 2. By (*) and (**),

$$r(a)=p(a)=a,r(b)=p(b)=b,\text{and}r(c)=p(c)=c.$$

Thus, $r(a)=a$, $r(b)=b$, and $r(c)=c$.

Now let $s(x)=r(x)-x$. Then $s(x)$ has degree at most 2. But $s(a)=r(a)-a=a-a=0$, $s(b)=r(b)-b=b-b=0$, and $s(c)=r(c)-c=c-c=0$, so that $s(x)$ has three distinct zeros. Thus, $s(x)=0$ for all real $x$, and $r(x)=x$ for all real $x$. Therefore, the desired remainder is the polynomial $x$.

What is the smallest positive integer $n$ for which ${136}^n$ will have more than $1000$ digits?

  (20 pts) 8. $n=469$

Solution: If we were to write out the number ${10}^{1000}$, we would write the digit 1 followed by 1000 zeros. Thus ${10}^{1000}$ is the smallest positive integer with more than 1000 digits. We see

	${136}^x$	$={10}^{1000}$		(*)
	${\log }_{10}{136}^x$	$={\log }_{10}{10}^{1000}$
	$x{\log }_{10}136$	$=1000$
	$x$	$={\displaystyle \frac{1000}{{\log }_{10}136}}\approx 468.7.$

Thus, , so that ${136}^{468}$ has fewer than 1001 digits, and ${136}^{469}$ has at least 1001 digits. So the smallest integer $n$ for which ${136}^n$ has more than 1000 digits is $n=469$.

Suppose that $p(x)$ is a polynomial of degree 2 such that $p(1)=1$, $p(2)=\frac{1}{2}$, and $p(3)=\frac{1}{3}$. Find $p(4)$.

  (20 pts) 9. $\frac{1}{2}$

Solution: Consider the polynomial $q(x)=xp(x)-1$, which has degree 3. Also note that

	$q(1)$	$=1p(1)-1=1\cdot 1-1=0$
	$q(2)$	$=2p(2)-1=2\cdot {\displaystyle \frac{1}{2}}-1=0$
	$q(3)$	$=3p(3)-1=3\cdot {\displaystyle \frac{1}{3}}-1=0,$

so that $q(x)=c(x-1)(x-2)(x-3)$ for some nonzero constant $c$. But $q(0)=0p(0)-1=-1=c(-1)(-2)(-3)=-6c$, so that $c=\frac{1}{6}$. Therefore,

$$p(4)=\frac{q(4)+1}{4}=\frac{\frac{1}{6}(4-1)(4-2)(4-3)+1}{4}=\frac{\frac{1}{6}\cdot 3\cdot 2\cdot 1+1}{4}=\frac{2}{4}=\frac{1}{2}.$$

(It is also possible to show that $p(x)=\frac{1}{6}{x}^2-x+\frac{11}{6}$.)

Consider a group of 6 people. Each of the six people knows exactly one piece of information, and all 6 pieces of information are different. Every time person “A” calls person “B” on the telephone, “A” tells “B” everything he or she knows, while “B” tells “A” nothing. What is the minimum number of telephone calls between pairs of people needed for everyone to know everything?

  (20 pts) 10. 10 calls

Solution: Let $p_i$ denote person $i$, $p_i\to p_j$ denote a call from $p_i$ to $p_j$, and $N$ denote the minimum number of calls which can leave everyone fully informed. The following sequence of calls leaves everyone informed:

	$p_1\to p_2,p_2\to p_3,p_3\to p_4,p_4\to p_5,p_5\to p_6,$
	$p_6\to p_1,p_1\to p_2,p_2\to p_3,p_3\to p_4,p_4\to p_5.$

This shows that $N\le 10$.