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Q1 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View

$$(3x-1)(x+1)+(3x-1)(x-2)=0$$
What is the sum of the real numbers $x$ that satisfy the equation?
A) $\frac{2}{3}$
B) $\frac{3}{4}$
C) $\frac{3}{5}$
D) $\frac{5}{6}$
E) $\frac{7}{6}$

Q2 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

$$f(x) = \frac{\left(1+x+x^{2}+x^{3}\right)(1-x)^{2}}{1-x-x^{2}+x^{3}}$$
Given this, what is the value of $f(\sqrt{2})$?
A) 1
B) 2
C) 3
D) 4
E) 5

Q3 Inequalities Solve Polynomial/Rational Inequality for Solution Set View

$$(2x-1)\left(4x^{2}-1\right)<0$$
Which of the following open intervals is the solution set of the inequality in real numbers?
A) $\left(-\infty, \frac{-1}{2}\right)$
B) $\left(\frac{-1}{2}, 0\right)$
C) $\left(\frac{-1}{2}, \frac{1}{2}\right)$
D) $\left(\frac{1}{4}, \frac{1}{2}\right)$
E) $\left(\frac{1}{2}, \infty\right)$

Q4 Number Theory GCD, LCM, and Coprimality View

The least common multiple of $b$ and $40$ is $120$.
Accordingly, how many different positive integers $b$ are there?
A) 6
B) 8
C) 10
D) 12
E) 14

Q5 Modulus function Domain or range of modulus-based functions View

$$f(x) = \sqrt{2-|x+3|}$$
Which of the following is the domain interval of the function?
A) $3 \leq x \leq 5$
B) $-1 \leq x \leq 5$
C) $-3 \leq x \leq 4$
D) $-3 \leq x \leq 0$
E) $-5 \leq x \leq -1$

Q6 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

A function defined from real numbers to a subset $K$ of real numbers $$f(x) = \begin{cases} -x+8, & \text{if } x < 3 \\ x+2, & \text{if } x \geq 3 \end{cases}$$ Given that the function is surjective, which of the following is the set $K$?
A) $[3, \infty)$
B) $[5, \infty)$
C) $[3,5]$
D) $(-\infty, 5)$
E) $(-\infty, 3)$

Q7 Inequalities Ordering and Sign Analysis from Inequality Constraints View

For given positive real numbers $a$, $c$ and negative real number $b$, $$a^{2}b > abc + c^{2}$$ Given that the inequality is satisfied, which of the following is necessarily true?
A) $a = |b|$
B) $a = c$
C) $c > |b|$
D) $a < c$
E) $c < a$

Q8 Groups Binary Operation Properties View

Binary operations $*$, $\oplus$, $\odot$ defined on the set of rational numbers
I. $a * b = a - b$ II. $a \oplus b = a + b + ab$ III. $a \odot b = \frac{a+b}{5}$
are defined as follows. Accordingly, which of these operations satisfy the associative property?
A) Only I
B) Only II
C) Only III
D) I and II
E) II and III

Q9 Factor & Remainder Theorem Divisibility and Factor Determination View

$$P(x) = 2x^{3} - (m+1)x^{2} - nx + 3m - 1$$
Given that the polynomial is completely divisible by $x^{2} - x$, what is $m - n$?
A) $\frac{-1}{3}$
B) $\frac{-1}{2}$
C) $\frac{3}{2}$
D) $2$
E) $3$

Q10 Curve Sketching Limit Reading from Graph View

Which of the following is the domain of the function $f$ whose graph is given above?
A) $[-3,0) \cup [4,7)$
B) $(-3,0) \cup (3,7]$
C) $[-3,2] \cup (3,7)$
D) $(-3,3) \cup (3,7]$
E) $[-3,2) \cup (4,7]$

Q11 Trig Graphs & Exact Values View

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $$f(x) = \begin{cases} 2\sin x, & \text{if } \sin x \geq 0 \\ 0, & \text{if } \sin x < 0 \end{cases}$$ Accordingly, which of the following is the image of the open interval $(-\pi, \pi)$ under $f$?
A) $[-2,2]$
B) $(-1,2)$
C) $[0,1]$
D) $(0,2)$
E) $[0,2]$

Q12 Groups Symmetric Group and Permutation Properties View

On the set $A = \{1,2,3,4,5\}$ $$f = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 5 & 2 & 4 \end{pmatrix}, \quad g = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 4 & 1 & 2 \end{pmatrix}$$ For the permutations, what is the value of $g f^{-1}(2)$?
A) 1
B) 2
C) 3
D) 4
E) 5

Q13 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View

$$f\left(\frac{x-1}{x+1}\right) = x^{2} - x + 2$$
Given this, what is the value of $f(3)$?
A) 5
B) 6
C) 7
D) 8
E) 11

Q14 Stationary points and optimisation Prove an inequality using calculus-based optimisation View

The function $f(x) = mx - 1 + \frac{1}{x}$ is given.
Accordingly, what is the smallest value of $m$ that satisfies the property $f(x) \geq 0$ for all $x > 0$?
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) $\frac{1}{4}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$

Q15 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View

Let $P(x)$ be a third-degree polynomial function such that $$P(-4) = P(-3) = P(5) = 0, \quad P(0) = 2$$ Given this, what is $P(1)$?
A) $\frac{7}{3}$
B) $\frac{8}{3}$
C) $\frac{7}{4}$
D) $\frac{9}{4}$
E) $\frac{8}{5}$

Q16 Inequalities Simultaneous/Compound Quadratic Inequalities View

The parabola $f(x)$ and the line $d$ are shown in the Cartesian coordinate plane above.
Accordingly, which of the following systems of inequalities has the shaded region as its solution set?
A) $\left.\begin{array}{l} y - x^{2} + 2x \leq 0 \\ y - x + 2 \geq 0 \end{array}\right\}$
B) $\left.\begin{array}{l} y - x^{2} + 2x \geq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$
C) $\left.\begin{array}{l} y - x^{2} + 4x \leq 0 \\ 2y - x + 2 \leq 0 \end{array}\right\}$
D) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 4 \leq 0 \end{array}\right\}$
E) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$

Q17 Probability Definitions Finite Equally-Likely Probability Computation View

Let $A = \{1,2,3,4\}$ and $B = \{-2,-1,0\}$. For any element $(a,b)$ taken from the Cartesian product set $A \times B$, what is the probability that the sum $a + b$ equals zero?
A) $\frac{1}{4}$
B) $\frac{1}{5}$
C) $\frac{1}{6}$
D) $\frac{1}{7}$
E) $\frac{2}{7}$

Q18 Quadratic trigonometric equations View

$$3\sin x - 4\cos x = 0$$
Given this, what is the value of $|\cos 2x|$?
A) $\frac{3}{4}$
B) $\frac{3}{5}$
C) $\frac{4}{5}$
D) $\frac{7}{25}$
E) $\frac{9}{25}$

Q19 Trig Proofs Trigonometric Identity Simplification View

$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$
Which of the following is this expression equal to?
A) $\frac{1}{\cos x}$
B) $\frac{1}{\sin x}$
C) $1$
D) $\arcsin x$
E) $\arccos x$

Q20 Addition & Double Angle Formulae Simplification of Trigonometric Expressions with Specific Angles View

$$\frac{\tan 60^{\circ}}{\sin 20^{\circ}} - \frac{1}{\cos 20^{\circ}}$$
Which of the following is this expression equal to?
A) 4
B) 2
C) 1
D) $\frac{\sqrt{3}}{2}$
E) $\frac{1}{2}$

Q21 Addition & Double Angle Formulae Simplification of Trigonometric Expressions with Specific Angles View

$$\frac{1+\cos 40^{\circ}}{\cos 55^{\circ} \cdot \cos 35^{\circ}}$$
Which of the following is this expression equal to?
A) $\cos 20^{\circ}$
B) $2\cos 20^{\circ}$
C) $4\cos 20^{\circ}$
D) $\cos 40^{\circ}$
E) $2\cos 40^{\circ}$

Q22 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

In the complex number plane $$|z-1| = |z+2|$$ Which of the following does this equation represent?
A) The line $x = 1$
B) The line $x = \frac{-1}{2}$
C) The line $x = 2$
D) The circle $(x-1)^{2} + y^{2} = 1$
E) The circle $x^{2} + (y+2)^{2} = 1$

Q23 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View

Let $\bar{z}$ denote the conjugate of $z$. For the complex number $z = 2 + i$, $$\frac{z}{\bar{z}-1}$$ Which of the following is this expression equal to?
A) $\frac{1}{2} + \frac{3}{2}i$
B) $\frac{2}{3} - \frac{3}{2}i$
C) $1 + 3i$
D) $2 - 3i$
E) $3 + i$

Q24 Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem View

$$z = 1 + i\sqrt{3}$$
Which of the following is this complex number equal to?
A) $2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$
B) $2\left(\cos\frac{\pi}{6} - i\sin\frac{\pi}{6}\right)$
C) $2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$
D) $4\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$
E) $4\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)$

Q25 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View

Let $b$ and $c$ be real numbers. One root of the polynomial $P(x) = x^{2} + bx + c$ is the complex number $3 - 2i$.
Accordingly, what is $P(-1)$?
A) 5
B) 10
C) 20
D) 25
E) 30

Q26 Laws of Logarithms Express One Logarithm in Terms of Another View

$$\log_{3} 5 = a$$
Given this, what is the value of $\log_{5} 15$?
A) $\frac{a}{a+1}$
B) $\frac{a+1}{a}$
C) $\frac{a}{a+3}$
D) $\frac{a+3}{a}$
E) $\frac{4a}{3}$

Q27 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

$$\frac{1}{\log_{2} 6} + \frac{1}{\log_{3} 6}$$
Which of the following is this expression equal to?
A) $\frac{1}{3}$
B) $1$
C) $2$
D) $\log_{6} 2$
E) $\log_{6} 3$

Q28 Laws of Logarithms Solve a Logarithmic Inequality View

$$0 \leq \log_{2}(x-5) \leq 2$$
How many integers $x$ satisfy these inequalities?
A) 2
B) 3
C) 4
D) 5
E) 6

Q29 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

For positive real numbers $a$, $b$, $c$ different from 1, $$\log_{a} b = \frac{1}{2}, \quad \log_{a} c = 3$$ Given this, what is the value of the expression $\log_{b}\left(\frac{b^{2}}{c\sqrt{a}}\right)$?
A) $\frac{3}{2}$
B) $\frac{5}{2}$
C) $\frac{5}{3}$
D) $-6$
E) $-5$

Q30 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View

$$\sum_{n=0}^{100} 3^{n}$$
What is the remainder when this sum is divided by 5?
A) 0
B) 1
C) 2
D) 3
E) 4

Q31 Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View

The sequences $\{a_{n}\}$ and $\{b_{n}\}$ are defined as follows. $$a_{n} = \begin{cases} 0, & \text{if } n \equiv 0 \pmod{3} \\ n, & \text{if } n \equiv 1 \pmod{3} \\ -n, & \text{if } n \equiv 2 \pmod{3} \end{cases}, \quad b_{n} = \sum_{k=0}^{n} a_{k}$$ Accordingly, what is $b_{4}$?
A) $-2$
B) $-1$
C) $0$
D) $2$
E) $3$

Q32 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

The angle formed by the lines $d_{1}$ and $d_{2}$ given above measures $30^{\circ}$. First, a perpendicular $A_{1}B_{1}$ is drawn from point $A_{1}$ on line $d_{1}$ to line $d_{2}$. Then, a perpendicular $B_{1}A_{2}$ is drawn from point $B_{1}$ to line $d_{1}$, and a perpendicular $A_{2}B_{2}$ is drawn from the foot of the perpendicular $A_{2}$ to line $d_{2}$, and this process continues.
Given that $|A_{1}B_{1}| = 12$ cm, what is the sum of the lengths of all perpendiculars drawn to line $d_{2}$ in this manner, $|A_{1}B_{1}| + |A_{2}B_{2}| + |A_{3}B_{3}| + \cdots$, in cm?
A) 32
B) 36
C) 38
D) 40
E) 48

Q33 3x3 Matrices Direct Determinant Computation View

$$\left|\begin{array}{rrr} 2 & -3 & 2 \\ 1 & 2 & 0 \\ 2 & 3 & 0 \end{array}\right|$$
What is the value of this determinant?
A) $-1$
B) $-2$
C) $-3$
D) $-4$
E) $-6$

Q34 Matrices Matrix Algebra and Product Properties View

$$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$
Given that $A^{t}$ is the transpose of the matrix and $A^{-1}$ is its inverse matrix, which of the following is the product $A^{t} \cdot A^{-1}$?
A) $\begin{bmatrix} \frac{5}{2} & -3 \\ \frac{9}{2} & -5 \end{bmatrix}$
B) $\begin{bmatrix} \frac{3}{2} & -2 \\ 1 & 3 \end{bmatrix}$
C) $\begin{bmatrix} -2 & \frac{-9}{2} \\ 3 & \frac{5}{2} \end{bmatrix}$
D) $\begin{bmatrix} \frac{9}{2} & 3 \\ \frac{-5}{2} & -1 \end{bmatrix}$
E) $\begin{bmatrix} -3 & -1 \\ \frac{5}{2} & -2 \end{bmatrix}$

Q35 3x3 Matrices Solving a 3×3 Linear System Explicitly View

$$\begin{array}{r} 2x + 2y - z = 1 \\ x + y + z = 2 \\ y - z = 1 \end{array}$$
In the solution of the system of equations above, what is $x$?
A) $-3$
B) $-2$
C) $-1$
D) $0$
E) $3$

Q36 Differentiation from First Principles View

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, $$f'(x) = 2x^{2} - 1, \quad f(2) = 4$$ Given this, what is the value of the limit $\displaystyle\lim_{x \rightarrow 2} \frac{f(x)-4}{x-2}$?
A) 3
B) 4
C) 5
D) 6
E) 7

Q37 Differentiating Transcendental Functions Limit involving transcendental functions View

$$\lim_{x \rightarrow 1} \frac{1-\sqrt{x}}{\ln x}$$
What is the value of this limit?
A) $\frac{-1}{2}$
B) $0$
C) $\frac{1}{2}$
D) $1$
E) $2$

Q38 Curve Sketching Limit Reading from Graph View

The graph of the function $f: \mathbb{R}\setminus\{-1\} \rightarrow \mathbb{R}\setminus\{2\}$ is shown in the figure above.
Accordingly, $$\lim_{x \rightarrow -\infty} f(x) + \lim_{x \rightarrow 0} f(x)$$ What is the sum of these limits?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $3$

Q39 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View

$$f(x) = \ln\left(\sin^{2} x + e^{2x}\right)$$
Given this, what is $f'(0)$?
A) $e$
B) $1$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $2$

Q40 Tangents, normals and gradients Determine unknown parameters from tangent conditions View

For the function $f(x) = 2x^{3} - ax^{2} + 3$, what should $a$ be so that the equation of the tangent line to the curve at some point is $y = 4$?
A) $-3$
B) $-1$
C) $0$
D) $1$
E) $3$

Q41 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

$$f(x) = x^{4} - 5x^{2} + 4$$
What is the maximum value of the function on the interval $\left[\frac{-1}{2}, \frac{1}{2}\right]$?
A) 8
B) 6
C) 4
D) 2
E) 0

Q42 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

$$f''(x) = 6x - 2, \quad f'(0) = 4, \quad f(0) = 1$$
For the function $f$ that satisfies these conditions, what is the value of $f(1)$?
A) 4
B) 5
C) 6
D) 7
E) 8

Q43 Tangents, normals and gradients Find tangent line with a specified slope or from an external point View

The tangent line drawn from a point $A(x, y)$ on the parabola $y^{2} = 4x$ has slope 1.
Accordingly, what is $x + y$, the sum of the coordinates of point $A$?
A) 1
B) 2
C) 3
D) 4
E) 5

Q44 Stationary points and optimisation Geometric or applied optimisation problem View

A workplace consisting of a corridor, kitchen, and study room has the model shown above as rectangle ABCD, and the perimeter of this rectangle is 72 meters.
For the kitchen in this workplace to have the largest area, what should $x$ be in meters?
A) 1
B) 2
C) 3
D) 4
E) 5

Q45 Tangents, normals and gradients Determine unknown parameters from tangent conditions View

If the line tangent to the parabola $y = x^{2} + bx + c$ at the point $x = 2$ is $y = x$, what is the sum $b + c$?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

Q46 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View

$$\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{\cos^{2} x}\, dx$$
What is the value of the integral?
A) 2
B) 1
C) 0
D) $-1$
E) $-2$

Q47 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

$$\int_{0}^{4} \frac{6x}{\sqrt{2x+1}}\, dx$$
What is the value of the integral?
A) 12
B) 15
C) 18
D) 20
E) 24

Q48 Areas Between Curves Compute Area Directly (Numerical Answer) View

What is the area in square units of the (finite) region bounded by the curve $y = x^{3}$ and the line $y = x$?
A) $\frac{1}{2}$
B) $\frac{3}{2}$
C) $1$
D) $\frac{1}{3}$
E) $\frac{2}{3}$

Q49 Integration by Parts Definite Integral Evaluation by Parts View

For the function $f$ whose graph is given above, $$\int_{1}^{3} \frac{x \cdot f'(x) - f(x)}{x^{2}}\, dx$$ What is the value of the integral?
A) $\frac{7}{2}$
B) $\frac{3}{2}$
C) $\frac{2}{3}$
D) $\frac{1}{3}$
E) $\frac{5}{4}$

Q50 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

$$f(x) = \begin{cases} 3 - x, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$$
What is the value of the integral $\displaystyle\int_{1}^{3} f(x+1)\, dx$?
A) 2
B) 4
C) 6
D) 8
E) 10

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