Question 1
The expression \( 4(2x + 19)(x - 5) - (9x + 13) \) can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. What is the value of \( b \)?
Solution: Expand the expression: \( 4(2x^2 - 10x + 19x - 95) - 9x - 13 \).
Simplify inside the parentheses: \( 4(2x^2 + 9x - 95) - 9x - 13 \).
Distribute the 4: \( 8x^2 + 36x - 380 - 9x - 13 \).
Combine like terms: \( 8x^2 + 27x - 393 \).
The coefficient \( b \) is 27.
Simplify inside the parentheses: \( 4(2x^2 + 9x - 95) - 9x - 13 \).
Distribute the 4: \( 8x^2 + 36x - 380 - 9x - 13 \).
Combine like terms: \( 8x^2 + 27x - 393 \).
The coefficient \( b \) is 27.
Question 2
The function \( h \) is defined by \( h(x) = 6x + 18 \). The graph of \( y = h(x) \) in the xy-plane has an x-intercept at \( (a, 0) \) and a y-intercept at \( (0, b) \), where \( a \) and \( b \) are constants. What is the value of \( a + b \)?
Solution: Find the x-intercept by setting \( y = 0 \): \( 0 = 6a + 18 \Rightarrow a = -3 \).
Find the y-intercept by setting \( x = 0 \): \( b = 6(0) + 18 \Rightarrow b = 18 \).
Calculate \( a + b = -3 + 18 = 15 \). The correct answer is D.
Find the y-intercept by setting \( x = 0 \): \( b = 6(0) + 18 \Rightarrow b = 18 \).
Calculate \( a + b = -3 + 18 = 15 \). The correct answer is D.
Question 3
\[ 4x^2 - px + w = -86 \]
In the given equation, \( p \) and \( w \) are integer constants. The equation has exactly one real solution. Which is NOT a possible value of \( w \)?
Solution: Rewrite the equation: \( 4x^2 - px + (w + 86) = 0 \).
For exactly one real solution, the discriminant \( b^2 - 4ac = 0 \).
\( (-p)^2 - 4(4)(w + 86) = 0 \Rightarrow p^2 = 16(w + 86) \).
Since \( p \) is an integer, \( p^2 \) is a perfect square. Because 16 is a perfect square, \( w + 86 \) must also be a perfect square.
Test the options: (A) -22+86=64 (Yes), (B) 14+86=100 (Yes), (C) 25+86=111 (No), (D) 314+86=400 (Yes).
The correct answer is C.
For exactly one real solution, the discriminant \( b^2 - 4ac = 0 \).
\( (-p)^2 - 4(4)(w + 86) = 0 \Rightarrow p^2 = 16(w + 86) \).
Since \( p \) is an integer, \( p^2 \) is a perfect square. Because 16 is a perfect square, \( w + 86 \) must also be a perfect square.
Test the options: (A) -22+86=64 (Yes), (B) 14+86=100 (Yes), (C) 25+86=111 (No), (D) 314+86=400 (Yes).
The correct answer is C.
Question 4
In the figure shown, \( d \), \( e \), and \( f \) are constants and the value of \( k \) is 3. If the measure of angles \( D \), \( E \), and \( F \) are \( 30^\circ \), \( x^\circ \), and \( 2x + 6^\circ \), respectively, what is the measure, in degrees, of angle \( S \)?
Solution: The sum of angles in \(\triangle DEF\) is \( 180^\circ \).
\( 30 + x + (2x + 6) = 180 \Rightarrow 3x + 36 = 180 \Rightarrow 3x = 144 \Rightarrow x = 48 \).
Angle \( F = 2(48) + 6 = 102^\circ \).
Because the triangles are similar, corresponding angles are equal. Angle \( S \) corresponds to Angle \( F \).
Angle \( S \) is 102 degrees.
\( 30 + x + (2x + 6) = 180 \Rightarrow 3x + 36 = 180 \Rightarrow 3x = 144 \Rightarrow x = 48 \).
Angle \( F = 2(48) + 6 = 102^\circ \).
Because the triangles are similar, corresponding angles are equal. Angle \( S \) corresponds to Angle \( F \).
Angle \( S \) is 102 degrees.
Question 5
Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 59 cm\(^2\), and the surface area of right rectangular prism Y is 1,475 cm\(^2\). The volume of right rectangular prism Y is 1,500 cm\(^3\). What is the sum of the volumes, in cm\(^3\), of right rectangular prism X and right rectangular prism Y?
Solution: Area ratio is \( 59 / 1475 = 1/25 \).
Scale factor \( k = \sqrt{1/25} = 1/5 \).
Volume ratio is \( k^3 = 1/125 \).
Volume X = \( 1500 \times (1/125) = 12 \).
Sum = \( 1500 + 12 = 1512 \).
Scale factor \( k = \sqrt{1/25} = 1/5 \).
Volume ratio is \( k^3 = 1/125 \).
Volume X = \( 1500 \times (1/125) = 12 \).
Sum = \( 1500 + 12 = 1512 \).