turkey-yks

Q1 Indices and Surds Numerical Arithmetic with Fractions and Decimals View

$$\frac { 6 ^ { 4 } - 4 ^ { 4 } } { 5 \cdot 2 ^ { 4 } }$$
What is the result of this operation?
A) 9
B) 12
C) 13
D) 14
E) 16

Q2 Indices and Surds Numerical Arithmetic with Fractions and Decimals View

$$\frac { 4 } { 9 - \frac { 49 } { 9 } } - \frac { 1 } { 8 }$$
What is the result of this operation?
A) 1
B) 2
C) 3
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 4 }$

Q3 Indices and Surds Simplifying Surd Expressions View

$$\frac { \sqrt { 48 } } { \frac { 1 } { \sqrt { 3 } } + \frac { 1 } { \sqrt { 27 } } }$$
What is the result of this operation?
A) 3
B) 5
C) 8
D) 9
E) 12

Q4 Permutations & Arrangements Factorial and Combinatorial Expression Simplification View

$$\frac { ( n + 1 ) ! + ( n - 1 ) ! } { n ^ { 3 } - 1 } = 24$$
Given this equality, what is n?
A) 3
B) 5
C) 6
D) 8
E) 9

Q5 Number Theory Congruence Reasoning and Parity Arguments View

The greatest common divisor of positive integers a and b is odd, and their least common multiple is even.
Accordingly, I. $a \cdot b$ II. $a + b$ III. $a ^ { b }$ Which of the following expressions always equals an odd number?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III

Q6 Number Theory Combinatorial Number Theory and Counting View

In the following table consisting of 100 unit squares numbered from 1 to 100, some squares will be painted.

1	2	3	$\ldots$	10
11	12	13	$\ldots$	20
.	.	.		.
.	.	.		.
.	.	.		.
91	92	93	..	100

Squares with even numbers are painted yellow, squares that are multiples of 3 are painted red, and squares that are multiples of 5 are painted blue.
For a square to be orange, it must be painted only yellow and red.
Accordingly, how many unit squares in the table are orange?
A) 8
B) 12
C) 13
D) 15
E) 18

Q7 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View

The sum of the prime divisors of a natural number A is;

3 less than the sum of the prime divisors of $12 \cdot A$.
5 less than the sum of the prime divisors of $70 \cdot A$.

Accordingly, what is the sum of the digits of the smallest value that A can take?
A) 4
B) 5
C) 6
D) 7
E) 8

Q8 Indices and Surds Ratio and Proportion Problems View

Positive real numbers $a$ and $b$ satisfy the equality
$$a ^ { 2 } - 2 a b - 3 b ^ { 2 } = 0$$
Accordingly, what is the value of the expression $\frac { a + b } { a - b }$?
A) 2
B) 3
C) 4
D) 5
E) 6

Q9 Inequalities Ordering and Sign Analysis from Inequality Constraints View

For integers a and b
$$16 ^ { a } \cdot 9 ^ { a } = 6 ^ { b } \cdot 8 ^ { 2 }$$
Given this equality, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 6
B) 9
C) 12
D) 15
E) 20

Q10 Modulus function Algebraic identities and properties of modulus View

Real numbers $x$ and $y$ satisfy the equality
$$\| x | + | y | | = | x + y |$$
Accordingly, which of the following inequalities is always true?
A) $x \cdot y \geq 0$
B) $x \cdot y \leq 0$
C) $x + y \geq 0$
D) $x + y \leq 0$
E) $x - y \leq 0$

Q11 Inequalities Integer Solutions of an Inequality View

$$\mathrm { A } = \left\{ \mathrm { n } ( - 1 ) ^ { \mathrm { n } } : \mathrm { n } = 1,2,3 , \ldots , \mathrm { k } \right\}$$
The difference between the largest and smallest elements of the set is 25. Accordingly, how many positive elements does set A have?
A) 4
B) 6
C) 8
D) 10
E) 12

Q12 Inequalities Integer Solutions of an Inequality View

Integers a and b satisfy the inequality
$$1 < a < b - a < 5$$
Accordingly, what is the sum of the values that b can take?
A) 11
B) 14
C) 15
D) 16
E) 18

Q13 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View

The smaller of two numbers is 3 less than the arithmetic mean of these two numbers, and the larger is 4 more than the geometric mean of these two numbers.
Accordingly, what is the sum of these two numbers?
A) 7
B) 9
C) 10
D) 12
E) 14

Q14 Number Theory Modular Arithmetic Computation View

$$1 ^ { 5 } + 2 ^ { 5 } + 3 ^ { 5 } + 4 ^ { 5 } + 5 ^ { 5 }$$
What is the remainder when this expression is divided by 7?
A) 4
B) 3
C) 2
D) 1
E) 0

Q15 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

Functions $f$ and $g$ defined on the set of real numbers satisfy the equalities
$$\begin{aligned} & ( f + g ) ( x ) = x ^ { 2 } \\ & ( f - g ) ( 2 x ) = x \end{aligned}$$
Accordingly, what is the product $f ( 4 ) \cdot g ( 4 )$?
A) 45
B) 51
C) 54
D) 60
E) 63

Q16 Composite & Inverse Functions Symmetry, Periodicity, and Parity from Composition Conditions View

Function f is defined for every $\mathrm { x } \in ( 0,3 ]$ as
$$f ( x ) = 2 x + 1$$
and satisfies the equality
$$f ( x ) = f ( x + 3 )$$
for every real number x. Accordingly, what is the sum $\mathbf { f } ( \mathbf { 6 } ) + \mathbf { f } ( \mathbf { 7 } ) + \mathbf { f } ( \mathbf { 8 } )$?
A) 8
B) 12
C) 15
D) 18
E) 21

Q17 Probability Definitions Set Operations View

Let N be the set of natural numbers. The sets
$$\begin{aligned} & C = \{ 2 n : n \in \mathbb { N } \} \\ & K = \left\{ n ^ { 2 } : n \in \mathbb { N } \right\} \end{aligned}$$
are given. Accordingly, which of the following is an element of the Cartesian product set
$$( \mathrm { K } \backslash \mathrm { C } ) \times ( \mathrm { C } \backslash \mathrm { K } )$$
?
A) $( 3,2 )$
B) $( 9,4 )$
C) $( 15,1 )$
D) $( 16,12 )$
E) $( 25,8 )$

Q18 Combinations & Selection Selection with Adjacency or Spacing Constraints View

A table consisting of two rows and 7 cells is given in the figure.
Patterns are created by painting 4 cells of this table black.
How many different patterns are there such that each row has at least one painted cell?
A) 26
B) 28
C) 30
D) 32
E) 34

Q19 Probability Definitions Finite Equally-Likely Probability Computation View

In the figure, 3 of the 6 edges of a regular tetrahedron are randomly painted.
Accordingly, what is the probability that all three painted edges are on the same face?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 5 }$
E) $\frac { 1 } { 6 }$

Q20 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

$$P ( x ) = ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) ^ { 4 }$$
What is the coefficient of the $x ^ { 4 }$ term in the polynomial?
A) 8
B) 10
C) 12
D) 14
E) 16

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

$$P ( x ) = x ^ { 3 } - m x + 1$$
The remainder when $P ( x - 1 )$ is divided by $x + 1$ equals the remainder when $P ( x + 1 )$ is divided by $x - 1$.
Accordingly, what is m?
A) 2
B) 4
C) 6
D) - 1
E) - 8

Q22 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View

A third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients and leading coefficient 1 satisfies the equalities
$$P ( 1 ) = P ( 3 ) = P ( 5 ) = 7$$
Accordingly, what is the value of $\mathbf { P } ( \mathbf { 0 } )$?
A) - 1
B) - 4
C) - 8
D) 4
E) 8

Q23 Discriminant and conditions for roots Root relationships and Vieta's formulas View

Let a be a real number. One root of the equation
$$a x ^ { 2 } - 18 x + 18 = 0$$
is 2 times the other. Accordingly, what is a?
A) 2
B) 3
C) 4
D) 5
E) 6

Q25 Trig Graphs & Exact Values View

$\cos x = \frac { \sqrt { 5 } } { 3 }$
Accordingly, I. $\sin \mathrm { x }$ II. $\sin 2 x$ III. $\cos 2 x$ Which of the following values equals a rational number?
A) Only I
B) Only III
C) I and II
D) I and III
E) II and III

Q26 Sine and Cosine Rules Find an angle using the cosine rule View

ABCD is a square, $\mathrm { AE } \cap \mathrm { BF } = \{ \mathrm { G } \}$, $| \mathrm { BC } | = 6$ units, $| \mathrm { DE } | = 4$ units, $| \mathrm { AF } | = 3$ units, $\mathrm { m } ( \widehat { \mathrm { FGE } } ) = \mathrm { x }$.
According to the given information above, what is the value of $\cot ( x )$?
A) $\frac { - 1 } { 4 }$
B) $\frac { - 5 } { 4 }$
C) $\frac { - 3 } { 8 }$
D) $\frac { - 1 } { 8 }$
E) $\frac { - 5 } { 8 }$

Q27 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View

Let z be a complex number satisfying the equality
$$i \cdot z + 1 = 2 ( 1 - \bar { z } )$$
What is the real part of the complex number z?
A) $\frac { 1 } { 6 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 5 } { 6 }$

Q28 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View

$$( 1 + i ) ^ { 4 } \cdot \left( 2 - \frac { 2 } { i } \right) ^ { 2 }$$
What is the result of this operation?
A) $4 i$
B) 16
C) $- 32 i$
D) - 8
E) 12

Q29 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View

Below, line segments $[ A B ]$ and $[ C D ]$ are given in the complex number plane.
For each complex number z taken on these line segments, the number $\mathrm { w } = \mathrm { z } \cdot \overline { \mathrm { z } }$ is defined.
Accordingly, in which of the following are the minimum and maximum values that w can take given respectively?
A) 5 and 20
B) 5 and 25
C) 5 and 30
D) 10 and 20
E) 10 and 25

Q30 Laws of Logarithms Prove a Logarithmic Identity View

Let t be a real number. The equalities
$$\begin{aligned} & x = e ^ { 2 \cos t } \\ & y = e ^ { 3 \sin t } \end{aligned}$$
are given.
Accordingly, which of the following gives the relationship between $x$ and y that is satisfied for every real number t?
A) $\ln ^ { 2 } x + \ln ^ { 2 } y = 4$
B) $\ln ^ { 2 } x + \ln ^ { 2 } y = 9$
C) $9 \ln ^ { 2 } x + 2 \ln ^ { 2 } y = 27$
D) $\ln ^ { 2 } x + 4 \ln ^ { 2 } y = 28$
E) $9 \ln ^ { 2 } x + 4 \ln ^ { 2 } y = 36$

Q31 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

$$\log _ { 2 } \sqrt { 8 \sqrt { 4 \sqrt { 2 } } }$$
What is the result of this operation?
A) $\frac { 13 } { 8 }$
B) $\frac { 15 } { 8 }$
C) $\frac { 17 } { 8 }$
D) $\frac { 23 } { 16 }$
E) $\frac { 27 } { 16 }$

Q32 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View

$$\left( \sum _ { k = 1 } ^ { 9 } k \right) \cdot \left( \sum _ { n = 1 } ^ { 8 } \frac { 1 } { n ( n + 1 ) } \right)$$
What is the result of this operation?
A) 27
B) 30
C) 32
D) 36
E) 40

Q33 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View

Let $(a_n)$ be a geometric sequence. The equality
$$\frac { a _ { 5 } - a _ { 1 } } { \left( a _ { 3 } \right) ^ { 2 } - \left( a _ { 1 } \right) ^ { 2 } } = \frac { 4 } { 9 }$$
is given. Given that $a _ { 2 } = \frac { 3 } { 2 }$, what is $a _ { 4 }$?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 27 } { 8 }$
E) $\frac { 27 } { 4 }$

Q34 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

In the first quadrant of the rectangular coordinate plane; a square $A _ { 1 }$ is drawn with two sides on the coordinate axes and one vertex on the line $\mathrm { d } : \mathrm { y } = 4 - \mathrm { x }$. Then, a square $A _ { 2 }$ adjacent to the square $A _ { 1 }$ with one side on the x-axis and one vertex on line d is drawn. Continuing in a similar manner, a sequence of squares $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 } , \mathrm {~A} _ { 3 } , \ldots$ is obtained as shown in the figure. Accordingly, what is the sum of the areas of all the squares $\mathbf { A } _ { \mathbf { n } }$ obtained in square units?
A) $\frac { 9 } { 2 }$
B) $\frac { 11 } { 2 }$
C) $\frac { 14 } { 3 }$
D) $\frac { 16 } { 3 }$
E) $\frac { 20 } { 3 }$

Q35 Matrices Linear System and Inverse Existence View

The inverse of matrix A is $A ^ { - 1 } = \left[ \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right]$. Given that
$$A \cdot \left[ \begin{array} { l } 1 \\ a \end{array} \right] = \left[ \begin{array} { l } b \\ 4 \end{array} \right]$$
what is the sum $\mathrm { a } + \mathrm { b }$?
A) 5
B) 7
C) 8
D) 9
E) 11

Q36 Matrices Determinant and Rank Computation View

$$A = \left[ \begin{array} { r r } 1 & 0 \\ - 1 & 3 \end{array} \right], \quad B = \left[ \begin{array} { r r } - 1 & 1 \\ 0 & m \end{array} \right]$$
The matrices satisfy the equality
$$\operatorname { det } ( A + B ) = \operatorname { det } ( A ) + \operatorname { det } ( B )$$
Accordingly, what is m?
A) - 3
B) - 1
C) 0
D) 2
E) 4

Q37 Matrices Linear System and Inverse Existence View

$$3 x - y = 2$$ $$5 x + 2 y = 3$$
The matrix representation of the linear equation system is
$$A \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 2 \\ 3 \end{array} \right]$$
Given that
$$A \cdot \left[ \begin{array} { l } 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } a \\ b \end{array} \right]$$
what is the sum $\mathbf { a + b }$?
A) 4
B) 6
C) 8
D) 10
E) 12

Q38 Small angle approximation View

$$f ( x ) = \left\{ \begin{array} { c c } \frac { a x } { x + 2 b } \cdot \cot x & , x \neq 0 \\ 2 & , x = 0 \end{array} \right.$$
The function is continuous at the point $x = 0$. Accordingly, what is the ratio $\frac { a } { b }$?
A) 1
B) 2
C) 4
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 6 }$

Q39 Modulus function Graph features and asymptotic behaviour of modulus functions View

$$f ( x ) = \left| \frac { 2 x - 1 } { x - 1 } \right|$$
The graph of the function intersects its horizontal asymptote at the point (a, b).
Accordingly, what is the sum $a + b$?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$

Q40 Differentiation from First Principles View

$$\lim _ { x \rightarrow 0 } \frac { \sqrt { x + 5 } - \sqrt { 5 } } { x }$$
What is the value of this limit?
A) $\frac { \sqrt { 5 } } { 5 }$
B) $\frac { 2 \sqrt { 5 } } { 5 }$
C) $\frac { \sqrt { 5 } } { 10 }$
D) 0
E) $2 \sqrt { 5 }$

Q41 Chain Rule Chain Rule with Composition of Explicit Functions View

Let $f ( x ) = e ^ { x }$. The function $g$ is defined as
$$g ( x ) = ( f \circ f ) ( x )$$
Accordingly, what is the value of the derivative of the $\mathbf { g }$ function at the point $\mathbf { x } = \boldsymbol { \ln } \mathbf { 2 }$, that is, $\mathbf { g } ^ { \prime } ( \ln 2 )$?
A) e
B) $\ln 2$
C) $2 \ln 2$
D) $e ^ { 2 }$
E) $2 e ^ { 2 }$

Q42 Tangents, normals and gradients Determine unknown parameters from tangent conditions View

Let a and b be real numbers. In the rectangular coordinate plane, the parabola
$$y = a x ^ { 2 } + b x$$
passes through the point $( 1,2 )$, and the tangent line to the parabola at this point intersects the y-axis at the point $( 0,1 )$.
Accordingly, what is the product $a \cdot b$?
A) - 3
B) - 2
C) - 1
D) 2
E) 4

Q43 Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View

In the rectangular coordinate plane
$$y ^ { 2 } + \sin \left( x ^ { 2 } - 1 \right) = 4$$
What is the slope of the tangent line to the curve given by this equation at the point $\mathbf { P } ( - \mathbf { 1 } , - \mathbf { 2 } )$?
A) - 1
B) $\frac { 1 } { 2 }$
C) 2
D) $\frac { - 1 } { 2 }$

Q44 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

Let $f$ be a function defined on the set of real numbers, and let the derivative of $f$ be denoted by $f ^ { \prime }$. The graph of the function $f ^ { \prime }$ is the parabolic curve shown in the figure.
Accordingly, regarding the function f: I. $f ( 0 ) < 0$ II. It is decreasing on the interval (-a, a). III. $f ( a )$ is a local minimum value.
Which of the following statements are definitely true?
A) Only II
B) Only III
C) I and II
D) II and III
E) I, II and III

Q45 Stationary points and optimisation Geometric or applied optimisation problem View

In the rectangular coordinate plane, rectangles are drawn such that two vertices lie on the x-axis and the other two vertices lie on the parabola $y = 27 - x ^ { 2 }$, and the rectangles lie between this parabola and the x-axis.
Accordingly, what is the perimeter of the rectangle with the largest area?
A) 40
B) 42
C) 44
D) 46
E) 48

Q46 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

$$\int _ { 4 } ^ { 9 } \frac { 3 x - 3 } { \sqrt { x } + 1 } d x$$
What is the value of the integral?
A) 13
B) 18
C) 23
D) 28
E) 33

Q47 Areas by integration View

The graph of a one-to-one and onto function f defined on the interval [2, 6] is given in the figure.
Given that the area of the shaded region is 13 square units,
$$\int _ { 2 } ^ { 6 } f ^ { - 1 } ( x ) d x$$
What is the value of the integral?
A) 18
B) 19
C) 20
D) 21
E) 22

Q48 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

The graph of a function f defined on the interval [-1, 7] is given in the rectangular coordinate plane divided into unit squares as shown in the figure.
Accordingly, what is the value of the integral $\int _ { - 1 } ^ { 7 } f ( x ) d x$?
A) 2
B) 4
C) 6
D) 8
E) 10

Q49 Areas Between Curves Find Parameter Given Area Condition View

Let k be a positive real number. The area of the bounded region between the line $\mathrm { y } = \mathrm { kx }$ and the parabola $y = x ^ { 2 }$ is $\frac { 9 } { 16 }$ square units.
Accordingly, what is the value of $\mathbf { k }$?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 7 } { 4 }$
D) $\frac { 7 } { 6 }$
E) $\frac { 8 } { 5 }$

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

In the rectangular coordinate plane, the region between the lines $y = - x + 5$, $y = x + 3$ and the coordinate axes is shown below.
What is the volume of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the y-axis?
A) $37 \pi$
B) $38 \pi$
C) $40 \pi$
D) $41 \pi$
E) $42 \pi$

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