In the figure below, \( AB \) and \( GE \) are parallel. Triangle \( ACD \) is isosceles with \( CA = CD \). The measures of angles \( FDE \) and \( GDH \) are \( 60^\circ \) and \( 65^\circ \) respectively. What is the measure of angle \( CAB \)?
B) \( 6^\circ \)
C) \( 10^\circ \)
D) \( 15^\circ \)
E) \( 5^\circ \)
If \( a \), \( b \), and \( y \) are positive real numbers (none equal to 1) and \( b^{2a + 6} = y^2 \), which of these must be true?
A) \( y = b^{a + 3} \)B) \( 2a + 6 = 2 \)
C) \( b = y \)
D) \( b = 2 \)
E) \( b(2a + 6) = 2y \)
What is true about the graph of the function \( f \) defined by:
\[ f(x) = 4 + \big| |x| - 3 \big| \](I) Symmetric with respect to the y-axis
(II) Has no x-intercepts
(III) Has a y-intercept at \( (0, 7) \)
B) (I) and (III) only
C) (I), (II), and (III)
D) None
E) (II) and (III) only
A number \( x \) is divided by \( 0.71 \) and the result decreased by \( 0.2 \). If the fifth root of the final result equals \( -0.15 \), what is \( x \) rounded to the nearest thousandth?
A) \( 0.141 \)B) \( 0.71 \)
C) \( 0.143 \)
D) \( 0.142 \)
E) \( 0.036 \)
Find the constant \( k \) so that the perpendicular bisector of the line segment with endpoints \( (k, 0) \) and \( (4, 6) \) has a slope of \( -3 \).
A) \( 4 \)B) \( -14 \)
C) \( 6 \)
D) \( 22 \)
E) \( 0 \)
Forty students are each registered for one or more of three courses. 23 students are in one course only, and 14 are in two courses only. How many are registered for all three courses?
A) \( 3 \)B) \( 11 \)
C) \( 37 \)
D) \( 17 \)
E) \( 26 \)
Which statements are true about the graphs of \( x^2 + y^2 = 9 \) and \( (x + 6)^2 + y^2 = 9 \)?
(I) Both graphs are circles.
(II) The graphs touch at one point.
(III) The graphs intersect at two distinct points.
B) (II) only
C) (III) only
D) (I) and (II) only
E) (I) and (III) only
If the perimeter of a regular hexagon is \( 6a \), its area is:
A) \( 6a^2 \)B) \( 3a^2 \)
C) \( 0.5 \sqrt{3} a^2 \)
D) \( \sqrt{3} a^2 \)
E) \( 1.5 \sqrt{3} a^2 \)
\( AB \) is a diameter of the circle below, and point \( C \) is on the circle. The diameter is 10 units, and side \( AC = 5 \). Find the measure of angle \( CBA \).
B) \( 30 \)
C) \( 40 \)
D) \( 45 \)
E) \( 43 \)
For what positive \( k \) does the equation have exactly one solution?
\[ (x + k)x = -4 \] A) \( 1 \)B) \( \frac{3}{2} \)
C) \( 3 \)
D) \( 4 \)
E) \( 5 \)
The average of the roots of a quadratic equation is 3, and their difference is 2. Which could be the equation?
A) \( x^2 - 5x + 6 = 0 \)B) \( x^2 + 5x + 6 = 0 \)
C) \( x^2 - 6x + 8 = 0 \)
D) \( x^2 + 6x + 8 = 0 \)
E) \( x^2 - 6x + 6 = 0 \)
Points \( M \), \( B(2, 6) \), and \( C(4, 8) \) form a right triangle with hypotenuse \( BC \). These three points lie on a circle of radius:
A) \( 2\sqrt{2} \)B) \( \sqrt{2} \)
C) \( 2 \)
D) \( 4 \)
E) \( 8 \)
A rectangular solid has volume \( 1000 \, \text{m}^3 \). If length, width, and height increase by 50%, the new volume is:
A) \( 3375 \, \text{m}^3 \)B) \( 3500 \, \text{m}^3 \)
C) \( 1500 \, \text{m}^3 \)
D) \( 5000 \, \text{m}^3 \)
E) \( 50000 \, \text{m}^3 \)
In the figure, angle \( ADE \) is right, and \( BC \parallel DE \). Given \( AC = 5 \) and \( BC = 4 \), find \( \tan(\angle CED) \).
B) \( \frac{4}{5} \)
C) \( \frac{4}{3} \)
D) \( 1 \)
E) \( \frac{3}{4} \)
What are the coordinates of the center of a circle entirely in quadrant II and tangent to the lines \( y = 8 \), \( y = 2 \), and \( x = -2 \)?
A) \( (-2, 8) \)B) \( (-5, 2) \)
C) \( (-2, 2) \)
D) \( (-5, 5) \)
E) \( (-5, 8) \)
In the figure, \( BC \parallel DE \), \( AC = x \), and \( CE = 4x \). If the area of triangle \( ABC \) is \( 100 \, \text{cm}^2 \), what is the area of triangle \( ADE \)?
B) \( 1600 \, \text{cm}^2 \)
C) \( 900 \, \text{cm}^2 \)
D) \( 500 \, \text{cm}^2 \)
E) \( 600 \, \text{cm}^2 \)
Find the volume of the cone generated by rotating right triangle \( ABC \) about side \( AB \).
B) \( 128\pi \)
C) \( 24\pi \)
D) \( 48\pi \)
E) \( 480\pi \)
If \( 3(x + y) = 27 \) and \( x, y \) are positive integers, which cannot be the value of \( \frac{x}{y} \)?
A) \( \frac{1}{2} \)B) \( 8 \)
C) \( 2 \)
D) \( \frac{1}{8} \)
E) \( \frac{1}{4} \)
Simplify:
\[ \frac{\cos(x) \sin(2x) - 2 \sin x}{\sin^2 x \cos(x)} \] A) \( 2 \cot x \)B) \( 2 \tan x \)
C) \( -2 \tan x \)
D) \( -2 \cot x \)
E) \( \cot x \)
Find the smallest positive integer \( x \) such that \( x^2 + 4 \) is divisible by 25.
A) \( 4 \)B) \( 7 \)
C) \( 6 \)
D) \( 11 \)
E) \( 8 \)
Find constant \( a \) such that \( x = 1 \) is a solution to:
\[ a\sqrt{x + 3} - 4a |2x - 1| = 4 \] A) \( -2 \)B) \( 2 \)
C) \( 1 \)
D) \( -1 \)
E) \( 5 \)
Given positive integers \( m > n \) and \( m^{2n} = 46656 \), find \( m \).
A) \( 9 \)B) \( 8 \)
C) \( 7 \)
D) \( 6 \)
E) \( 5 \)
The equation \( f(x) = 2 \) has root \( x = 5 \). The solution to \( f(-2x + 1) = 2 \) is:
A) \( -9 \)B) \( -\frac{1}{2} \)
C) \( 5 \)
D) \( 2 \)
E) \( -2 \)
For what value of \( b \) does the system have no solution?
\[ \begin{cases} 2x + 5y = 3 \\ -5x + by = 14 \end{cases} \] A) \( 5 \)B) \( -\frac{25}{2} \)
C) \( \frac{5}{2} \)
D) \( \frac{5}{3} \)
E) \( \frac{5}{14} \)
Find real numbers \( a \) and \( b \) such that \( 3a - bi = (2 - i)(4 + i) \), where \( i = \sqrt{-1} \).
A) \( a = -3, b = -2 \)B) \( a = -3, b = 2 \)
C) \( a = 3, b = 2 \)
D) \( a = 2, b = 4 \)
E) \( a = 8, b = -1 \)
A dealer increases an item's price by 20%, then by 30%. If \( x \) is the original price, the final price is:
A) \( 1.5x \)B) \( 1.56x \)
C) \( x + 0.5 \)
D) \( x + 0.56 \)
E) \( x + 6 \)
Two dice are thrown. What is the probability the sum is greater than 10?
A) \( \frac{1}{12} \)B) \( \frac{1}{36} \)
C) \( \frac{1}{6} \)
D) \( \frac{1}{4} \)
E) \( \frac{1}{2} \)
If \( 3x + 2y = 5 \) and \( 5x + 4y = 9 \), then \( 4x + 3y = \)
A) \( 0 \)B) \( 2 \)
C) \( 5 \)
D) \( 6 \)
E) \( 7 \)
The solution set of \( |2x - 4| > x + 1 \) is:
A) \( (5, +\infty) \)B) \( (-\infty, -1) \)
C) \( (-\infty, 1) \cup (5, +\infty) \)
D) \( (-\infty, -1) \cup (5, +\infty) \)
E) \( (-\infty, 0) \cup (5, +\infty) \)
If \( \frac{m}{n} = \frac{2}{3} \) for positive integers \( m, n \), which pair is impossible?
A) \( m = 12, n = 18 \)B) \( m = 60, n = 90 \)
C) \( m = 34, n = 51 \)
D) \( m = 7, n = 21 \)
E) \( m = 102, n = 153 \)
If 35 is the median of the data set \( \{21, 7, 45, 33, 62, x\} \), then \( x = \)
A) \( 3 \)B) \( 14 \)
C) \( 37 \)
D) \( 33 \)
E) \( 48 \)