Problem 1
Find the sum of all real solutions of $$x^4 - 5x^3 + 5x^2 + 5x - 6 = 0$$
A) \(5\)
B) \(6\)
C) \(4\)
D) \(3\)
E) \(7\)
Solution:
Test integer roots: \(x = 1, 2, 3, -1\) all satisfy the equation.
So it factors as \((x-1)(x-2)(x-3)(x+1)\) — all four roots are real.
Sum \(= 1 + 2 + 3 + (-1) = \boxed{5}\) (also equals \(-\) the \(x^3\) coefficient).
Problem 2
Let \(f(x) = 2x^2 - 3x + 1\) with roots \(a\) and \(b\). Find \(\dfrac{1}{a} + \dfrac{1}{b}\).
A) \(2\)
B) \(3\)
C) \(1.5\)
D) \(-2\)
E) \(6\)
Solution:
By Vieta's: \(a + b = \dfrac{3}{2}\), \(ab = \dfrac{1}{2}\).
\(\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{a+b}{ab} = \dfrac{3/2}{1/2}\)
\(= \boxed{3}\)
Problem 3
If \(x, y, z\) are the roots of \(t^3 - 6t^2 + 11t - 6 = 0\), find \(x^2 + y^2 + z^2\).
A) \(12\)
B) \(16\)
C) \(14\)
D) \(11\)
E) \(36\)
Solution:
By Vieta's: \(x+y+z = 6\) and \(xy+yz+zx = 11\).
\(x^2+y^2+z^2 = (x+y+z)^2 - 2(xy+yz+zx)\)
\(= 36 - 22 = \boxed{14}\)
Problem 4
If \(\log_2(x + 1) + \log_2(x - 1) = 3\), find \(x\).
A) \(2.5\)
B) \(4\)
C) \(2\)
D) \(3\)
E) \(-3\)
Solution:
Combine logs: \(\log_2[(x+1)(x-1)] = 3\)
\((x^2 - 1) = 2^3 = 8 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3\)
Domain needs \(x > 1\), so \(x = \boxed{3}\)
Problem 5
Simplify: $$\frac{x^2 - 9}{x^2 - 6x + 9} \cdot \frac{x^2 - 4x + 3}{x^2 + 6x + 9}$$
A) \(\dfrac{x+1}{x+3}\)
B) \(\dfrac{x-1}{x-3}\)
C) \(\dfrac{x+1}{x-3}\)
D) \(\dfrac{1}{x+3}\)
E) \(\dfrac{x-1}{x+3}\)
Solution:
Factor: \(\dfrac{(x+3)(x-3)}{(x-3)^2} \cdot \dfrac{(x-1)(x-3)}{(x+3)^2}\)
Combine: \(\dfrac{(x+3)(x-1)(x-3)^2}{(x-3)^2(x+3)^2}\)
Cancel \((x-3)^2\) and one \((x+3)\): \(\boxed{\dfrac{x-1}{x+3}}\)
Problem 6
Find the sum of all solutions of \(|2x - 4| = |x + 1|\).
A) \(6\)
B) \(4\)
C) \(8\)
D) \(5\)
E) \(7\)
Solution:
Case 1: \(2x - 4 = x + 1 \Rightarrow x = 5\)
Case 2: \(2x - 4 = -(x + 1) \Rightarrow 3x = 3 \Rightarrow x = 1\)
Sum \(= 5 + 1 = \boxed{6}\)
Problem 7
A quadratic \(f(x) = ax^2 + bx + c\) passes through \((0, 3)\), \((1, 6)\), and \((2, 13)\). Find \(f(3)\).
A) \(22\)
B) \(24\)
C) \(26\)
D) \(20\)
E) \(18\)
Solution:
\((0,3)\Rightarrow c=3\); \((1,6)\Rightarrow a+b=3\); \((2,13)\Rightarrow 2a+b=5\)
Solving: \(a = 2\), \(b = 1\), \(c = 3\)
\(f(3) = 2(9) + 3 + 3 = \boxed{24}\)
Problem 8
Solve for \(x\): $$\sqrt{x + 7} - \sqrt{x} = 1$$
A) \(12\)
B) \(6\)
C) \(9\)
D) \(8\)
E) \(16\)
Solution:
\(\sqrt{x+7} = 1 + \sqrt{x}\); square: \(x + 7 = 1 + 2\sqrt{x} + x\)
\(6 = 2\sqrt{x} \Rightarrow \sqrt{x} = 3 \Rightarrow x = 9\)
Check: \(\sqrt{16} - \sqrt{9} = 4 - 3 = 1\) ✓ → \(\boxed{9}\)
Problem 9
If \((x + y)^3 = 216\) and \((x - y)^3 = 8\), find \(xy\).
A) \(15\)
B) \(12\)
C) \(10\)
D) \(8\)
E) \(6\)
Solution:
Cube roots: \(x + y = 6\) and \(x - y = 2\)
Add/subtract: \(x = 4\), \(y = 2\)
\(xy = 4 \cdot 2 = \boxed{8}\)
Problem 10
The parabola \(y = x^2 + 6x + k\) passes through \((2, -4)\). Find the \(x\)-coordinate of its vertex.
A) \(-2\)
B) \(-4\)
C) \(3\)
D) \(-6\)
E) \(-3\)
Solution:
Through \((2,-4)\): \(-4 = 4 + 12 + k \Rightarrow k = -20\)
Vertex \(x = -\dfrac{b}{2a} = -\dfrac{6}{2}\)
\(x = \boxed{-3}\) (independent of \(k\))
Problem 11
Simplify \(i^{2023}\), where \(i = \sqrt{-1}\).
A) \(-i\)
B) \(i\)
C) \(-1\)
D) \(1\)
E) \(0\)
Solution:
Powers of \(i\) cycle every 4: \(i, -1, -i, 1\).
\(2023 \div 4\) leaves remainder \(3\).
\(i^{2023} = i^3 = \boxed{-i}\)
Problem 12
What is the discriminant of \(2x^2 - 4x + 5 = 0\)?
A) \(24\)
B) \(-24\)
C) \(-16\)
D) \(16\)
E) \(-8\)
Solution:
Discriminant \(= b^2 - 4ac\)
\(= (-4)^2 - 4(2)(5) = 16 - 40\)
\(= \boxed{-24}\) (no real solutions)
Problem 13
Find the remainder when \(P(x) = x^3 - 2x^2 + 3x - 4\) is divided by \((x - 2)\).
A) \(-2\)
B) \(4\)
C) \(2\)
D) \(0\)
E) \(6\)
Solution:
Remainder Theorem: remainder \(= P(2)\)
\(P(2) = 8 - 8 + 6 - 4\)
\(= \boxed{2}\)
Problem 14
Find the sum of the geometric series \(3 + 6 + 12 + \cdots\) for the first \(6\) terms.
A) \(192\)
B) \(186\)
C) \(195\)
D) \(189\)
E) \(96\)
Solution:
\(a = 3\), \(r = 2\), \(n = 6\)
\(S_n = \dfrac{a(r^n - 1)}{r - 1} = \dfrac{3(64 - 1)}{1}\)
\(= 3 \cdot 63 = \boxed{189}\)
Problem 15
Find the sum of the infinite geometric series \(16 + 8 + 4 + 2 + \cdots\)
A) \(24\)
B) \(28\)
C) \(30\)
D) \(16\)
E) \(32\)
Solution:
\(a = 16\), \(r = \dfrac{1}{2}\) (\(|r| < 1\), so it converges)
\(S = \dfrac{a}{1 - r} = \dfrac{16}{1 - \tfrac{1}{2}}\)
\(= \dfrac{16}{1/2} = \boxed{32}\)
Problem 16
Evaluate \(\log_3 81\).
A) \(4\)
B) \(3\)
C) \(27\)
D) \(9\)
E) \(5\)
Solution:
Ask: \(3\) to what power equals \(81\)?
\(3^4 = 81\)
\(\log_3 81 = \boxed{4}\)
Problem 17
If \(f(x) = 2x + 3\) and \(g(x) = x^2\), find \(f(g(2))\).
A) \(7\)
B) \(11\)
C) \(14\)
D) \(19\)
E) \(49\)
Solution:
Inner first: \(g(2) = 2^2 = 4\)
Then \(f(4) = 2(4) + 3\)
\(= \boxed{11}\)
Problem 18
Find the inverse of \(f(x) = \dfrac{x - 3}{2}\).
A) \(\dfrac{x+3}{2}\)
B) \(2x - 3\)
C) \(2x + 3\)
D) \(\dfrac{x-3}{2}\)
E) \(\dfrac{x}{2} + 3\)
Solution:
Let \(y = \dfrac{x-3}{2}\), then swap \(x\) and \(y\): \(x = \dfrac{y-3}{2}\)
\(2x = y - 3 \Rightarrow y = 2x + 3\)
\(f^{-1}(x) = \boxed{2x + 3}\)
Problem 19
Solve for \(x\): \(2^{\,x+1} = 32\).
A) \(5\)
B) \(3\)
C) \(6\)
D) \(4\)
E) \(16\)
Solution:
Write \(32\) as a power of \(2\): \(32 = 2^5\)
\(2^{x+1} = 2^5 \Rightarrow x + 1 = 5\)
\(x = \boxed{4}\)
Problem 20
Solve \(x^2 = -9\) over the complex numbers.
A) \(\pm 9i\)
B) \(\pm 3\)
C) \(\pm i\sqrt{3}\)
D) \(3i\)
E) \(\pm 3i\)
Solution:
\(x = \pm\sqrt{-9} = \pm\sqrt{9}\,\sqrt{-1}\)
\(\sqrt{-1} = i\)
\(x = \boxed{\pm 3i}\)
Problem 21
What is the coefficient of \(x^2\) in the expansion of \((x + 2)^4\)?
A) \(24\)
B) \(16\)
C) \(12\)
D) \(32\)
E) \(8\)
Solution:
Binomial term: \(\binom{4}{k} x^{4-k} 2^k\); for \(x^2\), \(k = 2\)
\(\binom{4}{2} \cdot 2^2 = 6 \cdot 4\)
\(= \boxed{24}\)
Problem 22
Write \(y = x^2 - 4x + 7\) in vertex form. What is the \(y\)-coordinate of the vertex?
A) \(7\)
B) \(3\)
C) \(-2\)
D) \(2\)
E) \(4\)
Solution:
Complete the square: \(x^2 - 4x + 7 = (x^2 - 4x + 4) + 3\)
\(= (x - 2)^2 + 3\), so the vertex is \((2, 3)\)
\(y\)-coordinate \(= \boxed{3}\)
Problem 23
Solve the system \(y = x^2\) and \(y = x + 2\). What is the sum of the \(x\)-values of the solutions?
A) \(-1\)
B) \(3\)
C) \(1\)
D) \(2\)
E) \(0\)
Solution:
Set equal: \(x^2 = x + 2 \Rightarrow x^2 - x - 2 = 0\)
\((x - 2)(x + 1) = 0 \Rightarrow x = 2, -1\)
Sum \(= 2 + (-1) = \boxed{1}\)
Problem 24
Rationalize and simplify: \(\dfrac{3}{\sqrt{5} - \sqrt{2}}\).
A) \(\sqrt{5} - \sqrt{2}\)
B) \(\dfrac{\sqrt{5} + \sqrt{2}}{3}\)
C) \(3(\sqrt{5} + \sqrt{2})\)
D) \(\sqrt{5} + \sqrt{2}\)
E) \(\sqrt{7}\)
Solution:
Multiply by the conjugate: \(\dfrac{3}{\sqrt5 - \sqrt2} \cdot \dfrac{\sqrt5 + \sqrt2}{\sqrt5 + \sqrt2}\)
Denominator: \((\sqrt5)^2 - (\sqrt2)^2 = 5 - 2 = 3\)
\(= \dfrac{3(\sqrt5 + \sqrt2)}{3} = \boxed{\sqrt{5} + \sqrt{2}}\)
Problem 25
Evaluate the determinant \(\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix}\).
A) \(2\)
B) \(-22\)
C) \(22\)
D) \(14\)
E) \(-2\)
Solution:
\(2\times2\) determinant \(= ad - bc\)
\(= (2)(5) - (3)(4) = 10 - 12\)
\(= \boxed{-2}\)
Problem 26
Find the sum of the first \(20\) positive even integers \(2 + 4 + 6 + \cdots + 40\).
A) \(420\)
B) \(400\)
C) \(380\)
D) \(441\)
E) \(210\)
Solution:
Arithmetic series: \(S = \dfrac{n}{2}(\text{first} + \text{last})\)
\(= \dfrac{20}{2}(2 + 40) = 10 \cdot 42\)
\(= \boxed{420}\)
Problem 27
Solve \(\log_2 x + \log_2 (x - 2) = 3\).
A) \(2\)
B) \(4\)
C) \(6\)
D) \(8\)
E) \(-2\)
Solution:
Combine: \(\log_2[x(x-2)] = 3 \Rightarrow x^2 - 2x = 8\)
\(x^2 - 2x - 8 = 0 \Rightarrow (x-4)(x+2) = 0\)
Domain needs \(x > 2\), so \(x = \boxed{4}\)
Problem 28
For what positive value of \(x\) is \(\dfrac{x}{x^2 - 4}\) undefined?
A) \(4\)
B) \(-2\)
C) \(2\)
D) \(0\)
E) \(1\)
Solution:
Undefined where the denominator is \(0\): \(x^2 - 4 = 0\)
\(x = \pm 2\)
The positive value is \(x = \boxed{2}\)
Problem 29
Which quadratic equation has roots \(3\) and \(-5\)?
A) \(x^2 - 2x - 15 = 0\)
B) \(x^2 + 2x + 15 = 0\)
C) \(x^2 - 2x + 15 = 0\)
D) \(x^2 + 2x - 15 = 0\)
E) \(x^2 + 8x - 15 = 0\)
Solution:
Sum of roots \(= 3 + (-5) = -2\); product \(= 3 \cdot (-5) = -15\)
\(x^2 - (\text{sum})x + (\text{product}) = x^2 - (-2)x + (-15)\)
\(= \boxed{x^2 + 2x - 15 = 0}\)
Problem 30
A population starts at \(500\) and doubles every year. What is the population after \(3\) years?
A) \(3000\)
B) \(1500\)
C) \(4500\)
D) \(8000\)
E) \(4000\)
Solution:
Exponential growth: \(A = 500 \cdot 2^t\) with \(t = 3\)
\(A = 500 \cdot 2^3 = 500 \cdot 8\)
\(= \boxed{4000}\)