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Q7 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

Let $a$ and $b$ be real numbers. For the functions $f$ and $g$ defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{x}{2} + 1 \\ & g(x) = 2x - 3 \end{aligned}$$
the equalities
$$\begin{aligned} & (f + g)(a) = f(a) \\ & (f - g)(b) = g(b) \end{aligned}$$
are satisfied. Accordingly, what is the value of $(f \circ g)(a \cdot b)$?
A) $\frac{1}{2}$ B) $\frac{5}{2}$ C) $\frac{9}{2}$ D) $\frac{13}{2}$ E) $\frac{17}{2}$

Q11 Discriminant and conditions for roots Condition for repeated (equal/double) roots View

Let $a$ and $b$ be positive real numbers. The equations
$$\begin{aligned} & x^{2} + ax + b = 0 \\ & ax^{2} + (b + 3)x + a = 0 \end{aligned}$$
are given. Given that the solution set of each of these equations has exactly 1 element, what is the product of the different values that the sum $a + b$ can take?
A) 24 B) 32 C) 45 D) 72 E) 120

Q12 Polynomial Division & Manipulation Polynomial Construction from Root/Value Conditions View

Let $P(x)$ and $Q(x)$ be polynomials with real coefficients such that $P(x) + Q(x)$ is a second-degree polynomial and
$$\begin{aligned} & P(x) \cdot Q(x) = -4 \cdot (x-1)^{4} \cdot (x-2)^{2} \\ & P(3) = -16 \end{aligned}$$
are satisfied. Accordingly, what is the value of $Q(4)$?
A) 12 B) 24 C) 36 D) 48 E) 54

Q13 Arithmetic Sequences and Series Find Specific Term from Given Conditions View

For an arithmetic sequence $(a_{n})$,
$$\begin{aligned} & a_{1} \cdot a_{2} \cdot a_{3} = 2 \\ & a_{2} \cdot a_{3} \cdot a_{4} = 14 \end{aligned}$$
are given. Accordingly, what is the product $a_{3} \cdot a_{4} \cdot a_{5}$?
A) 28 B) 35 C) 42 D) 49 E) 56

Q14 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

Let $n$ be a positive integer. In the expansion of
$$\left(x^{2} + x\right)^{n}$$
both the coefficient of the term containing $x^{19-n}$ and the coefficient of the term containing $x^{16-n}$ equal a positive integer $k$. Accordingly, what is $k$?
A) 6 B) 12 C) 15 D) 18 E) 21

Q17 Laws of Logarithms Solve a Logarithmic Equation View

Let $a$ and $b$ be non-consecutive positive integers. The equality
$$\ln(a!) = \ln(b!) + 3 \cdot \ln 2 + 2 \cdot \ln 3 + \ln 7$$
is satisfied. Accordingly, what is the sum $a + b$?
A) 10 B) 13 C) 15 D) 18 E) 20

Q18 Laws of Logarithms Solve a Logarithmic Equation View

Let $a, b, c$ and $d$ be distinct positive real numbers. The sets $A$ and $B$ are defined as
$$\begin{aligned} & A = \left\{ \log_{2} a, \log_{2} b, \log_{2} c, \log_{2} d \right\} \\ & B = \left\{ \log_{\frac{1}{2}} a, \log_{\frac{1}{2}} b, \log_{\frac{1}{2}} c, \log_{\frac{1}{2}} d \right\} \end{aligned}$$
$$\begin{aligned} & s(A \cap B) = 3 \\ & a \cdot b \cdot c \cdot d = \frac{7}{5} \\ & a + b + c + d = \frac{38}{5} \end{aligned}$$
Given that, what is the sum $a^{2} + b^{2} + c^{2} + d^{2}$?
A) 20 B) 22 C) 24 D) 26 E) 28

Q19 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

Let $a$ be a non-zero real number, and $b$ and $c$ be real numbers. For the function $f(x) = ax + b$ defined on the set of real numbers and its inverse function $f^{-1}$,
$$\begin{aligned} & \lim_{x \rightarrow b} \frac{f(x)}{f^{-1}(x)} = c \\ & f(1) = 3 \end{aligned}$$
are given. Accordingly, what is the sum of the different values that $c$ can take?
A) 6 B) 7 C) 10 D) 11 E) 14

Q20 Curve Sketching Determining coefficients from given conditions on function values or geometry View

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = \begin{cases} x^{2} - ax + 6 & , x \leq a \\ 2x + a & , a < x \leq b \\ 11 - 2x + b & , x > b \end{cases}$$
is continuous on its domain.
Accordingly, what is the product $a \cdot b$?
A) 4 B) 6 C) 8 D) 10 E) 12

Q21 Chain Rule Finding Composition Parameters from Derivative Conditions View

Let $n$ be a positive integer and $a$ be a non-zero real number. For a polynomial function $f$ with degree $n$ and leading coefficient $a$,
$$\left((f(x))^{3}\right)' = \left(f'(x)\right)^{4}$$
is satisfied.
Accordingly, what is the product $a \cdot n$?
A) $\frac{1}{4}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $1$ E) $2$

Q22 Chain Rule Chain Rule with Composition of Explicit Functions View

In the rectangular coordinate plane, for a function $y \geq f(x)$,

the tangent line at the point $(2, f(2))$ is $y = 3x - 1$
the tangent line at the point $(5, f(5))$ is $y = 2x + 4$

Accordingly, for the function $$g(x) = x^{2} \cdot (f \circ f)(x)$$
what is the value of $g'(2)$?
A) 64 B) 72 C) 80 D) 88 E) 96

Q23 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

In the rectangular coordinate plane, the graph of the derivative $f'$ of a continuous function $f$ defined on the set of real numbers is shown in the figure.
$$f(5) = f(20) = 0$$
Given that, what is the local minimum value of the function $f$?
A) $-18$ B) $-15$ C) $-12$ D) $-9$ E) $-6$

Q25 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

For a continuous function $f$ defined on the set of real numbers and the function $g(x) = 2x + 2$ defined as,
$$\begin{aligned} & \int_{-1}^{1} f(g(x))\, dx = 18 \\ & \int_{2}^{4} g(f(x))\, dx = 18 \end{aligned}$$
are satisfied. Accordingly, what is the value of the integral $\int_{0}^{2} f(x)\, dx$?
A) 20 B) 23 C) 26 D) 29 E) 32

Q26 Areas by integration Integral Inequalities and Limit of Integral Sequences View

Let $m$ be a positive real number. In the rectangular coordinate plane, the region between the graph of a function $f$ defined on the closed interval $[-m, m]$ and the $x$-axis is divided into four regions and these regions are colored as shown in the figure. The areas of these regions, which are different from each other, are denoted by $A, B, C$ and $D$ as shown in the figure.
$$\int_{-m}^{m} |f(x)|\, dx = \int_{-m}^{m} f(x)\, dx + \int_{0}^{m} 2 \cdot f(x)\, dx$$
Given that, which of the following is the integral $\int_{-m}^{m} f(x)\, dx$ equal to?
A) $A + B$ B) $A + C$ C) $A + D$ D) $B + C$ E) $C + D$

Q27 Reciprocal Trig & Identities View

The simplified form of the expression
$$\frac{1 - \cos(4x)}{\sin(4x) + 2 \cdot \sin(2x)}$$
is which of the following?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\sec x$ E) $\operatorname{cosec} x$

Q28 Trig Graphs & Exact Values Evaluate trigonometric expression given a constraint View

Let $x, y$ and $z$ be distinct elements of the set $\left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\right\}$ such that
$$\sin x < \tan y < \sec z$$
Which of the following is the correct ordering of $x$, $y$ and $z$?
A) $x < y < z$ B) $y < x < z$ C) $y < z < x$ D) $z < x < y$ E) $z < y < x$

Q29 Sine and Cosine Rules Geometric Configuration with Trigonometric Identities View

For a triangle $ABC$ with side lengths $|BC| = a$ units, $|AC| = b$ units and $|AB| = c$ units,
$$2a^{2} = 2b^{2} + 2c^{2} + 3bc$$
is satisfied. Let $m(\widehat{BAC}) = x$. What is the value of $\tan x$?
A) $-\frac{\sqrt{2}}{3}$ B) $-\frac{\sqrt{3}}{3}$ C) $-\frac{\sqrt{5}}{3}$ D) $-\frac{\sqrt{6}}{3}$ E) $-\frac{\sqrt{7}}{3}$

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

Let $0 < x < \frac{\pi}{2}$. Given that
$$2 \cdot \cos^{2} x + 9 \cdot \sin^{2} x + 2 \cdot \sin(2x) = 9$$
what is the value of $\cot x$?
A) $\frac{4}{7}$ B) $\frac{7}{6}$ C) $\frac{3}{5}$ D) $\frac{2}{3}$ E) $\frac{5}{2}$

Q31 Straight Lines & Coordinate Geometry Geometric Configuration with Trigonometric Identities View

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$ B) $\cot A \cdot \cot B$ C) $\cot A - \tan B$ D) $1 + \cot A \cdot \tan B$ E) $1 - \tan A \cdot \cot B$

Q35 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

In the rectangular coordinate plane, a triangle $OAB$ with one vertex at the origin and the other two vertices on the axes, and the line segment $[PR]$ connecting the points $P(6, -3)$ and $R(-2, 9)$ are drawn. The line segment $[PR]$ passes through the midpoints of both $[OA]$ and $[OB]$.
According to this, what is the area of triangle $OAB$ in square units?
A) 36 B) 42 C) 48 D) 54 E) 60

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

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the region between the lines $y = -\sqrt{3}x$ and $y = ax + b$ and the $x$-axis forms an equilateral triangle with area $9\sqrt{3}$ square units.
Accordingly, what is the product $a \cdot b$?
A) 18 B) 24 C) 27 D) 30 E) 36

Q37 Circles Distance from Center to Line View

A circle drawn in the rectangular coordinate plane

has one common point with the line $d_{1}: y - \frac{4x}{3} - 46 = 0$,
has two common points with the line $d_{2}: y - \frac{4x}{3} - 6 = 0$,
has no common points with the line $d_{3}: y - \frac{4x}{3} - 1 \geqslant 0$.

It is known that. Accordingly, which of the following could be the radius of this circle in units?
A) 11 B) 13 C) 15 D) 17 E) 19

Q38 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

In the rectangular coordinate plane, when point $A$ is translated 15 units in the negative direction along the $x$-axis, the resulting point lies on the line $d: 4x - 3y + 24 = 0$.
Accordingly, if point $A$ is translated how many units in the positive direction along the $y$-axis, the resulting point will lie on line $d$?
A) 9 B) 12 C) 16 D) 20 E) 25

Q39 Circles Chord Length and Chord Properties View

Let $m$ and $n$ be real numbers. In the rectangular coordinate plane, a circle passing through point $A(4, 1)$ is drawn with equation
$$x^{2} + y^{2} - 2x + 6y = n$$
The line $y = mx$ drawn in the plane intersects this circle at points $B$ and $C$. Given that $m(\widehat{BAC}) = 90^{\circ}$, what is the sum $m + n$?
A) 8 B) 9 C) 10 D) 11 E) 12

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