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Q1 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View

In the set of complex numbers
$$\frac { ( 4 - 2 i ) \cdot ( 6 + 3 i ) } { ( 1 - i ) \cdot ( 1 + i ) }$$
What is the result of this operation?
A) 15
B) 12
C) 10
D) 9
E) 6

Q2 Proof Counting or Combinatorial Problems on APs View

When all of the numbers 1, 2, 3, 4, 5, 6, and 7 are placed in 7 boxes with addition or subtraction symbols between them, with one number in each box, the result of the operation obtained is 4.
$$\square + \square + \square + \square + \square - \mathrm { A } - \mathrm { B } = 4$$
Accordingly, what is the product A · B?
A) 15
B) 24
C) 28
D) 30
E) 35

Q4 Number Theory GCD, LCM, and Coprimality View

Let p, r, and t be different prime numbers;

Integer multiples of p form set A,
Integer multiples of r form set B,
Integer multiples of t form set C.

It is known that two of the numbers 220, 245, 330, and 350 are elements of the blue-colored set, and the other two are elements of the yellow-colored set. Accordingly, what is the sum $\mathbf { p } + \mathbf { r } + \mathbf { t }$?
A) 10
B) 14
C) 15
D) 21
E) 23

Q7 Laws of Logarithms Verify Truth of Logarithmic Statements View

In a Mathematics lesson, Canan performed operations by following the steps below in order. I. $\operatorname { step } \quad : \quad 6 = 1 \cdot 2 \cdot 3 = \mathrm { e } ^ { \ln 1 } \cdot \mathrm { e } ^ { \ln 2 } \cdot \mathrm { e } ^ { \ln 3 }$ II. $\operatorname { step } : \quad e ^ { \ln 1 } \cdot e ^ { \ln 2 } \cdot e ^ { \ln 3 } = e ^ { \ln 1 + \ln 2 + \ln 3 }$ III. step: $\quad e ^ { \ln 1 + \ln 2 + \ln 3 } = e ^ { \ln 6 }$ IV. $\operatorname { step } : : \mathrm { e } ^ { \ln 6 } = \mathrm { e } ^ { \ln ( 2 + 4 ) }$ V. step: $\mathrm { e } ^ { \ln ( 2 + 4 ) } = \mathrm { e } ^ { \ln 2 + \ln 4 }$ VI. $\operatorname { step } : \quad e ^ { \ln 2 + \ln 4 } = e ^ { \ln 2 } \cdot e ^ { \ln 4 }$ VII. step: $e ^ { \ln 2 } \cdot e ^ { \ln 4 } = 2 \cdot 4 = 8$
At the end of these steps, Canan obtained the result $6 = 8$. Accordingly, in which of the numbered steps did Canan make an error?
A) II
B) III
C) IV
D) V
E) VI

Q8 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View

A function f defined on the set of real numbers satisfies the equality
$$f ( x + y ) = f ( x ) + f ( y )$$
for every real numbers x and y. Given that $\mathbf { f } ( \mathbf { 2 } ) - \mathbf { f } ( \mathbf { 1 } ) = \mathbf { 1 0 }$,
what is the result of the operation $$\frac { f ( 3 ) \cdot f ( 4 ) } { f ( 5 ) }$$?
A) 15
B) 16
C) 18
D) 21
E) 24

Q9 Curve Sketching Limit Reading from Graph View

In the rectangular coordinate plane, the graph of a function f defined on the interval $[ 0,2 ]$ is given below.
Accordingly, I. $( f \circ f ) ( x ) = 2$ II. $( f \circ f ) ( x ) = 1$ III. $( f \circ f ) ( x ) = 0$ Which of these equalities are satisfied for exactly two different values of x?
A) Only I
B) Only II
C) Only III
D) I and II
E) II and III

Q10 Inequalities Solve Polynomial/Rational Inequality for Solution Set View

Let a be a real number. Regarding the inequality $x + 1 \leq a$, the following are known.

$\mathrm { x } = 0$ satisfies this inequality.
$x = 4$ does not satisfy this inequality.

Accordingly, what is the widest interval expressing the values that the number a can take?
A) $( 0,4 ]$
B) $[ 0,4 )$
C) $[ 1,4 ]$
D) $( 1,5 ]$
E) $[ 1,5 )$

Q11 Inequalities Ordering and Sign Analysis from Inequality Constraints View

For distinct real numbers $a , b$ and $c$,
$$\begin{aligned} & a + b = | a | \\ & b + c = | b | \end{aligned}$$
equalities are given. Accordingly; what is the correct ordering of the numbers $\mathbf { a , b }$ and c?
A) a < b < c
B) a $<$ c $<$ b
C) b $<$ a $<$ c
D) b $<$ c $<$ a
E) c $<$ a $<$ b

Q12 Polynomial Division & Manipulation View

It is known that a fourth-degree polynomial whose leading coefficient is 1 has roots that are all integers. Some parts of this polynomial's graph where it intersects the axes in the rectangular coordinate plane are given below.
Accordingly, what is the sum of the coefficients of this polynomial?
A) 72
B) 80
C) 84
D) 92
E) 96

Q13 Discriminant and conditions for roots Quadratic equation with parametric or self-referential conditions View

A second-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients whose leading coefficient is 1 has two distinct roots that are $P ( 0 )$ and $P ( - 1 )$. Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 5 } { 2 }$
D) 1
E) 2

Q14 Laws of Logarithms Solve a Logarithmic Equation View

Let x be an integer greater than 1.

$\frac { 64 } { \mathrm { x } }$ is an integer,
$\frac { \ln 64 } { \ln x }$ is not an integer.

Accordingly, what is the sum of the values that x can take?
A) 40
B) 42
C) 48
D) 54
E) 56

Q15 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

Ada calculates the value of $\log _ { 2 } n$ on her scientific calculator for every positive integer n where $\mathrm { n } \leq 32$, and observes that each value is either an integer or a decimal number. Ada writes down either the number itself if the value displayed on the screen is an integer, or the integer part of the number if it is a decimal, and then finds the sum of these numbers she wrote down. Accordingly, what is the result of the sum that Ada found?
A) 94
B) 97
C) 100
D) 103
E) 106

Q16 Sequences and series, recurrence and convergence Summation of sequence terms View

For a sequence $a_n$ where the sum of any three consecutive terms is equal to each other,
$$a _ { 2 } + a _ { 3 } = a _ { 4 } = 2$$
equality is satisfied.
Accordingly, $$a _ { 1 } + a _ { 2 } + \ldots + a _ { 25 }$$
what is the result of the sum?
A) 34
B) 35
C) 36
D) 37
E) 38

Q17 Curve Sketching Determining coefficients from given conditions on function values or geometry View

Let $0 < x _ { 1 } < x _ { 2 }$. A function f defined on the set of real numbers as
$$f ( x ) = \left( x - x _ { 1 } \right) \left( x - x _ { 2 } \right)$$
The parabola represented by this function intersects the axes at different points A and B in the rectangular coordinate plane as shown in the figure.
The distances from points A and B to the origin are equal, and this parabola takes its minimum value when $x = \frac { 3 } { 5 }$. Accordingly, what is the ratio $\frac { \mathbf { x } _ { \mathbf { 2 } } } { \mathbf { x } _ { \mathbf { 1 } } }$?
A) 2
B) 3
C) 4
D) 5
E) 6

Q18 Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let A and B be non-empty sets consisting of digits. If
$$A \cap B = A \cap \{ 0,2,4,6,8 \}$$
equality is satisfied, then A is called the common-intersection set of B. Given that set A is the common-intersection set of
$$B = \{ 0,1,2,3,4 \}$$
how many different sets A are there?
A) 3
B) 7
C) 15
D) 31
E) 63

Q19 Permutations & Arrangements Circular Arrangement View

Six friends named Ayça, Büşra, Ceyda, Deniz, Erdem and Furkan attending a party have a table with 6 chairs around it as shown in the figure.
Ayça and Büşra, who are on bad terms, do not want to sit in chairs that are next to each other or facing each other at this table. Accordingly, in how many different ways can these six friends sit in these chairs around the table?
A) 432
B) 384
C) 360
D) 288
E) 240

Q20 Probability Definitions Finite Equally-Likely Probability Computation View

Ege's bag contains four cards of the same size: an identity card, a student card, a meal card, and a bus card. Ege draws a card randomly from his bag to find the bus card. If he draws the wrong card, he keeps it in his hand and draws another card randomly from his bag, and continues this way until he finds the bus card. What is the probability that Ege finds the bus card on the third attempt?
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 1 } { 16 }$
E) $\frac { 3 } { 16 }$

Q21 Composite & Inverse Functions Limit Evaluation Involving Composition or Substitution View

A function f is defined on a subset of the set of real numbers as
$$f ( x ) = \frac { x ^ { 2 } - 4 x + 4 } { x - 2 } + \frac { x ^ { 2 } - 6 x + 9 } { 2 x - 6 }$$
Accordingly, $$\lim _ { x \rightarrow 2 } f ( x ) + \lim _ { x \rightarrow 3 } f ( x )$$
what is the value of this expression?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 4 }$

Q22 Composite & Inverse Functions Qualitative Analysis of DE Solutions View

Let a be a real number. A function f is defined on the set of real numbers as
$$f ( x ) = \left\{ \begin{array} { c c c } a - x & , & x < 1 \\ 5 x - 4 & , & 1 \leq x \leq 5 \\ ( x - a ) ^ { 2 } + 12 & , & x > 5 \end{array} \right.$$
If there is only one point where the function f is not continuous, what is the value of
$$f ( 7 ) - f ( 0 )$$?
A) 3
B) 4
C) 5
D) 6
E) 7

Q23 Chain Rule Finding Composition Parameters from Derivative Conditions View

Let k be a real number. For differentiable functions f and g defined on subsets of the set of real numbers,
$$f ( x ) = g \left( x ^ { 2 } \right) + k x ^ { 3 }$$
equality is satisfied.
Given that $$f ^ { \prime } ( - 1 ) = g ^ { \prime } ( 1 ) = 2$$
what is k?
A) 2
B) 1
C) 0
D) - 1
E) - 2

Q24 Stationary points and optimisation Chain Rule with Composition of Explicit Functions View

A function f is defined on the set of real numbers as
$$f ( x ) = x ^ { 2 } + x - 4$$
A function g defined and continuous on the set of real numbers has a derivative $g ^ { \prime }$ such that $g ^ { \prime } ( x ) = 0$ only for $x = 2$. Accordingly, the product of the x values satisfying
$$( g \circ f ) ^ { \prime } ( x ) = 0$$
is what?
A) 0
B) 1
C) 3
D) 4
E) 6

Q25 Tangents, normals and gradients Ordering and Comparing Exponential Values View

Below; the graphs of linear functions $f$, $g$ and $h$ are shown in Figure 1 on a rectangular coordinate plane divided into unit squares, and the derivatives of these functions are shown in Figure 2.
Accordingly; what is the correct ordering of $f ( 0 ) , g ( 0 )$ and $h ( 0 )$?
A) $\mathrm { f } ( 0 ) < \mathrm { h } ( 0 ) < \mathrm { g } ( 0 )$
B) $g ( 0 ) < f ( 0 ) < h ( 0 )$
C) $g ( 0 ) < h ( 0 ) < f ( 0 )$
D) $h ( 0 ) < f ( 0 ) < g ( 0 )$
E) $h ( 0 ) < g ( 0 ) < f ( 0 )$

Q26 Tangents, normals and gradients Common tangent line to two curves View

In the rectangular coordinate plane, the tangent line drawn to the graph of the function $f ( x ) = x ^ { 2 } + a x$ at the point $( 2 , f ( 2 ) )$ is tangent to the graph of the function $g ( x ) = b x ^ { 3 }$ at the point $( 1 , g ( 1 ) )$. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 2
B) 4
C) 6
D) 8
E) 10

Q27 Areas by integration View

In the rectangular coordinate plane, the line $y = \frac { x } { 2 }$ and the graph of the function $y = f ( x )$ are given below.
$$\begin{aligned} & \int _ { 0 } ^ { 4 } f ( x ) d x = 8 \\ & \int _ { 4 } ^ { 6 } f ( x ) d x = 3 \end{aligned}$$
Given that, what is the sum of the areas of the shaded regions in square units?
A) 3
B) 4
C) 5
D) 6
E) 8

Q28 Areas by integration View

Let c be a positive real number. In the rectangular coordinate plane, the line $y = c$ and the graph of the function $y = f ( x )$ are given below.
The area of the blue region is 2 square units more than the area of the yellow region.
$$\int _ { 1 } ^ { 4 } f ( 2 x ) d x = 28$$
Given that, what is the value of c?
A) 8
B) 9
C) 10
D) 11
E) 12

Q29 Areas by integration View

Let a be a positive integer. In the rectangular coordinate plane, the triangular region between the line $x + y = 2$ and the axes is divided into two regions by the curve $y = x ^ { a }$ as shown in the figure.
In the figure; the area of region $A _ { 2 }$ is 2 times the area of region $A _ { 1 }$. Accordingly, what is the value of a?
A) 2
B) 3
C) 4
D) 5
E) 6

Q30 Stationary points and optimisation MCQ on derivative and graph interpretation View

A function f is continuous on the closed interval $[ 0,6 ]$ and differentiable on each of the open intervals $( 0,3 ) , ( 3,4 ) , ( 4,6 )$. The graph of its derivative $f ^ { \prime }$ is given in the rectangular coordinate plane below.
$$\begin{gathered} \text{Let } 0 < c < 2 \text{ and } \\ f ( 0 ) = 5 \end{gathered}$$
Accordingly, which of the following could be the value of f(6)?
A) 5,5
B) 7,3
C) 10,1
D) 12,7
E) 14,9

Q31 Trig Graphs & Exact Values View

Let $\mathrm { a } \in \left( \frac { \pi } { 12 } , \frac { \pi } { 6 } \right)$.
$$\begin{aligned} & x = \sin ( 3 a ) \\ & y = \cos ( 3 a ) \\ & z = \tan ( 3 a ) \end{aligned}$$
What is the correct ordering of the numbers?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $y < z < x$
E) $z < x < y$

Q32 Reciprocal Trig & Identities View

Let $0 < \mathrm { x } < \frac { \pi } { 2 }$. $\sec x \cdot \tan x \cdot ( 1 - \sin x ) = \frac { 1 } { 4 }$ Accordingly, what is the value of $\csc x$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) 2
E) 3

Q33 Trig Proofs Trigonometric Identity Simplification View

Two right triangles $A B C$ and $B C D$ with one side coinciding are drawn as shown in the figure, and the resulting two regions are painted yellow and blue.
$$\mathrm { m } ( \widehat { \mathrm { DCA } } ) = \mathrm { m } ( \widehat { \mathrm { BAC } } ) = \mathrm { x }$$
Accordingly, what is the expression in terms of x for the ratio of the area of the yellow region to the area of the blue region?
A) $\sin 2 x$
B) $\cos 2 x$
C) $\sin ^ { 2 } x$
D) $\cot ^ { 2 } x$
E) $\csc ^ { 2 } x$

Q34 Straight Lines & Coordinate Geometry Collinearity and Concurrency View

Let m be a real number. In the rectangular coordinate plane,

the slope of a line passing through the point $( 0,1 )$ is $m$,
the slope of a line passing through the point $( 0,0 )$ is $2 m$,
the slope of a line passing through the point $( 1,0 )$ is $3 m$, and these three lines intersect at one point.

Accordingly, what is the value of $m$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 5 }$
E) $\frac { 4 } { 5 }$

Q37 Circles Circle Equation Derivation View

In the rectangular coordinate plane, a circle divided into two equal parts by the line $x + y = 4$ intersects the x-axis at a single point and the y-axis at two different points. Given that the distance between the points where the circle intersects the y-axis is 4 units, what is the circumference of the circle in units?
A) $4 \pi$
B) $5 \pi$
C) $6 \pi$
D) $7 \pi$
E) $8 \pi$

Q38 Circles Area and Geometric Measurement Involving Circles View

In a plane, three circles with radius r are constructed with the vertices of a right triangle $ABC$ as centers, and these circles do not intersect each other. The lengths of the parts on the sides of the triangle that are not inside these circles are given as 2 units, 3 units, and 5 units. Accordingly, what is the total area of the regions inside the circles but outside the triangle in square units?
A) $6 \pi$
B) $8 \pi$
C) $9 \pi$
D) $\frac { 9 \pi } { 2 }$
E) $\frac { 15 \pi } { 2 }$

Q39 Vectors Introduction & 2D Vector Word Problem / Physical Application View

In the rectangular coordinate plane, a point $P ( a , b )$ is rotated counterclockwise by $90 ^ { \circ }$ around the origin, and then the resulting point is translated 3 units in the positive direction along the x-axis and 1 unit in the positive direction along the y-axis, yielding the point $P ( a , b )$ again. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 0
B) 1
C) 2
D) 3
E) 4

Q40 Circles Volume by Displacement or Composite Solid View

A quarter-circle slice is cut out from a circular piece of paper with radius 8 units. The remaining part is joined together as shown in the figure with the red lines coinciding to form a right circular cone.
Accordingly, what is the height of the formed cone in units?
A) $2 \sqrt { 3 }$
B) $2 \sqrt { 5 }$
C) $2 \sqrt { 7 }$
D) $3 \sqrt { 2 }$
E) $3 \sqrt { 3 }$

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