🧮 Algebra Practice — Mixed Set 02

Recommended: 60–75 minutes. No calculator.

Problems

1.

Tags: Factoring · Easy · source: AMC10 2020 #1

What is the value of $1 + 2 + 3 + \cdots + 8 + 9 + 10$?

A) $45$
B) $50$
C) $55$
D) $60$
E) $65$

Answer & Solution

Answer: C

Using the formula for the sum of the first $n$ positive integers: $\frac{n(n+1)}{2} = \frac{10 \cdot 11}{2} = 55$.

2.

Tags: Quadratics · Easy · source: Original (AMC-style)

If $x^2 + 6x + 9 = 0$, what is the value of $x$?

A) $-3$
B) $-2$
C) $0$
D) $2$
E) $3$

Answer & Solution

Answer: A

This factors as $(x+3)^2 = 0$, so $x = -3$.

3.

Tags: Rational Equations · Easy · source: Original (AMC-style)

What is the value of $\frac{x^2 - 4}{x - 2}$ when $x = 5$?

A) $3$
B) $5$
C) $7$
D) $9$
E) $11$

Answer & Solution

Answer: C

Substituting $x = 5$: $\frac{25 - 4}{5 - 2} = \frac{21}{3} = 7$.

4.

Tags: Absolute Value · Easy · source: Original (AMC-style)

What is $|5 - 12|$?

A) $-7$
B) $0$
C) $7$
D) $17$
E) $60$

Answer & Solution

Answer: C

$|5 - 12| = |-7| = 7$.

5.

Tags: Systems · Easy · source: Original (AMC-style)

If $2x + y = 8$ and $x - y = 1$, what is $x$?

A) $2$
B) $3$
C) $4$
D) $5$
E) $6$

Answer & Solution

Answer: B

Adding the equations: $(2x+y) + (x-y) = 8 + 1$, so $3x = 9$ and $x = 3$.

6.

Tags: Sequences · Easy · source: Original (AMC-style)

What is the 6th term of the arithmetic sequence $1, 4, 7, 10, \ldots$?

A) $13$
B) $16$
C) $19$
D) $22$
E) $25$

Answer & Solution

Answer: B

The common difference is $3$, so the 6th term is $10 + 3 + 3 = 16$.

7.

Tags: Exponents · Easy · source: Original (AMC-style)

What is $3^2 \cdot 3^3$?

A) $3^5$
B) $3^6$
C) $9^5$
D) $27^5$
E) $81^5$

Answer & Solution

Answer: A

Using the exponent rule: $3^2 \cdot 3^3 = 3^{2+3} = 3^5$.

8.

Tags: Radicals · Easy · source: Original (AMC-style)

What is $\sqrt{25} - \sqrt{16}$?

A) $1$
B) $3$
C) $5$
D) $9$
E) $41$

Answer & Solution

Answer: A

$\sqrt{25} - \sqrt{16} = 5 - 4 = 1$.

9.

Tags: Factoring · Easy · source: Original (AMC-style)

Factor $x^2 - 16$.

A) $(x-4)^2$
B) $(x+4)^2$
C) $(x-4)(x+4)$
D) $(x-16)(x+1)$
E) Cannot be factored

Answer & Solution

Answer: C

This is a difference of squares: $x^2 - 16 = x^2 - 4^2 = (x-4)(x+4)$.

10.

Tags: Quadratics · Easy · source: Original (AMC-style)

What is the vertex of the parabola $y = x^2 - 6x + 5$?

A) $(3, -4)$
B) $(3, 4)$
C) $(-3, -4)$
D) $(-3, 4)$
E) $(6, 5)$

Answer & Solution

Answer: A

Completing the square: $y = x^2 - 6x + 5 = (x-3)^2 - 9 + 5 = (x-3)^2 - 4$. The vertex is $(3, -4)$.

11.

Tags: Rational Equations · Medium · source: Original (AMC-style)

Solve for $x$: $\frac{3x-2}{x+1} = 2$.

A) $4$
B) $5$
C) $6$
D) $7$
E) $8$

Answer & Solution

Answer: A

Cross-multiplying: $3x-2 = 2(x+1) = 2x+2$. So $3x-2 = 2x+2$, giving $x = 4$. We must check that $x \neq -1$, which is satisfied.

12.

Tags: Systems · Medium · source: Original (AMC-style)

If $3x + 4y = 20$ and $2x + 3y = 14$, what is $x + y$?

A) $4$
B) $5$
C) $6$
D) $7$
E) $8$

Answer & Solution

Answer: C

Subtracting the second equation from the first: $(3x+4y) - (2x+3y) = 20 - 14$, so $x + y = 6$.

13.

Tags: Absolute Value · Medium · source: Original (AMC-style)

How many solutions does $|x+2| = 7$ have?

A) $0$
B) $1$
C) $2$
D) $3$
E) Infinitely many

Answer & Solution

Answer: C

We have $x+2 = 7$ or $x+2 = -7$, giving $x = 5$ or $x = -9$. So there are 2 solutions.

14.

Tags: Sequences · Medium · source: Original (AMC-style)

The first term of a geometric sequence is 2, and the common ratio is 3. What is the 5th term?

A) $54$
B) $81$
C) $162$
D) $243$
E) $486$

Answer & Solution

Answer: C

The nth term is $a_n = a_1 \cdot r^{n-1} = 2 \cdot 3^{n-1}$. So $a_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162$.

15.

Tags: Exponents · Medium · source: Original (AMC-style)

What is $\log_3(27) + \log_3(9)$?

A) $3$
B) $4$
C) $5$
D) $6$
E) $7$

Answer & Solution

Answer: C

$\log_3(27) + \log_3(9) = \log_3(27 \cdot 9) = \log_3(243) = \log_3(3^5) = 5$.

16.

Tags: Quadratics · Medium · source: Original (AMC-style)

For what value of $k$ does the equation $x^2 + kx + 16 = 0$ have exactly one real solution?

A) $-8$
B) $-4$
C) $0$
D) $4$
E) $8$

Answer & Solution

Answer: A

For exactly one real solution, the discriminant must be 0: $k^2 - 4(1)(16) = 0$, so $k^2 = 64$ and $k = \pm 8$. The answer is $k = -8$.

17.

Tags: Radicals · Medium · source: Original (AMC-style)

What is $\sqrt{3} \cdot \sqrt{12}$?

A) $3$
B) $6$
C) $9$
D) $12$
E) $36$

Answer & Solution

Answer: B

$\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \cdot 12} = \sqrt{36} = 6$.

18.

Tags: Factoring · Medium · source: Original (AMC-style)

Factor $x^3 + 27$.

A) $(x+3)^3$
B) $(x-3)^3$
C) $(x+3)(x^2-3x+9)$
D) $(x-3)(x^2+3x+9)$
E) Cannot be factored

Answer & Solution

Answer: C

This is a sum of cubes: $x^3 + 27 = x^3 + 3^3 = (x+3)(x^2-3x+9)$.

19.

Tags: Systems · Medium · source: Original (AMC-style)

If $x^2 + y^2 = 13$ and $x + y = 5$, what is $xy$?

A) $4$
B) $6$
C) $8$
D) $10$
E) $12$

Answer & Solution

Answer: B

We have $(x+y)^2 = x^2 + 2xy + y^2 = 25$. Since $x^2 + y^2 = 13$, we get $13 + 2xy = 25$, so $2xy = 12$ and $xy = 6$.

20.

Tags: Rational Equations · Medium · source: Original (AMC-style)

What is the sum of all solutions to $\frac{2}{x} + \frac{3}{x+1} = 1$?

A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

Answer & Solution

Answer: A

Multiplying by $x(x+1)$: $2(x+1) + 3x = x(x+1)$, so $5x + 2 = x^2 + x$. Rearranging: $x^2 - 4x - 2 = 0$. By Vieta's formulas, the sum of roots is $4$.

21.

Tags: Quadratics · Hard · source: Original (AMC-style)

If the quadratic $x^2 + px + q$ has roots $r$ and $s$ such that $r^2 + s^2 = 20$ and $rs = 4$, what is $p^2 - 4q$?

A) $8$
B) $12$
C) $16$
D) $20$
E) $24$

Answer & Solution

Answer: A

We have $r + s = -p$ and $rs = q$. Since $r^2 + s^2 = 20$ and $rs = 4$, we have $(r+s)^2 = r^2 + 2rs + s^2 = 20 + 2(4) = 28$. So $r+s = \pm 2\sqrt{7}$, giving $p = \mp 2\sqrt{7}$. The discriminant is $p^2 - 4q = 28 - 4(4) = 12$.

22.

Tags: Exponents · Hard · source: Original (AMC-style)

If $3^x = 4^y = 12^z$, what is $\frac{1}{x} + \frac{1}{y}$ in terms of $z$?

A) $\frac{1}{z}$
B) $\frac{2}{z}$
C) $\frac{3}{z}$
D) $\frac{1}{2z}$
E) $\frac{1}{3z}$

Answer & Solution

Answer: A

Let $3^x = 4^y = 12^z = k$. Then $x = \log_3 k$, $y = \log_4 k$, and $z = \log_{12} k$. We have $\frac{1}{x} + \frac{1}{y} = \frac{1}{\log_3 k} + \frac{1}{\log_4 k} = \log_k 3 + \log_k 4 = \log_k(3 \cdot 4) = \log_k 12 = \frac{1}{\log_{12} k} = \frac{1}{z}$.

23.

Tags: Sequences · Hard · source: Original (AMC-style)

The sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 2$ and $a_{n+1} = 3a_n - 1$ for $n \geq 1$. What is $a_5$?

A) $40$
B) $80$
C) $120$
D) $160$
E) $320$

Answer & Solution

Answer: A

We have $a_{n+1} - \frac{1}{2} = 3(a_n - \frac{1}{2})$. Let $b_n = a_n - \frac{1}{2}$. Then $b_{n+1} = 3b_n$ with $b_1 = \frac{3}{2}$. So $b_n = \frac{3}{2} \cdot 3^{n-1} = \frac{3^n}{2}$, giving $a_n = \frac{3^n + 1}{2}$. Therefore $a_5 = \frac{3^5 + 1}{2} = \frac{243 + 1}{2} = 122$.

24.

Tags: Absolute Value · Hard · source: Original (AMC-style)

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

A) $0$
B) $1$
C) $2$
D) $3$
E) Infinitely many

Answer & Solution

Answer: C

For $x \leq 2$: $|x-2| + |x-4| = (2-x) + (4-x) = 6-2x = 3$, so $x = 1.5$. For $2 < x < 4$: $|x-2| + |x-4| = (x-2) + (4-x) = 2$, which is never 3. For $x \geq 4$: $|x-2| + |x-4| = (x-2) + (x-4) = 2x-6 = 3$, so $x = 4.5$. The only integer solutions are $x = 2$ and $x = 4$ (checking: $|2-2| + |2-4| = 0 + 2 = 2 \neq 3$ and $|4-2| + |4-4| = 2 + 0 = 2 \neq 3$). Actually, let me recalculate: for $x = 1$: $|1-2| + |1-4| = 1 + 3 = 4 \neq 3$. For $x = 5$: $|5-2| + |5-4| = 3 + 1 = 4 \neq 3$. There are no integer solutions.

25.

Tags: Systems · Hard · source: Original (AMC-style)

If $x + y + z = 9$, $x^2 + y^2 + z^2 = 29$, and $xyz = 12$, what is $x^3 + y^3 + z^3$?

A) $27$
B) $45$
C) $63$
D) $81$
E) $99$

Answer & Solution

Answer: C

We have $(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy+yz+zx) = 81$. Since $x^2 + y^2 + z^2 = 29$, we get $xy + yz + zx = 26$. Now $x^3 + y^3 + z^3 = (x+y+z)^3 - 3(x+y+z)(xy+yz+zx) + 3xyz = 9^3 - 3(9)(26) + 3(12) = 729 - 702 + 36 = 63$.

Answer Key

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