December 2025 Int.D Module 2

\(\begin{aligned}
&\text{A total of 215 toothpicks of equal length were used to construct two types of figures:}\\
&\text{triangles and squares. The triangles and squares were constructed so that no two figures}\\
&\text{has a common side. The equation }3x+4y=215\text{ represents this situation, where }x\mathrm{~is~the}\\&\text{number of triangles constructed and }y\text{ is the number of squares constructed. What is the}\\&\text{best interpretation of }(x,y)=(25,35)\text{ in this context?}\end{aligned}\)

\(\begin{aligned}
&\text{The function }f\text{ is defined by }f(x)=2x+b,\mathrm{~where~}b\text{ is a constant and }f(4)=1.\\
&\text{What is the value of }b?\end{aligned}\)

\(\begin{aligned}
&\text{The graph of }y=f(x)\text{ is shown, where the function }f\text{ is defined by }f(x)=ax^3+bx^2+cx+d\\
&\text{and }a,b,c,\mathrm{~and~}d\text{ are constants. For how many values of }x\mathrm{~does~}f(x)=0?\end{aligned}\)

\(\mathrm{If~}3(x-4)^2=243\text{, what is the value of }x^2-8x?\)

\(\text{The scatterplot shows the relationship between two variables }x\text{ and }y.\)

\(\text{Which of the following equations is the most appropriate quadratic model for the data shown?}\)

\(\begin{aligned}&\text{A right circular cylinder has a volume of 36,501}\pi\text{ cubic inches. The height of the cylinder}\\
&\text{is 3 times its radius. What is the radius, in inches, of the cylinder?}\end{aligned}\)

\(\begin{aligned}
&\text{The function }g\text{ is defined by }g(x)=16x+39.\text{ For what value of }x\text{ does }\\
&g(x)=43?\end{aligned}\)

\(\begin{aligned}
&\text{Circle }R\text{ in the }xy\text{-plane is represented by the equation }(x+7)^2+(y+19)^2=100.\text{ Circle }S\\
&\text{is obtained by shifting circle }R\text{ to the right 2 units. An equation representing circle }G\text{ is }\\&(x+h)^2+(y+k)^2=100,\text{where }h\text{ and }k\text{ are constants. What is the value of }h?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\begin{array}{|c|c|}\hline x&y\\\hline-26&t\\\hline-13&t+29\\\hline0&t+58\\\hline\end{array}\\
&\text{For a linear function between }x\mathrm{~and~}y,\text{ the table gives three values of }x\text{ and their}\\
&\text{corresponding values of }y,\mathrm{~where~}t\text{ is a constant. Which equation represents this }\\
&\text{relationship?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5(7x)+8(6y)=266\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5(7x)-8(6y)=-406\\
&\text{The solution to the given system of equations is }(x,y).\text{ What is the value of }7x+6y?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y<13x+12\\
&\text{For which of the following tables are all the values of }x\text{ and their corresponding values of}\\
&y\text{ solutions to the given inequality?}\end{aligned}\)

\(\begin{aligned}
&\text{The function }f\text{ is defined by the equation }f(x)=9^{x}+b,\mathrm{~where~}b\text{ is a constant. }\\
&\text{In the }xy\text{-plane, the graph of }y=f(x)\text{ contains the points }(0,27)\text{ and }(3,p+16),\\
&\text{where }p\text{ constant. What is the value of }p?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{-20x-37+m}{2}=k(x+6)\\
&\text{In the given equation, }k\text{ and }m\text{ are constants. The equation has infinitely many solutions.}\\
&\text{What is the value of }m?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f(x)=5,000(1.003)^{2x}\\
&\text{The given function }f\text{ models the balance of a bank account, in dollars, }x\text{ years after it is}\\
&\text{opened. Which statement is the best interpretation of }(1.003)^{2x}?\end{aligned}\)

\(\begin{aligned}&\text{At a hotel, guests used the pool for a total of }t\text{ hours in March. At the same hotel, guests}\\&\text{used the pool for 3.78t hours in April. What is the percent in the number of hours guests}\\&\text{used the pool frm March to April?}\end{aligned}\)

\(\begin{aligned}
&\text{A researcher selected 23 guinea pigs from one habitat and 19 wild cavies from another}\\
&\text{habitat at random in Venezuela. A field of a specific size within each habitat was divided}\\
&\text{into equally-sized virtual squares, and the animals were allowed to wander in the field for}\\
&\text{a fixed amount of time. To observe their behavior, the researcher counted the number}\\
&\text{of times the guinea pigs and wild cavies crossed the virtual squares in their field during}\\
&\text{early adolescence and again during late adolescence. The researcher found that in early}\\
&\text{adolescence, the number of virtual squares the wild cavies crossed was significantly more}\\
&\text{than the number of virtual squares the guinea pigs crossed. The researcher also found}\\
&\text{that the number of virtual squares crossed by both the wild cavies and the guinea pigs}\\
&\text{decreased from early adolescence to late adolescence. Based on the researcher’s findings,}\\
&\text{which of the following statements is an appropriate conclusion that can be drawn from}\\&\text{this study?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{1}{7xy}+xyz=\frac{1}{9yz}\\
&\text{In the given equation, }x,~y,\text{ and }z\text{ are positive numbers. Which expression is equivalent to }y?\end{aligned}\)

\(\begin{aligned}&\text{The quadratic function }g\text{ models the depth, in meters, below the surface of the water of}\\&\text{a Weddell seal }t\text{ minutes after the seal entered the water during a dive. The function}\\&\text{estimated that the seal reached its maximum depth of 409.6 meters 8 minutes after it}\\&\text{entered the water and then reached the surface of the water 16 minutes after it entered}\\&\text{the water. Based on the function, what was the estimated depth, to the nearest meter,}\\&\text{of the seal 11 minutes after it entered the water?}\end{aligned}\)

\(\begin{aligned}&\text{The measure of angle }G\mathrm{~is~}\frac{\pi}{10}\text{ radians. If the measure of angle }G\mathrm{~is~}9n\text{ degrees, where }n\\
&\text{is a constant, what is the value of }n?\end{aligned}\)

\(\begin{aligned}&\text{In triangle }XYZ,\text{ the measure of angle }X\mathrm{~is~}90°.\mathrm{~Point~}W\text{ lies on segment }YZ,\mathrm{~and}\\&\mathrm{segment~}WX\text{ is a perpendicular to segment }YZ.\text{ The length of segment }WY\mathrm{~is~}684,\\
&\text{and the length of segment }WX\mathrm{~is~}513.\text{ What is the value of }\tan Z?\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r^2+qr=4r-55\\&\text{In the given equation, }q\text{ is an integer constants. The given equation has no real solutions.}\\&\text{What is the largest possible value of }q?\end{aligned}\)

\(\begin{aligned}&\text{In triangle }ABC,\text{ the measure of angle }B\mathrm{~is~}90°\mathrm{~and~}BD\text{ is an altitude of the triangle.}\\&\mathrm{The~length~of~\overline{AB}~is~15~and~the~length~of~\overline{AC}~is~17~greater~than~the~length~of~\overline{AB}.~What}\\&\text{is the value of }\frac{BC}{BD}?\end{aligned}\)