Problem 1
Find the exact value of \(\sin\dfrac{\pi}{3}\).
A) \(\dfrac{\sqrt{3}}{2}\)
B) \(\dfrac{1}{2}\)
C) \(\dfrac{\sqrt{2}}{2}\)
D) \(\sqrt{3}\)
E) \(\dfrac{\sqrt{3}}{3}\)
Solution:
\(\dfrac{\pi}{3} = 60^\circ\).
From the special triangle, \(\sin 60^\circ = \dfrac{\sqrt3}{2}\).
\(= \boxed{\dfrac{\sqrt{3}}{2}}\)
Problem 2
Find the exact value of \(\cos\dfrac{5\pi}{6}\).
A) \(\dfrac{\sqrt{3}}{2}\)
B) \(-\dfrac{\sqrt{3}}{2}\)
C) \(-\dfrac{1}{2}\)
D) \(\dfrac{1}{2}\)
E) \(-\dfrac{\sqrt{2}}{2}\)
Solution:
\(\dfrac{5\pi}{6} = 150^\circ\) lies in Quadrant II, where cosine is negative.
Reference angle \(30^\circ\): \(\cos 30^\circ = \dfrac{\sqrt3}{2}\).
\(\cos 150^\circ = \boxed{-\dfrac{\sqrt{3}}{2}}\)
Problem 3
Find the exact value of \(\tan 135^\circ\).
A) \(1\)
B) \(\sqrt{3}\)
C) \(-1\)
D) \(-\sqrt{3}\)
E) \(0\)
Solution:
\(135^\circ\) is in Quadrant II with reference angle \(45^\circ\).
Tangent is negative in Quadrant II; \(\tan 45^\circ = 1\).
\(\tan 135^\circ = \boxed{-1}\)
Problem 4
What is the amplitude of \(y = 3\sin(2x)\)?
A) \(2\)
B) \(6\)
C) \(\dfrac{1}{2}\)
D) \(3\)
E) \(\pi\)
Solution:
For \(y = A\sin(Bx)\), the amplitude is \(|A|\).
Here \(A = 3\).
Amplitude \(= \boxed{3}\)
Problem 5
What is the period of \(y = 2\cos(3x)\)?
A) \(3\)
B) \(2\pi\)
C) \(\dfrac{\pi}{3}\)
D) \(6\pi\)
E) \(\dfrac{2\pi}{3}\)
Solution:
For \(y = A\cos(Bx)\), the period is \(\dfrac{2\pi}{|B|}\).
Here \(B = 3\).
Period \(= \boxed{\dfrac{2\pi}{3}}\)
Problem 6
Find the sum of all solutions of \(\sin x = \dfrac{1}{2}\) on \([0, 2\pi)\).
A) \(\pi\)
B) \(\dfrac{\pi}{2}\)
C) \(\dfrac{2\pi}{3}\)
D) \(\dfrac{\pi}{3}\)
E) \(\dfrac{5\pi}{6}\)
Solution:
\(\sin x = \tfrac12\) at \(x = \dfrac{\pi}{6}\) and \(x = \dfrac{5\pi}{6}\) in \([0,2\pi)\).
Sum \(= \dfrac{\pi}{6} + \dfrac{5\pi}{6} = \dfrac{6\pi}{6}\).
\(= \boxed{\pi}\)
Problem 7
The terminal side of \(\dfrac{5\pi}{6}\) meets the unit circle at point \(P\). Find the coordinates of \(P\).
A) \(\left(-\dfrac{1}{2}, \dfrac{\sqrt3}{2}\right)\)
B) \(\left(-\dfrac{\sqrt3}{2}, \dfrac{1}{2}\right)\)
C) \(\left(\dfrac{\sqrt3}{2}, \dfrac{1}{2}\right)\)
D) \(\left(-\dfrac{\sqrt3}{2}, -\dfrac{1}{2}\right)\)
E) \(\left(\dfrac{1}{2}, -\dfrac{\sqrt3}{2}\right)\)
Solution:
On the unit circle, \(P = (\cos\theta, \sin\theta)\).
\(\cos\dfrac{5\pi}{6} = -\dfrac{\sqrt3}{2}\), \(\sin\dfrac{5\pi}{6} = \dfrac{1}{2}\).
\(P = \boxed{\left(-\dfrac{\sqrt3}{2}, \dfrac{1}{2}\right)}\)
Problem 8
In the triangle below, two sides are 5 and 7 with an included angle of \(60^\circ\). Find the third side \(c\).
A) \(\sqrt{74}\)
B) \(8\)
C) \(\sqrt{39}\)
D) \(\sqrt{109}\)
E) \(\sqrt{14}\)
Solution:
Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos C\).
\(c^2 = 25 + 49 - 2(5)(7)\cos 60^\circ = 74 - 70\cdot\tfrac12 = 39\).
\(c = \boxed{\sqrt{39}}\)
Problem 9
In the triangle below, \(\angle A = 30^\circ\), \(\angle B = 45^\circ\), and side \(a = 6\) (opposite \(A\)). Find side \(b\) (opposite \(B\)).
A) \(6\)
B) \(3\sqrt{2}\)
C) \(12\)
D) \(6\sqrt{2}\)
E) \(6\sqrt{3}\)
Solution:
Law of Sines: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\).
\(b = \dfrac{a\sin B}{\sin A} = \dfrac{6\sin 45^\circ}{\sin 30^\circ} = \dfrac{6\cdot \tfrac{\sqrt2}{2}}{\tfrac12}\).
\(= \boxed{6\sqrt{2}}\)
Problem 10
If \(\sin\theta = \dfrac{3}{5}\) and \(\cos\theta = \dfrac{4}{5}\), find \(\sin 2\theta\).
A) \(\dfrac{12}{25}\)
B) \(\dfrac{7}{25}\)
C) \(\dfrac{24}{5}\)
D) \(\dfrac{48}{25}\)
E) \(\dfrac{24}{25}\)
Solution:
Double-angle identity: \(\sin 2\theta = 2\sin\theta\cos\theta\).
\(= 2\cdot \dfrac{3}{5}\cdot \dfrac{4}{5} = \dfrac{24}{25}\).
\(= \boxed{\dfrac{24}{25}}\)
Problem 11
Find \(\arctan(1)\) (principal value, in radians).
A) \(\dfrac{\pi}{4}\)
B) \(\dfrac{\pi}{3}\)
C) \(\dfrac{\pi}{6}\)
D) \(\dfrac{\pi}{2}\)
E) \(1\)
Solution:
We need the angle in \(\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)\) whose tangent is \(1\).
\(\tan\dfrac{\pi}{4} = 1\).
\(\arctan(1) = \boxed{\dfrac{\pi}{4}}\)
Problem 12
Evaluate \((1 + i)^8\), where \(i = \sqrt{-1}\).
A) \(8\)
B) \(16\)
C) \(256\)
D) \(-16\)
E) \(1\)
Solution:
In polar form, \(1 + i = \sqrt{2}\,\text{cis}\,45^\circ\).
By De Moivre: \((1+i)^8 = (\sqrt2)^8\,\text{cis}(8\cdot 45^\circ) = 16\,\text{cis}\,360^\circ\).
\(= 16(1) = \boxed{16}\)
Problem 13
Convert \(2\,\text{cis}\,\dfrac{\pi}{2}\) (that is, \(2(\cos\tfrac{\pi}{2} + i\sin\tfrac{\pi}{2})\)) to rectangular form.
A) \(2\)
B) \(-2i\)
C) \(2i\)
D) \(-2\)
E) \(0\)
Solution:
\(\cos\dfrac{\pi}{2} = 0\) and \(\sin\dfrac{\pi}{2} = 1\).
\(2(0 + i\cdot 1) = 2i\).
\(= \boxed{2i}\)
Problem 14
Find the magnitude of the vector \(\vec{v} = \langle 3, 4 \rangle\).
A) \(7\)
B) \(\sqrt{7}\)
C) \(25\)
D) \(5\)
E) \(12\)
Solution:
\(|\vec{v}| = \sqrt{x^2 + y^2}\).
\(= \sqrt{3^2 + 4^2} = \sqrt{25}\).
\(= \boxed{5}\)
Problem 15
Which graph represents \(y = 2\sin x\) over \([0, 2\pi]\)?
Solution:
\(y = 2\sin x\) starts at the origin (value 0) and rises first.
Its amplitude is 2 (peak at \(+2\), trough at \(-2\)) and period \(2\pi\).
The correct graph is the sine curve peaking at 2, not cosine, not reflected, and not compressed.
Problem 16
Find the sum of the infinite geometric series \(4 + \dfrac{4}{3} + \dfrac{4}{9} + \cdots\)
A) \(6\)
B) \(12\)
C) \(\dfrac{16}{3}\)
D) \(3\)
E) \(8\)
Solution:
\(a = 4\), \(r = \dfrac{1}{3}\) (\(|r| < 1\)).
\(S = \dfrac{a}{1 - r} = \dfrac{4}{1 - \tfrac13} = \dfrac{4}{\tfrac23}\).
\(= \boxed{6}\)
Problem 17
An arithmetic sequence has first term 3 and common difference 4. Find the 10th term.
A) \(36\)
B) \(39\)
C) \(43\)
D) \(35\)
E) \(40\)
Solution:
\(a_n = a_1 + (n-1)d\).
\(a_{10} = 3 + (10-1)(4) = 3 + 36\).
\(= \boxed{39}\)
Problem 18
Find the coefficient of \(x^3\) in the expansion of \((x + 2)^5\).
A) \(80\)
B) \(10\)
C) \(40\)
D) \(32\)
E) \(20\)
Solution:
General term: \(\binom{5}{k}x^k 2^{\,5-k}\); for \(x^3\), \(k = 3\).
\(\binom{5}{3}\cdot 2^{2} = 10 \cdot 4\).
\(= \boxed{40}\)
Problem 19
Solve for \(x\): \(3^{\,2x} = 27\).
A) \(3\)
B) \(9\)
C) \(\dfrac{2}{3}\)
D) \(\dfrac{3}{2}\)
E) \(6\)
Solution:
\(27 = 3^3\), so \(3^{2x} = 3^3\).
\(2x = 3\).
\(x = \boxed{\dfrac{3}{2}}\)
Problem 20
Evaluate \(\log_2 32 + \log_3 27\).
A) \(6\)
B) \(5\)
C) \(15\)
D) \(9\)
E) \(8\)
Solution:
\(\log_2 32 = 5\) since \(2^5 = 32\).
\(\log_3 27 = 3\) since \(3^3 = 27\).
\(5 + 3 = \boxed{8}\)
Problem 21
Find the horizontal asymptote of \(f(x) = \dfrac{2x^2 + 1}{x^2 - 4}\).
A) \(y = 2\)
B) \(y = 0\)
C) \(y = 1\)
D) \(y = 4\)
E) \(y = -2\)
Solution:
Numerator and denominator have equal degree (2).
The horizontal asymptote is the ratio of leading coefficients: \(\dfrac{2}{1}\).
\(\boxed{y = 2}\)
Problem 22
Where are the vertical asymptotes of \(f(x) = \dfrac{x + 1}{x^2 - 9}\)?
A) \(x = 3\) only
B) \(x = 3\) and \(x = -3\)
C) \(x = 9\)
D) \(x = -1\)
E) \(x = 0\)
Solution:
Vertical asymptotes occur where the denominator is 0 (and numerator is not).
\(x^2 - 9 = 0 \Rightarrow x = \pm 3\); the numerator \(x+1\) is nonzero there.
\(\boxed{x = 3 \text{ and } x = -3}\)
Problem 23
If \(f(x) = x^2 - 1\) and \(g(x) = 2x + 3\), find \(f(g(1))\).
A) \(8\)
B) \(15\)
C) \(24\)
D) \(10\)
E) \(48\)
Solution:
Inner first: \(g(1) = 2(1) + 3 = 5\).
Then \(f(5) = 5^2 - 1 = 24\).
\(= \boxed{24}\)
Problem 24
Find the inverse of \(f(x) = \dfrac{2x - 1}{3}\).
A) \(\dfrac{2x + 1}{3}\)
B) \(\dfrac{3x - 1}{2}\)
C) \(\dfrac{3}{2x - 1}\)
D) \(\dfrac{3x + 1}{2}\)
E) \(\dfrac{x + 1}{2}\)
Solution:
Let \(y = \dfrac{2x-1}{3}\) and swap variables: \(x = \dfrac{2y-1}{3}\).
\(3x = 2y - 1 \Rightarrow 2y = 3x + 1\).
\(f^{-1}(x) = \boxed{\dfrac{3x + 1}{2}}\)
Problem 25
Find the remainder when \(P(x) = 2x^3 - x + 5\) is divided by \((x + 1)\).
A) \(-2\)
B) \(6\)
C) \(2\)
D) \(8\)
E) \(4\)
Solution:
Remainder Theorem: remainder \(= P(-1)\).
\(P(-1) = 2(-1)^3 - (-1) + 5 = -2 + 1 + 5\).
\(= \boxed{4}\)
Problem 26
Evaluate \(\displaystyle\sum_{k=1}^{4} (2k + 1)\).
A) \(24\)
B) \(20\)
C) \(16\)
D) \(28\)
E) \(10\)
Solution:
Terms: \(k=1{:}\,3,\; k=2{:}\,5,\; k=3{:}\,7,\; k=4{:}\,9\).
\(3 + 5 + 7 + 9\).
\(= \boxed{24}\)
Problem 27
Simplify \((3 + 2i)(1 - i)\).
A) \(1 - i\)
B) \(5 - i\)
C) \(5 + i\)
D) \(1 + 5i\)
E) \(3 - i\)
Solution:
FOIL: \(3 - 3i + 2i - 2i^2\).
Since \(i^2 = -1\): \(3 - i + 2\).
\(= \boxed{5 - i}\)
Problem 28
Find the modulus \(|3 - 4i|\).
A) \(7\)
B) \(\sqrt{7}\)
C) \(5\)
D) \(25\)
E) \(1\)
Solution:
\(|a + bi| = \sqrt{a^2 + b^2}\).
\(= \sqrt{3^2 + (-4)^2} = \sqrt{25}\).
\(= \boxed{5}\)
Problem 29
For the ellipse \(\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\), find the length of the major axis.
A) \(5\)
B) \(6\)
C) \(25\)
D) \(10\)
E) \(8\)
Solution:
\(a^2 = 25\) (the larger denominator), so \(a = 5\).
The major axis has length \(2a = 2(5)\).
\(= \boxed{10}\)
Problem 30
Find the constant term in the expansion of \(\left(x + \dfrac{1}{x}\right)^6\).
A) \(15\)
B) \(6\)
C) \(64\)
D) \(1\)
E) \(20\)
Solution:
General term: \(\binom{6}{k}x^{6-k}\left(\dfrac1x\right)^k = \binom{6}{k}x^{6-2k}\).
Constant term needs \(6 - 2k = 0 \Rightarrow k = 3\).
\(\binom{6}{3} = \boxed{20}\)