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Q2 Solving quadratics and applications Finding a ratio or relationship between variables from an equation View

For positive real numbers $a, x$ and $y$
$$\begin{aligned} & -2x^{2} + y^{2} = 2a \\ & 3x^{2} - 2y^{2} = -6a \end{aligned}$$
what is the ratio $\dfrac{y}{x}$?
A) 1 B) $\sqrt{2}$ C) $\sqrt{3}$ D) 2 E) 3

Q5 Inequalities Integer Solutions of an Inequality View

Let $a$ be an integer. There are exactly 4 integer values of $x$ satisfying
$$0 < \left| x^{2} - 2x + 2 \right| - x^{2} - x < a$$
What is the sum of the different integer values that $a$ can take?
A) 33 B) 36 C) 39 D) 42 E) 45

Q6 Curve Sketching Limit Reading from Graph View

In the rectangular coordinate plane, the graphs of functions $f$ and $g$ defined on the closed interval $[0,1]$ and the line $y = x$ are given below.
For real numbers $a, b$ and $c$ in the open interval $(0,1)$
$$\begin{aligned} & a < f(a) < g(a) \\ & g(b) < b < f(b) \\ & c < g(c) < f(c) \end{aligned}$$
If these inequalities are satisfied, which of the following orderings is correct?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $c < a < b$ E) $c < b < a$

Q7 Curve Sketching Limit Reading from Graph View

In the rectangular coordinate plane, the graphs of linear functions $f$, $g$ and $h$ are shown in the figure.
Regarding these functions, the following equalities are given:
$$\begin{aligned} & f(x-5) = g(x) \\ & h(x) = -f(x) \end{aligned}$$
Which of the following orderings is correct for the values $f(0)$, $g(0)$ and $h(0)$?
A) $g(0) < f(0) < h(0)$ B) $f(0) < h(0) < g(0)$ C) $f(0) < g(0) < h(0)$ D) $g(0) < h(0) < f(0)$ E) $h(0) < g(0) < f(0)$

Q10 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View

Let $a$ and $b$ be positive real numbers. For each of the equations
$$\begin{aligned} & ax^{2} - 2x + b = 0 \\ & bx^{2} - 3bx + a = 0 \end{aligned}$$
the sum of roots is 1 more than the product of roots.
Which of the following could be the quadratic equation whose roots are $a$ and $b$?
A) $9x^{2} + 8x + 18 = 0$ B) $9x^{2} - 14x + 8 = 0$ C) $9x^{2} - 18x + 14 = 0$ D) $9x^{2} - 8x + 14 = 0$ E) $9x^{2} - 18x + 8 = 0$

Q11 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = x^{3} + 9x^{2} + ax + b$$
takes positive values on positive real numbers and negative values on negative real numbers.
What is the smallest integer value that $a$ can take?
A) 9 B) 13 C) 17 D) 21 E) 25

Q12 Arithmetic Sequences and Series Find Common Difference from Given Conditions View

For an arithmetic sequence $(a_{n})$ with common difference $r$
$$\begin{aligned} & a_{30} - a_{25} < 53 \\ & a_{25} - a_{8} > 53 \end{aligned}$$
the inequalities are given.
What is the sum of the integer values that $r$ can take?
A) 49 B) 56 C) 64 D) 72 E) 84

Q13 Laws of Logarithms Solve a Logarithmic Equation View

Let $a$, $x$ and $y$ be positive real numbers. The numbers
$$\log_{a} x, \quad \log_{a} y, \quad \log_{a}(x+y)$$
arranged from smallest to largest are consecutive integers. What is the value of $\log_{a}(2a+1)$?
A) $-2$ B) $-1$ C) $2$ D) $3$ E) $4$

Q14 Combinations & Selection Selection with Adjacency or Spacing Constraints View

Melisa will stack 10 circular cardboards with radii $1\text{ cm}, 2\text{ cm}, 3\text{ cm}, \cdots, 10\text{ cm}$ on a table with their centers coinciding, where each has a different natural number radius.
In how many different ways can Melisa arrange them so that when viewed from above, exactly one of the circles is completely hidden?
A) 36 B) 40 C) 45 D) 48 E) 54

Q15 Probability Definitions Conditional Probability and Bayes' Theorem View

A certain bus arrives at the bus stop near Duru's house with probability $\dfrac{7}{10}$ at exactly 09:02 and with probability $\dfrac{3}{10}$ at exactly 09:03. Duru leaves home at exactly 09:00 to catch this bus. The time it takes for Duru to reach the stop is 100 seconds with probability $\dfrac{1}{2}$, 150 seconds with probability $\dfrac{3}{10}$, and 250 seconds with probability $\dfrac{1}{5}$.
What is the probability that Duru is at the stop when the bus arrives?
A) $\dfrac{55}{100}$ B) $\dfrac{59}{100}$ C) $\dfrac{63}{100}$ D) $\dfrac{67}{100}$ E) $\dfrac{71}{100}$

Q16 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

Let $m$ and $n$ be natural numbers. If the constant term in the expansion of
$$\left(x + \frac{5}{x^{m}}\right)^{n}$$
is 60, what is $m + n$?
A) 36 B) 35 C) 31 D) 27 E) 23

Q17 Chain Rule Limit Evaluation Involving Composition or Substitution View

$$\lim_{x \rightarrow 1} \frac{(1 - \sqrt{x}) \cdot (\sqrt[3]{x} - 2)}{-x^{2} + 9x - 8}$$
What is the value of this limit?
A) 1 B) $\dfrac{1}{2}$ C) $\dfrac{1}{7}$ D) $\dfrac{1}{14}$ E) $\dfrac{1}{18}$

Q18 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

The graph of a function $f$ in the rectangular coordinate plane is given below.
A function $g$ defined on the set of real numbers has a limit at all points where it is defined, and $\lim_{x \rightarrow 3} g(x) = 14$ is calculated.
If the function $f \cdot g$ is continuous on the set of real numbers, what is the value of $g(3)$?
A) 4 B) 6 C) 8 D) 10

Q19 Discriminant and conditions for roots Condition for repeated (equal/double) roots View

Let $a$ and $b$ be real numbers. In the rectangular coordinate plane, the parabola $y = x^{2} + ax + b$ is tangent to the $x$-axis and to the line $y = x$.
What is the product $a \cdot b$?
A) $\dfrac{1}{2}$ B) $\dfrac{1}{4}$ C) $\dfrac{1}{8}$ D) $\dfrac{1}{16}$ E) $\dfrac{1}{32}$

Q20 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $a$ and $b$ be real numbers. The function $f$ defined as
$$f(x) = ax^{3} + bx^{2} + x + 7$$
is always increasing.
If $f(-1) = 0$, what is the sum of the different integer values that $b$ can take?
A) 11 B) 13 C) 15 D) 17 E) 19

Q21 Stationary points and optimisation Determine parameters from given extremum conditions View

Let $k$ and $m$ be real numbers. The functions $f$ and $g$ defined on the set of real numbers are
$$\begin{aligned} & f(x) = 2x^{3} - 9x^{2} - mx - k \\ & g(x) = x^{3} \cdot f(x) \end{aligned}$$
The functions $f$ and $g$ have local extrema at $x = -1$.
What is the sum $k + m$?
A) 31 B) 33 C) 35 D) 37 E) 39

Q22 Applied differentiation Applied modeling with differentiation View

In the rectangular coordinate plane, the shaded region between the lines $y = 2 + 3x$ and $y = 12 - x$ and the positive $x$ and $y$-axes is given below.
Rectangles are constructed in this region with one side on the $x$-axis and one vertex each on the lines $y = 2 + 3x$ and $y = 12 - x$.
What is the length of the side on the $x$-axis of the rectangle with the largest area?
A) 6 B) $\dfrac{19}{3}$ C) $\dfrac{20}{3}$ D) 7 E) $\dfrac{22}{3}$

Q23 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

$$\int_{1}^{2} (x+2) \cdot \sqrt[3]{x^{2} + 4x - 4}\, dx$$
What is the value of this integral?
A) $\dfrac{45}{8}$ B) $\dfrac{47}{8}$ C) $\dfrac{49}{8}$ D) $\dfrac{45}{4}$ E) $\dfrac{47}{4}$

Q24 Areas by integration View

For the function $f$ whose graph is given above in the rectangular coordinate plane
$$\begin{aligned} & \int_{a}^{c} |f(x)|\, dx = 20 \\ & \int_{a}^{c} f(x)\, dx = 8 \end{aligned}$$
the equalities are satisfied.
What is the value of $$\int_{a/2}^{b/2} f(2x)\, dx$$ ?
A) $-3$ B) $-4$ C) $-5$ D) $-6$ E) $-7$

Q25 Areas by integration View

Let $a$ be a real number. In the rectangular coordinate plane, the graphs of the functions $y = a\sqrt{x}$ and $y = \sqrt{x}$ are given below.
Let the area of the blue shaded region be $A_{1}$ and the area of the yellow shaded region be $A_{2}$. If
$$A_{1} \cdot A_{2} = 96$$
what is $a$?
A) 2 B) 3 C) 4 D) 5 E) 6

Q26 Integration by Parts Differentiation Under the Integral Sign Combined with Parts View

The functions $f$ and $g$ are defined and differentiable on the set of real numbers and satisfy
$$\begin{aligned} & \int_{1}^{2} f^{\prime}(3x)\, dx = 4 \\ & \int f(2x)\, dx = g(x) + C, \quad (C \text{ constant}) \end{aligned}$$
If $f(3) = 5$, what is the value of the derivative $g^{\prime}(3)$?
A) 1 B) 5 C) 9 D) 13 E) 17

Q27 Trig Graphs & Exact Values View

Let $a = \sin(40^{\circ})$
$$\begin{aligned} & b = \sec(40^{\circ}) \\ & c = \tan(40^{\circ}) \end{aligned}$$
Which of the following is the correct ordering of the numbers $a$, $b$ and $c$?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $b < c < a$ E) $c < a < b$

Q28 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View

$$\frac{\cos(2x+y) + \sin(2x-y)}{\cos(2x) + \sin(2x)}$$
Which of the following is the simplified form of this expression?
A) $\cos y - \sin y$ B) $\cos y + \sin y$ C) $\cos x - \sin y$ D) $\sin x - \cos y$ E) $\sin x - \cos x$

Q29 Standard trigonometric equations Evaluate trigonometric expression given a constraint View

Let $0 < a < \dfrac{\pi}{2}$ and
$$\cos^{2} a - \cos(2a) = \sin(2a)$$
For the value of $a$ satisfying this equality, which of the following is correct?
A) $\tan a = \dfrac{1}{5}$ B) $\cot a = \dfrac{2}{\sqrt{5}}$ C) $\cos a = \dfrac{1}{\sqrt{5}}$ D) $\operatorname{cosec} a = \sqrt{5}$ E) $\sin(2a) = \dfrac{3}{5}$

Q30 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

Let $0 \leq x \leq \pi$ and
$$\sqrt{2}\sin(4x) - \cos(8x) = 1$$
What is the sum of the $x$ values satisfying this equality?
A) $\dfrac{\pi}{3}$ B) $\dfrac{3\pi}{4}$ C) $\pi$ D) $\dfrac{3\pi}{2}$ E) $2\pi$

Q31 Radians, Arc Length and Sector Area View

O is the center of a semicircle, ABCD is a rectangle.
A, O and B are collinear.
$|AE| = |ED| = \dfrac{1}{2}$ unit, $m(\widehat{AOE}) = x$
Points C and D lie on the semicircle with center O.
What is the length of a diagonal of rectangle ABCD in terms of $x$?
A) $\tan x$ B) $\operatorname{cosec} x$ C) $\sec x$ D) $\sin x$ E) $\cos x$

Q32 Sine and Cosine Rules Compute area of a triangle or related figure View

In a triangle $ABC$, the length of side $AB$ is equal to half the length of side $BC$.
If two of the altitudes of this triangle have lengths 4 units and 10 units, which of the following could be the length of the other altitude?
I. 2 units II. 5 units III. 8 units
A) Only I B) Only II C) Only III D) I and II E) II and III

Q33 Radians, Arc Length and Sector Area View

For each pizza at a restaurant, a circular pizza dough is first rolled out. Then, an orange-colored ingredient section is created to form a concentric circle with this dough. The top view of the pizza ordered by Ali and Ayşe at this restaurant is shown in the figure.
Ayşe divides the pizza shown in the figure into 8 equal slices; she takes 3 of these slices and gives 5 to Ali. While Ali eats all of his slices, Ayşe eats only the orange-colored portions of her slices as shown in the figure.
In the end, the area of the portions Ali ate is calculated to be 2.4 times the area of the portions Ayşe ate.
What is the ratio of the radius of this pizza to the radius of the orange section?
A) $\dfrac{3}{2}$ B) $\dfrac{4}{3}$ C) $\dfrac{5}{4}$ D) $\dfrac{6}{5}$ E) $\dfrac{7}{6}$

Q34 Radians, Arc Length and Sector Area View

A wire in the shape of a circular arc with a central angle of $120^{\circ}$ has its ends mounted on the top edge of a rectangular board. This board is hung on a nail in the wall as shown in Figure 1, with the midpoint of the wire aligned with the nail, so that the long sides of the board are parallel to the ground.
Then this wire is removed, bent into a semicircle shape, and mounted on the board again. The board is hung on the same nail as shown in Figure 2, with the midpoint of the wire again aligned with the nail, so that the long sides of the board are again parallel to the ground. As a result, the height of the board from the ground decreases by 8 units compared to the initial position.
What is the length of the wire?
A) $24\pi$ B) $28\pi$ C) $32\pi$ D) $36\pi$ E) $40\pi$

Q35 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View

In a rectangular coordinate plane, points $A(9,2)$, $B(10,1)$, $C$, $D(4,13)$, $E(3,6)$ and $F$ are given.
Given that the centroid of triangle $ABC$ and the centroid of triangle $DEF$ are the same point, what is the distance between points $C$ and $F$ in units?
A) 10 B) 13 C) 15 D) 17 E) 20

Q36 Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View

In the rectangular coordinate plane below, a red square with one side on the $y$-axis and a blue square with one side on the $x$-axis share a common vertex.
One vertex of each of the red and blue squares lies on the line $\dfrac{x}{2} + \dfrac{y}{3} = 1$.
According to this, what is the side length of the red square in units?
A) $\dfrac{14}{15}$ B) $\dfrac{15}{16}$ C) $\dfrac{16}{17}$ D) $\dfrac{17}{18}$ E) $\dfrac{18}{19}$

Q37 Simultaneous equations View

In a rectangular coordinate plane, what is the area of the triangular region bounded by the lines $2x - y = 0$, $x + 2y = 0$ and $x - 8y + 30 = 0$ in square units?
A) 9 B) 12 C) 15 D) 18 E) 21

Q38 Straight Lines & Coordinate Geometry Reflection and Image in a Line View

In a rectangular coordinate plane, point $A(a, b)$; its reflection with respect to point $B(3, 0)$ is point $C$, and its reflection with respect to the $y$-axis is point $D$.
Given that the equation of the line passing through points $C$ and $D$ is $y = -x - 1$, what is the sum $a + b$?
A) 7 B) 13 C) 15 D) 19 E) 24

Q39 Circles Circle Equation Derivation View

In a rectangular coordinate plane, point $A(11, 9)$ is located in the interior of a circle that is tangent to the line $y = x$ at point $B(7, 7)$.
Accordingly, what is the smallest integer value that the radius of this circle can take in units?
A) 6 B) 8 C) 10 D) 12 E) 14

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