Question 1
What is the greatest common factor (GCF) of 45, 135 and 270?
Solution
Use prime factorization:
\[ 45 = 1 \times 3^2 \times 5 \]
\[ 135 = 1 \times 3^3 \times 5 \]
\[ 270 = 1 \times 2 \times 3^3 \times 5 \]
The GCF \( = 5 \times 3^2 = 45 \)
Question 2
What is the value of \( 2x + \frac{1}{2} \) when \( 5x - 3 = -3x + 5 \)?
Solution
Solve the equation:
\[ 5x + 3x = 5 + 3 \]
\[ 8x = 8 \]
\[ x = 1 \]
Now calculate:
\[ 2x + \frac{1}{2} = 2(1) + \frac{1}{2} = 2\frac{1}{2} \]
Question 3
The width \( W \) of a rectangle is 2 inches less than half its length \( L \). Express the perimeter \( P \) of the rectangle in terms of the length \( L \).
Solution
Express width in terms of length:
\[ W = \frac{L}{2} - 2 \]
Perimeter formula:
\[ P = 2(W + L) = 2\left(\frac{L}{2} - 2 + L\right) = L - 4 + 2L = 3L - 4 \]
Question 4
The total surface area of all six sides of the rectangular box below is equal to 80 square inches. What is the volume of the rectangular box in inches cubed?
Solution
Total surface area calculation:
Left and right sides: \( 4x + 4x = 8x \)
Front and back sides: \( 4x + 4x = 8x \)
Top and bottom sides: \( 16 + 16 = 32 \)
Total area: \( 16x + 32 = 80 \)
Solve for \( x \):
\[ 16x = 80 - 32 \]
\[ 16x = 48 \]
\[ x = 3 \text{ in} \]
Volume calculation:
\[ V = 4 \times 4 \times 3 = 48 \text{ inches}^3 \]
Question 5
The points \( A(-4, -4) \), \( B(-1, -2) \) and \( C(x, -8) \) are the vertices of a right triangle with the right angle at \( (-1, -2) \). Find the value of \( x \).
Solution
Since the right angle is at point \( B \), \( BA \) and \( BC \) are perpendicular. Using distance formula:
\[ AB^2 = (-1 + 4)^2 + (-2 + 4)^2 = 13 \]
\[ BC^2 = (x + 1)^2 + (-8 + 2)^2 = (x + 1)^2 + 36 \]
\[ AC^2 = (x + 4)^2 + (-8 + 4)^2 = (x + 4)^2 + 16 \]
By Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
\[ 13 + (x + 1)^2 + 36 = (x + 4)^2 + 16 \]
Simplify:
\[ 13 + x^2 + 2x + 1 + 36 = x^2 + 8x + 16 + 16 \]
\[ -6x = -18 \]
\[ x = 3 \]
Question 6
In the figure below, \( L_1 \) and \( L_2 \) are parallel lines. The correct relationship between angles \( y \) and \( x \) is:
Solution
The angle supplementary to angle \( x \) on the right side of \( L_3 \) is equal to \( y \) since they are corresponding angles. Therefore:
\[ x + y = 180^\circ \]
Question 7
If \( (x + y)^2 = 144 \) and \( x^2 - y^2 = 24 \), then what is \( x \) if \( x \) and \( y \) are both positive?
Solution
From first equation:
\[ x + y = 12 \quad \text{(taking positive root)} \]
Factor second equation:
\[ (x + y)(x - y) = 24 \]
Substitute \( x + y = 12 \):
\[ 12(x - y) = 24 \]
\[ x - y = 2 \]
Solve system:
\[ x + y = 12 \]
\[ x - y = 2 \]
Add equations:
\[ 2x = 14 \]
\[ x = 7 \]
Question 8
How many 3-digit numbers can we make using the digits 4, 5, 7 and 9, where repetition is allowed?
Solution
For each of the 3 positions, we have 4 choices:
\[ 4 \times 4 \times 4 = 4^3 = 64 \]
Question 9
A boat travels 10 miles East and then 24 miles South to an island. How many miles are there from the point of departure of the boat to the island?
Solution
Using Pythagorean theorem:
\[ d^2 = 10^2 + 24^2 = 676 \]
\[ d = \sqrt{676} = 26 \text{ miles} \]
Question 10
What is the slope of any line perpendicular to the line \( -5x + 3y = 9 \)?
Solution
Find slope of given line:
\[ -5x + 3y = 9 \]
\[ 3y = 5x + 9 \]
\[ y = \frac{5}{3}x + 3 \]
Slope \( m_1 = \frac{5}{3} \)
Perpendicular slope \( m_2 \) satisfies:
\[ m_1 \cdot m_2 = -1 \]
\[ \frac{5}{3} \cdot m_2 = -1 \]
\[ m_2 = -\frac{3}{5} \]
Question 11
Which of the following is a factor of the polynomial \( -2x^2 + 7x - 6 \)?
A) \( -2x - 3 \)
B) \( 2x + 2 \)
C) \( x - 6 \)
D) \( 2x - 2 \)
E) \( -2x + 3 \)
Solution
Factor the polynomial:
\[ -2x^2 + 7x - 6 = (-2x + 3)(x - 2) \]
Thus \( -2x + 3 \) is a factor.
Question 12
\( \sqrt{(-9)(-4)} + \sqrt{(-4)} = ? \)
Solution
\[ \sqrt{(-9)(-4)} = \sqrt{36} = 6 \]
\[ \sqrt{(-4)} = \sqrt{-1 \cdot 4} = \sqrt{-1} \cdot \sqrt{4} = i \cdot 2 = 2i \]
\[ 6 + 2i, \quad \text{where } i = \sqrt{-1} \]
Question 13
Find the linear function \( f \) such that \( f(2) = 5 \) and \( f(3) = -5 \).
Solution
Let \( f(x) = mx + b \)
From given conditions:
\[ f(2) = 2m + b = 5 \]
\[ f(3) = 3m + b = -5 \]
Subtract first equation from second:
\[ (3m + b) - (2m + b) = -5 - 5 \]
\[ m = -10 \]
Substitute to find \( b \):
\[ 2(-10) + b = 5 \]
\[ -20 + b = 5 \]
\[ b = 25 \]
\[ f(x) = -10x + 25 \]
Question 14
A circular garden has an area of \( 100\pi \) square feet. What is the circumference of the garden in feet?
Solution
\[ \pi R^2 = 100\pi \]
\[ R^2 = 100 \]
\[ R = 10 \text{ feet} \]
Circumference:
\[ C = 2\pi R = 20\pi \text{ feet} \]
Question 15
If \( \frac{5}{x} = 10 \) and \( \frac{2}{y} = 6 \), then \( \frac{x}{y} = ? \)
Solution
From first equation:
\[ \frac{5}{x} = 10 \Rightarrow x = \frac{5}{10} = \frac{1}{2} \]
From second equation:
\[ \frac{2}{y} = 6 \Rightarrow y = \frac{2}{6} = \frac{1}{3} \]
Therefore:
\[ \frac{x}{y} = \frac{1/2}{1/3} = \frac{3}{2} \]
Question 16
If \( \frac{2^{m-3}}{4^{2m}} = 8 \), then \( 2m - 1 = ? \)
Solution
Rewrite in terms of base 2:
\[ 4^{2m} = (2^2)^{2m} = 2^{4m} \]
\[ 8 = 2^3 \]
Substitute:
\[ \frac{2^{m-3}}{2^{4m}} = 2^3 \]
\[ 2^{m-3-4m} = 2^3 \]
\[ 2^{-3m-3} = 2^3 \]
Equate exponents:
\[ -3m - 3 = 3 \]
\[ -3m = 6 \]
\[ m = -2 \]
\[ 2m - 1 = 2(-2) - 1 = -5 \]
Question 17
In the figure below, \( ABC \) is a right triangle. Points \( B, C, D \) are collinear; points \( D, E, F \) are collinear and points \( B, A, F \) are collinear. The length of segments \( DC \) and \( DE \) are equal. What is the size, in degrees, of angle \( AFE \)?
Solution
In right triangle \( ABC \):
\[ \angle BCA = 90^\circ - 50^\circ = 40^\circ \]
Since \( DC = DE \), triangle \( DCE \) is isosceles:
\[ \angle DEC = \angle DCE = 40^\circ \]
Vertical angles: \( \angle AEF = \angle DEC = 40^\circ \)
Since \( \angle FAE = 90^\circ \):
\[ \angle AFE = 90^\circ - 40^\circ = 50^\circ \]
Question 18
Which of the following is an equation of a line perpendicular to the line with equation \( 3x - 6y = 9 \)?
A) \( y = 2 \)
B) \( 3x + 6y = 9 \)
C) \( x - 2y = 3 \)
D) \( 2x + 2y = 3 \)
E) \( 2x + y = 7 \)
Question 19
If \( a \) and \( b \) are any real numbers, then which of the following expressions is always positive?
A) \( |a| \)
B) \( |a + b| \)
C) \( |a - b| + \frac{1}{2} \)
D) \( a^2 + b^2 \)
E) \( (a + b)^2 \)
Question 20
The geometric figure below consists of a right triangle and 2 semicircles. The diameters of the semicircles are the sides of the triangle. What is the area of the shaded region in square centimeters if the length of the hypotenuse of the triangle is 8 centimeters?
A) 64
B) \( 8\pi \)
C) \( 64\pi \)
D) \( 10\pi \)
E) 16
Question 21
The mean of the numbers \( a, b, c, d, e \) is 23. The mean of the numbers \( a, b, c, d, e, f \) is 22. What is the value of \( f \)?
A) 23
B) 18
C) 22
D) 22.5
E) 20
Question 22
Which of the following equations corresponds to the graph shown below?
A) \( y = (x - 3)^2 - 1 \)
B) \( y = -(x - 3)^2 + 1 \)
C) \( y = (x - 3)^2 + 1 \)
D) \( y = x - 3 \)
E) \( y = -(x - 3)^2 - 1 \)
Question 23
Functions \( f \) and \( g \) are defined by \( f(x) = x^2 + x \) and \( g(x) = \sqrt{x + 6} \). What is the value of \( g(f(2)) \)?
A) 3
B) -3
C) 7
D) 6
E) -6
Question 24
The sum of three consecutive integers is equal to 192. What is the product of these numbers?
A) 216000
B) 7077888
C) 576
D) 110592
E) 262080
Question 25
What are the x-coordinates of the points intersection of the line with equation \( y = x + 1 \) and the circle with equation \( x^2 + y^2 = 5 \)?
A) -2, 0
B) 1, 2
C) -2, 1
D) -2, -1
E) 1, 3
Question 26
\( \frac{1}{2}\sin(2x)(1 + \cot^2(x)) = \)
A) \( \tan(x) \)
B) \( \sin(x) \)
C) \( \cos(x) \)
D) \( \cot(x) \)
E) \( \sec(x) \)
Question 27
Find the area of the rectangle ABCD shown in the figure below.
A) \( \frac{2500}{\sqrt{2}} \)
B) 2500
C) \( \frac{2500}{\sqrt{3}} \)
D) 1250
E) 5000
Question 28
Solve for \( x \): \( \log_x(1024) = -5 \)
A) \( \frac{1}{4} \)
B) 4
C) \( \frac{1}{2} \)
D) \( \frac{1}{8} \)
E) 2
Question 29
Simplify: \( 6\sqrt[3]{32} + 2\sqrt[3]{108} \)
A) 32
B) \( 18\sqrt[3]{2} \)
C) \( 36\sqrt[3]{2} \)
D) \( 18\sqrt[3]{4} \)
E) \( 36\sqrt[3]{4} \)
Question 30
Evaluate: \( \frac{1}{(-5)^2} \)
A) \( -\frac{1}{25} \)
B) \( \frac{1}{25} \)
C) 25
D) -25
E) \( \frac{1}{10} \)