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Q1 Indices and Surds Evaluating Expressions Using Index Laws View

$$3 ^ { 2 } \cdot \frac { 1 - 3 ^ { - 4 } } { 1 - 3 ^ { - 2 } }$$
What is the result of this operation?
A) 4
B) 5
C) 8
D) 9
E) 10

Q2 Inequalities Simplifying a rational or algebraic expression and solving View

$$\frac { \sqrt { 2 - 2 x } } { \sqrt { 3 + 3 x } } = \frac { 1 } { 2 }$$
Given that this holds, what is x?
A) $\frac { 2 } { 7 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 4 } { 9 }$
D) $\frac { 5 } { 11 }$
E) $\frac { 7 } { 12 }$

Q3 Proof Factorial and Combinatorial Expression Simplification View

$$\frac { ( 10 ! ) ^ { 2 } - ( 9 ! ) ^ { 2 } } { 11 ! - 10 ! - 9 ! }$$
Which of the following is this operation equal to?
A) $8 !$
B) $9 !$
C) $10 !$
D) $8 \cdot 8 !$
E) $8 \cdot 9 !$

Q4 Indices and Surds Simplifying a rational or algebraic expression and solving View

$$\frac { \frac { 4 } { 3 } + \frac { 3 } { 4 } } { \frac { 2 } { 3 } - \frac { 1 } { 4 } }$$
What is the result of this operation?
A) 5
B) 10
C) 15
D) 20
E) 25

Q5 Number Theory GCD, LCM, and Coprimality View

$\mathbf { a } < \mathbf { b } < \mathbf { c }$ are positive integers and
$$\begin{aligned} & \gcd ( a , b ) = 5 \\ & \gcd ( b , c ) = 4 \end{aligned}$$
Given this, what is the minimum value that the sum $\mathbf { a + b + c }$ can take?
A) 27
B) 35
C) 39
D) 45
E) 49

Q6 Number Theory Evaluating an algebraic expression given a constraint View

Let $\mathrm { a } , \mathrm { b }$ and c be prime numbers such that
$$\mathrm { ab } + \mathrm { ac } = 4 \mathrm { a } ^ { 2 } + 8$$
Given this, what is the product $\mathbf { a } \cdot \mathbf { b } \cdot \mathbf { c }$?

Q7 Solving quadratics and applications Linear Diophantine Equations View

$$\frac { x + \frac { 1 } { x + 2 } } { 1 - \frac { 1 } { x + 2 } } = \frac { 1 } { 4 }$$
What is the value of $x$ that satisfies this equality?
A) $\frac { - 3 } { 2 }$
B) $\frac { - 3 } { 4 }$
C) $\frac { - 1 } { 4 }$
D) $\frac { - 5 } { 4 }$
E) $\frac { - 3 } { 8 }$

Q8 Solving quadratics and applications Finite Geometric Sum and Term Relationships View

Let x be a positive integer such that
$$\frac { 10 x } { x + 3 }$$
is equal to the square of an integer. What is the sum of the values that x can take?
A) 26
B) 27
C) 29
D) 31
E) 32

Q9 Arithmetic Sequences and Series Quadratic Diophantine Equations and Perfect Squares View

Let a, b and c be three consecutive even integers arranged from smallest to largest such that the geometric mean of b and c is $\sqrt { 2 }$ times the geometric mean of a and b.
Accordingly, what is the sum $\mathrm { a } + \mathrm { b } + \mathrm { c }$?
A) 12
B) 18
C) 24
D) 30
E) 36

Q10 Arithmetic Sequences and Series Counting solutions satisfying modulus conditions View

What is the sum of all natural numbers such that when divided by 6, the quotient and remainder are equal to each other?
A) 84
B) 91
C) 96
D) 105
E) 112

Q11 Modulus function Ordering and Sign Analysis from Inequality Constraints View

Let $a , b , c$ be real numbers and $a \cdot b \cdot c > 0$ such that
$$\begin{aligned} & a \cdot b = - 2 | a | \\ & \frac { b } { c } = 3 | b | \end{aligned}$$
Given that $\mathbf { a } + \mathbf { b } + \mathbf { c } = \mathbf { 0 }$, what is a?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 7 } { 3 }$
E) $\frac { 8 } { 3 }$

Q12 Inequalities Basic Combination Computation View

Let $a , b , c$ be real numbers and $0 < b < 1$ such that
$$\begin{aligned} & a = b \cdot c \\ & a + c = b \end{aligned}$$
Given this, which of the following orderings is correct?
A) $a < b < c$
B) $a < c < b$
C) $b < a < c$
D) $c < a < b$
E) $c < b < a$

Q13 Combinations & Selection Combinatorial Counting (Non-Probability) View

Let $\mathbf { A } = \{ \mathbf { a } , \mathbf { b } , \mathbf { c } , \mathbf { d } \}$. For non-empty subsets $X , Y$ of A
$$\begin{aligned} & X \cap Y = \emptyset \\ & X \cup Y = A \end{aligned}$$
How many ordered pairs (X, Y) are there such that these conditions hold?
A) 6
B) 8
C) 10
D) 12
E) 14

Q14 Probability Definitions Combinatorial Probability View

Four identical matches are taken, each with only one flammable end. These matches are randomly arranged along all sides of a square whose side length is the same as the length of one match, with the ends touching each other.
What is the probability that there are no flammable ends in contact with each other in this arrangement?
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 1 } { 16 }$
E) $\frac { 3 } { 16 }$

Q16 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

The function f on the set of real numbers is defined for every real number x as
$$f ( x ) = \left\{ \begin{array} { c c } x + 2 , & x < 0 \\ x , & x \geq 0 \end{array} \right.$$
Accordingly, what is the value of the sum $\sum _ { k = - 3 } ^ { 4 } f ( k )$?
A) 8
B) 10
C) 12
D) 14
E) 16

Q17 Function Transformations Set Operations View

Below is the graph of a function $f$. $( a > 2 , b < 1 )$
Accordingly, which of the following could be the graph of the function $| \mathbf { f } ( \mathbf { x } + \mathbf { 2 } ) | - \mathbf { 1 }$?
A) [graph A]
B) [graph B]
C) [graph C]
D) [graph D]
E) [graph E]

Q19 Binomial Theorem (positive integer n) Set Operations View

Let m and n be real numbers. In the expansion of
$$\left( \frac { \mathrm { m } } { \mathrm { nx } } + \frac { \mathrm { nx } ^ { 2 } } { \mathrm {~m} } \right) ^ { 3 }$$
arranged according to powers of x, the constant term is 6.
Accordingly, what is the ratio $\frac { m } { n }$?
A) 1
B) 2
C) 3
D) 4
E) 5

Q20 Factor & Remainder Theorem Recover a Function from a Composition or Functional Equation View

Let $\mathrm { P } ( \mathrm { x } )$ be a second-degree polynomial and $\mathrm { Q } ( \mathrm { x } ) = \mathrm { k }$ be a constant polynomial such that
$$\begin{aligned} & P ( x ) + Q ( x ) = 2 x ^ { 2 } + 3 \\ & P ( Q ( x ) ) = 9 \end{aligned}$$
Accordingly, what is the sum of the values that k can take?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 3 } { 4 }$

Q21 Factor & Remainder Theorem Remainder Theorem with Composed or Shifted Arguments View

The third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with leading coefficient 1 is divisible without remainder by $x ^ { 2 } + 4$. The remainder obtained from dividing the polynomial $P ( 2 x )$ by $2 x - 3$ is 52.
Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) 20
B) 22
C) 24
D) 26
E) 28

Q22 Discriminant and conditions for roots Determine coefficients or parameters from root conditions View

Let b and c be non-zero real numbers such that the roots of the equation
$$x ^ { 2 } + b x + c = 0$$
are b and c. Accordingly, what is the product $b \cdot c$?
A) $- 6$
B) $- 5$
C) $- 4$
D) $- 3$
E) $- 2$

Q23 Chain Rule Full function study (variation table, limits, asymptotes) View

In the open interval $(1, e)$, I. The function $\sin ( \ln ( x ) )$ is increasing. II. The function $\cos ( \ln ( x ) )$ is increasing. III. The function $\tan ( \ln ( \mathrm { x } ) )$ is increasing. Which of these statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Q24 Reciprocal Trig & Identities Find an angle using the cosine rule View

$| \mathrm { AB } | = 6$ units $| \mathrm { BH } | = 2$ units $[ \mathrm { BC } ] \cap [ \mathrm { GF } ] = \mathrm { H }$ $\mathrm { m } ( \widehat { \mathrm { GHC } } ) = \mathrm { x }$
In the figure, $ABCD$ and $AEFG$ are congruent squares. Accordingly, what is the value of $\tan ( x )$?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 5 } { 4 }$

Q25 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

For $0 \leq \mathrm { x } \leq \pi$,
$$\frac { \sin x \cdot \tan x } { 3 } = 1 - \cos x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equation?
A) $\frac { \pi } { 3 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 4 \pi } { 3 }$
D) $\pi$
E) $2 \pi$

Q26 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View

In the set of complex numbers, the result of the operation
$$( 3 - i ) ( 2 - i ) ( 1 + i ) ( 2 + i ) ( 3 + i )$$
is $\mathbf { a } + \mathbf { b i }$. What is the sum $a + b$?
A) 80
B) 84
C) 90
D) 96
E) 100

Q27 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View

Let z be a complex number such that
$$\begin{aligned} & | z - 1 | = | z - 2 | \\ & | z | = \sqrt { 3 } \end{aligned}$$
What is the value of $| z - 3 |$?
A) 2
B) $\sqrt { 2 }$
C) $\sqrt { 3 }$
D) $1 + \sqrt { 2 }$
E) $\sqrt { 3 } - 1$

Q28 Complex Numbers Arithmetic Geometric Interpretation in the Complex Plane View

In the complex plane, the vertices of the quadrilateral formed by the roots of the equation
$$z ^ { 4 } = 16$$
have what area in square units?
A) 8
B) 12
C) 16
D) $4 \sqrt { 3 }$
E) $6 \sqrt { 2 }$

Q29 Laws of Logarithms Solve a Logarithmic Equation View

$\log _ { 4 } x$ and $\log _ { 4 } \left( x ^ { 2 } \right)$ are consecutive two positive even integers.
Accordingly, what is the value of $\log _ { x } 4$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 16 }$
D) 1
E) 2

Q30 Laws of Logarithms Solve a Logarithmic Equation View

Let k be a positive real number such that for the function
$$f ( x ) = \log _ { x } ( x - k )$$
we have $f ( 3 k ) = \frac { 2 } { 3 }$. What is k?
A) $\frac { 3 } { 8 }$
B) $\frac { 9 } { 8 }$
C) $\frac { 27 } { 8 }$
D) $\frac { 2 } { 9 }$
E) $\frac { 4 } { 9 }$

Q31 Sequences and series, recurrence and convergence Telescoping or Non-Standard Summation Involving an AP View

$$\sum _ { n = 5 } ^ { 14 } \frac { 1 } { 1 + 2 + \cdots + n }$$
What is the value of this sum?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 3 } { 5 }$
D) $\frac { 2 } { 15 }$
E) $\frac { 4 } { 15 }$

Q32 Number Theory Prime Number Properties and Identification View

Let n be an integer greater than 2, and let the largest prime divisor of n be denoted by $\tilde{n}$. The terms of the sequence $(a_n)$ are defined for $n \geq 2$ as
$$a _ { n } = \left\{ \begin{aligned} 1 & , \tilde{n} < 10 \\ - 1 & , \tilde{n} > 10 \end{aligned} \right.$$
Accordingly, what is the sum $\sum _ { n = 15 } ^ { 30 } a _ { n }$?
A) 2
B) 3
C) 4
D) 5
E) 6

Q33 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

In the rectangular coordinate plane, isosceles triangles are drawn with base vertices at consecutive even natural numbers on the x-axis and apex on the curve $y = 2 ^ { - x }$.
Accordingly, what is the sum of the areas of all the triangles drawn in square units?
A) $\frac { 3 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 1
E) 2

Q34 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View

Let $M = \left[ \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right]$ and $X = \left[ \begin{array} { l } 1 \\ 2 \end{array} \right]$ such that
$$\begin{aligned} & \mathrm { M } \cdot \mathrm { X } = \mathrm { aX } \\ & \mathrm { M } ^ { - 1 } \cdot \mathrm { X } = \mathrm { bX } \end{aligned}$$
For real numbers a and b satisfying these equalities, what is the sum $a + b$?
A) $\frac { 1 } { 3 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 10 } { 3 }$

Q35 3x3 Matrices Determinant and Rank Computation View

Let A be a $2 \times 2$ matrix and $I$ be the $2 \times 2$ identity matrix such that
$$A ^ { 2 } = \left[ \begin{array} { l l } 2 & 1 \\ 1 & 5 \end{array} \right]$$
What is the value of the determinant $| ( \mathbf { A } - \mathbf { I } ) ( \mathbf { A } + \mathbf { I } ) |$?
A) 2
B) 3
C) 4
D) 5
E) 6

Q36 Matrices Linear System and Inverse Existence View

Let $A$ and $B$ be $2 \times 1$ matrices and $t$ be a variable such that for all $x$ and $y$ values satisfying
$$x - y = 3$$
we have
$$\left[ \begin{array} { l } x \\ y \end{array} \right] = t A + B$$
Accordingly, which of the following could matrices A and B be, respectively?
A) $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
B) $\left[ \begin{array} { l } 0 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
C) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 1 \end{array} \right]$
D) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
E) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ - 3 \end{array} \right]$

Q37 Curve Sketching Geometric or real-world application leading to a quadratic equation View

$$\lim _ { x \rightarrow 0 ^ { + } } ( \sin x ) \cdot ( \ln x )$$
Which of the following is this limit equal to?
A) $- 1$
B) 0
C) 1
D) $\infty$
E) $- \infty$

Q38 Curve Sketching Asymptote Determination View

The function
$$f ( x ) = \frac { a x } { | b x + 2 | }$$
defined on a subset of the set of positive real numbers has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.
Accordingly, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5

Q39 Curve Sketching Multi-Statement Verification (Remarks/Options) View

Below is the graph of the function $f$.
Accordingly, regarding the function f: I. The function f does not have an absolute maximum value on the interval $[ 0,4 ]$. II. There exists $a \in [ 0,4 ]$ such that $f ( a ) = 2$. III. $\lim _ { x \rightarrow 1 ^ { - } } ( f \circ f ) ( x ) = 1$.
Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

Q40 Differentiation from First Principles Differentiability of functions involving modulus View

I. $f ( x ) = x - 1$ II. $g ( x ) = | x - 1 |$ III. $h ( x ) = \sqrt [ 3 ] { ( x - 1 ) ^ { 2 } }$ Which of the following functions do not have a derivative at the point $x = 1$?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

Q41 Composite & Inverse Functions Derivative of an Inverse Function View

For a function f defined on the set of positive real numbers with $f ( 3 ) = 2$, the derivative of the function f is given as
$$f ^ { \prime } ( x ) = x ^ { 2 } + x$$
For the function $\mathbf { g } ( \mathbf { x } ) = \mathbf { f } ^ { - \mathbf { 1 } } ( \mathbf { 2 x } )$, what is the value of $\mathbf { g } ^ { \prime } ( \mathbf { 1 } )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 6 }$

Q42 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given below.
Accordingly, regarding the function f: I. It is decreasing. II. $f ( a )$ is a local maximum value. III. $f ^ { \prime \prime } ( a )$ is not defined.
Which of the following statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Q43 Applied differentiation Geometric or applied optimisation problem View

In the rectangular coordinate plane, the graph of the curve $y = e ^ { \left( - x ^ { 2 } \right) }$ is given.
In this plane, a rectangle with one side on the x-axis and two vertices on the curve is drawn with the maximum possible area.
What is the area of this rectangle in square units?
A) $\sqrt { \mathrm { e } }$
B) $\sqrt { 2 e }$
C) $\frac { \sqrt { e } } { 2 }$
D) $\sqrt { \frac { 2 } { \mathrm { e } } }$
E) $2 \sqrt { e }$

Q44 Tangents, normals and gradients Prove a given line is tangent to a curve View

The line $y = 4 x - 2$ is tangent to the graph of the function $f ( x ) = x ^ { 4 } + 1$ at the point $\mathrm { P } ( \mathrm { a } , \mathrm { b } )$.
Accordingly, what is the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7

Q45 Integration by Parts Definite Integral Evaluation by Parts View

For a function f defined on the set of real numbers and twice differentiable,
$$\begin{aligned} & f ( 1 ) = f ( 2 ) = 2 \\ & f ^ { \prime } ( 1 ) = f ^ { \prime } ( 2 ) = - 1 \end{aligned}$$
the following equalities are given.
Accordingly, what is the value of the integral $\int _ { 1 } ^ { 2 } x \cdot f ^ { \prime \prime } ( x ) d x$?
A) $- 1$
B) $- 2$
C) $- 3$
D) $\frac { - 1 } { 2 }$
E) $\frac { - 2 } { 3 }$

Q46 Areas by integration View

In the rectangular coordinate plane, the shaded region between the curve $y = x ^ { 2 }$, the x-axis, and the line $x = 3$ is shown.
This shaded region is divided into three equal-area sub-regions by the lines $x = a$ and $x = b$.
Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) $5 \sqrt { 2 }$
B) $4 \sqrt { 3 }$
C) $6 \sqrt { 3 }$
D) $3 \sqrt [ 3 ] { 6 }$
E) $2 \sqrt [ 3 ] { 9 }$

Q47 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

Let a be a positive real number. For every second-degree polynomial $P ( x )$ with real coefficients and leading coefficient 1,
$$\int _ { - 1 } ^ { 1 } \mathrm { P } ( \mathrm { x } ) \mathrm { dx } = \mathrm { P } ( \mathrm { a } ) + \mathrm { P } ( - \mathrm { a } )$$
the equality is satisfied. Accordingly, what is the value of a?
A) $\sqrt { 2 }$
B) $\sqrt { 3 }$
C) $\sqrt { 6 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) $\frac { \sqrt { 3 } } { 3 }$

Q48 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

$$\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } } { x ^ { 2 } - 1 } d x$$
What is the value of the integral?
A) $1 + \ln \left( \frac { 4 } { 3 } \right)$
B) $1 + \ln \left( \frac { 9 } { 2 } \right)$
C) $2 + \ln \left( \frac { 3 } { 2 } \right)$
D) $2 + \ln \left( \frac { 5 } { 3 } \right)$
E) $3 + \ln \left( \frac { 1 } { 3 } \right)$

Q49 Areas by integration Geometric Series with Trigonometric or Functional Terms View

Let $R$ be the set of real numbers. For every natural number n,
$$\begin{aligned} & f _ { n } : [ n \pi , ( n + 1 ) \pi ] \rightarrow R \\ & f _ { n } ( x ) = \frac { 1 } { 5 ^ { n } } | \sin x | \end{aligned}$$
What is the sum of the areas of the regions between the functions defined in this form and the x-axis in square units?
A) $\frac { 7 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 9 } { 5 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$

Q50 Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View

In the rectangular coordinate plane, the region between the parabola $y = x ^ { 2 }$, the line $x = 1$, and the line $y = 0$ is shown.
What is the volume in cubic units of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the line $\mathbf { y = - 1 }$?
A) $\frac { 3 \pi } { 4 }$
B) $\frac { 5 \pi } { 8 }$
C) $\frac { 7 \pi } { 10 }$
D) $\frac { 11 \pi } { 12 }$
E) $\frac { 13 \pi } { 15 }$

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