turkey-yks

Q1 Indices and Surds Numerical Arithmetic with Fractions and Decimals View

$$\frac { 3 } { 0,2 } - ( 0,25 ) ^ { - 2 }$$
What is the result of this operation?
A) $\frac { - 2 } { 5 }$
B) $\frac { 3 } { 10 }$
C) $\frac { 1 } { 15 }$
D) - 1
E) - 3

Q2 Inequalities Ordering and Sign Analysis from Inequality Constraints View

$$\sqrt { 2 } < x < \sqrt { 3 }$$
Given this, which of the following can x be?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 7 } { 4 }$
E) $\frac { 6 } { 5 }$

Q3 Factor & Remainder Theorem Sum of Coefficients and Coefficient Relationships View

Given that $t ^ { 3 } - 2 = 0$, which of the following is the equivalent of $\frac { 1 } { t ^ { 2 } + t + 1 }$ in terms of $t$?
A) $t + 1$
B) $\mathrm { t } - 2$
C) $t - 1$
D) $t ^ { 2 } + 1$
E) $t ^ { 2 } + 3$

Q4 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View

The geometric mean of numbers a and b is 3, and their arithmetic mean is 6.
Accordingly, what is the arithmetic mean of $\mathrm { a } ^ { 2 }$ and $\mathrm { b } ^ { 2 }$?
A) 67
B) 65
C) 63
D) 61
E) 57

Q5 Solving quadratics and applications Evaluating an algebraic expression given a constraint View

Given that $x - 2 y = 3$, what is the value of
$$x ^ { 2 } + 4 y ^ { 2 } - 4 x y - 2 y + x - 3$$
?
A) 4
B) 5
C) 8
D) 9
E) 15

Q6 Solving quadratics and applications Polynomial identity or factoring to simplify a given expression View

Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 3 } - 3 x ^ { 2 } y = 3 \\ & y ^ { 3 } - 3 x y ^ { 2 } = 11 \end{aligned}$$
Accordingly, what is the difference $x - y$?
A) 3
B) 2
C) 1
D) - 2
E) - 3

Q7 Permutations & Arrangements Factorial and Combinatorial Expression Simplification View

For two-digit positive integers a and b
$$\frac { a ! } { b ! } = 132$$
Given this, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 22
B) 23
C) 24
D) 25
E) 26

Q8 Indices and Surds Simplifying Algebraic Expressions with Indices or Factoring View

$$\frac { a ^ { 4 } - a ^ { 3 } } { a ^ { 4 } + a ^ { 2 } } \cdot \frac { a ^ { 2 } + 1 } { a ^ { 2 } - a }$$
Which of the following is the simplified form of this expression?
A) $a - 1$
B) a
C) 1
D) $a + 1$
E) $a ^ { 2 } + 1$

Q9 Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View

$$\frac { 2 ( x - y ) } { x - y - 1 } + \frac { x - y - 1 } { x - y - 2 } = 3$$
Given this, what is the difference $x - y$?
A) $\frac { - 1 } { 2 }$
B) $\frac { - 2 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 5 } { 3 }$
E) $\frac { 5 } { 4 }$

Q10 Number Theory Combinatorial Number Theory and Counting View

$$\begin{aligned} & A = \left\{ n \in Z ^ { + } \mid n \leq 100 ; n \text{ is divisible by } 3 \right\} \\ & B = \left\{ n \in Z ^ { + } \mid n \leq 100 ; n \text{ is divisible by } 5 \right\} \end{aligned}$$
The sets are given. Accordingly, how many elements does the difference set $\mathrm { A } \backslash \mathrm { B }$ have?
A) 33
B) 32
C) 30
D) 28
E) 27

Q11 Number Theory GCD, LCM, and Coprimality View

Let p and q be distinct prime numbers such that
$$\begin{aligned} & a = p ^ { 4 } \cdot q ^ { 2 } \\ & b = p ^ { 2 } \cdot q ^ { 3 } \end{aligned}$$
Which of the following is the greatest common divisor of numbers a and b?
A) $p ^ { 5 } \cdot q ^ { 4 }$
B) $p ^ { 4 } \cdot q ^ { 3 }$
C) $p ^ { 3 } \cdot q ^ { 4 }$
D) $p ^ { 2 } \cdot q ^ { 2 }$
E) $p ^ { 2 } \cdot q ^ { 3 }$

Q12 Number Theory Modular Arithmetic Computation View

$$\begin{aligned} & 2 ^ { x } \equiv 1 ( \bmod 7 ) \\ & 3 ^ { y } \equiv 4 ( \bmod 7 ) \end{aligned}$$
For the smallest positive integers x and y satisfying these congruences, what is the difference $y - x$?
A) 5
B) 4
C) 3
D) 2
E) 1

Q13 Inequalities Simultaneous/Compound Quadratic Inequalities View

$$\left. \begin{array} { l } x ( 3 - x ) > 0 \\ ( 2 x + 1 ) ( x - 2 ) < 0 \end{array} \right\}$$
If the solution set of the inequality system given above is the open interval $(\mathbf { a } , \mathbf { b })$, what is the difference $\mathbf { a - b }$?
A) - 2
B) 0
C) 1
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

Q14 Groups Binary Operation Properties View

The $\Delta$ operation on the set $\mathrm { A } = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } \}$ is defined by the table below. For example, a $\Delta \mathrm { d } = \mathrm { c }$ and $\mathrm { d } \Delta \mathrm { a } = \mathrm { a }$.

$\Delta$	a	b	c	d	e
a	a	b	a	c	d
b	c	b	b	a	e
c	a	b	c	d	e
d	a	a	d	d	b
e	e	e	e	d	a

According to this table, which of the following subsets of set A

$\mathrm { K } = \{ \mathrm { b } , \mathrm { c } , \mathrm { d } \}$
$\mathrm { L } = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } \}$
$\mathrm { M } = \{ \mathrm { c } , \mathrm { d } , \mathrm { e } \}$

are closed under the $\Delta$ operation?
A) Only K
B) Only L
C) K and L
D) K and M
E) L and M

Q15 Modulus function Solving inequalities involving modulus View

Let x be a real number with $| x | \leq 4$, and
$$2 x + 3 y = 1$$
What is the sum of the integer values of y that satisfy this equation?
A) - 1
B) 0
C) 1
D) 2
E) 3

Q16 Factor & Remainder Theorem Sum of Coefficients and Coefficient Relationships View

Real coefficient polynomials $P ( x ) , Q ( x )$ and $R ( x )$ are given. For the polynomial $\mathrm { P } ( \mathrm { x } )$ whose constant term is nonzero,
$$P ( x ) = Q ( x ) \cdot R ( x + 1 )$$
the equality is satisfied. If the constant term of P is twice the constant term of Q, what is the sum of the coefficients of R?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) 1
E) 2

Q17 Roots of polynomials Determine coefficients or parameters from root conditions View

The leading coefficient is 1, and the fourth-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with real coefficients has roots $-i$ and $2i$. What is $\mathbf { P } ( \mathbf { 0 } )$?
A) 2
B) 4
C) 6
D) 7
E) 8

Q18 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

$$P ( x ) = ( x + 2 ) ^ { 4 } + 3 ( x + 1 ) ^ { 3 }$$
In this polynomial, what is the coefficient of the $\mathbf { x }$ term?
A) 41
B) 39
C) 37
D) 35
E) 33

Q19 Combinations & Selection Combinatorial Probability View

From a group of 6 girls and 7 boys, 2 representatives are selected.
What is the probability that one of the two selected representatives is a girl and the other is a boy?
A) $\frac { 3 } { 4 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 2 } { 13 }$
D) $\frac { 7 } { 13 }$
E) $\frac { 9 } { 13 }$

Q20 Complex Numbers Arithmetic True/False or Property Verification Statements View

For complex numbers $z = a + b i ( b \neq 0 )$ and $w = c + d i$, if the sum $\mathbf { Z } + \mathbf { W }$ and the product $\mathbf { Z } \cdot \mathbf { W }$ are both real numbers, then I. $z$ and $w$ are conjugates of each other. II. $\mathrm { z } - \mathrm { w }$ is real. III. $z ^ { 2 } + w ^ { 2 }$ is real. 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

Q21 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View

The function f on the set of complex numbers is
$$f ( z ) = \sum _ { k = 0 } ^ { 101 } z ^ { k }$$
is defined in this form. Accordingly, what is the value of f(i)?
A) $1 + i$
B) $1 - \mathrm { i }$
C) i
D) - i
E) 1

Q22 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View

If $\bar { z }$ denotes the conjugate of $z$, what is the non-zero complex number $z$ that satisfies the equation $z ^ { 2 } = \bar { z }$ and whose argument is between $\frac { \pi } { 2 }$ and $\pi$?
A) $\frac { - 1 } { 2 } + ( \sqrt { 3 } ) \mathrm { i }$
B) $\frac { - 1 } { 2 } + \left( \frac { \sqrt { 3 } } { 2 } \right) \mathrm { i }$
C) $\frac { - \sqrt { 2 } } { 2 } + \left( \frac { 1 } { 2 } \right) \mathrm { i }$
D) $\frac { - \sqrt { 2 } } { 2 } + \left( \frac { \sqrt { 2 } } { 2 } \right) i$
E) $\frac { - \sqrt { 3 } } { 2 } + \left( \frac { 1 } { 2 } \right) \mathrm { i }$

Q23 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View

$$2 ^ { 2 x } - 2 \cdot 2 ^ { x } - 8 = 0$$
Given this equation, which of the following is x?
A) 2
B) 1
C) $\ln 2$
D) $\ln 4$
E) $2 \ln 4$

Q24 Laws of Logarithms Solve a Logarithmic Equation View

$$\log _ { 9 } \left( x ^ { 2 } + 2 x + 1 \right) = t \quad ( x > - 1 )$$
Given this equation, which of the following is the expression for x in terms of t?
A) $3 ^ { t } - 1$
B) $3 ^ { \mathrm { t } - 1 }$
C) $3 - 2 ^ { t }$
D) $2 \cdot 3 ^ { \mathrm { t } - 1 }$
E) $3 ^ { t } - 2$

Q25 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

$$f ( x ) = \arcsin \left( \frac { x } { 3 } + 2 \right)$$
Which of the following is the inverse function $\mathbf { f } ^ { \mathbf { - 1 } } ( \mathbf { x } )$ of this function?
A) $2 \sin ( x ) - 6$
B) $2 \sin ( x ) + 3$
C) $3 \sin ( x ) - 6$
D) $\sin ( 2 x - 6 )$
E) $\sin ( 2 x ) - 3$

Q26 Function Transformations View

The graph of the function $f ( x ) = x ^ { 2 } - 2 x + 3$ is translated $a$ units to the right and $b$ units downward to obtain the graph of the function $g ( x ) = x ^ { 2 } - 8 x + 14$.
Accordingly, what is the value of the expression $| \mathbf { a } | + | \mathbf { b } |$?
A) 4
B) 5
C) 6
D) 7
E) 8

Q27 Reciprocal Trig & Identities View

Given that $0 < x < \frac { \pi } { 2 }$ and $\cot \mathrm { x } - 3 \tan \mathrm { x } = \frac { 1 } { \sin 2 \mathrm { x } }$, what is $\sin ^ { 2 } x$?
A) $\frac { 1 } { 9 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 1 } { 7 }$

Q28 Addition & Double Angle Formulae Direct Double Angle Evaluation View

$\cos \mathrm { x } = \frac { - 4 } { 5 }$ Given this, what is $\cos 2 \mathrm { x }$?
A) $\frac { 3 } { 5 }$
B) $\frac { 5 } { 13 }$
C) $\frac { 12 } { 13 }$
D) $\frac { 24 } { 25 }$
E) $\frac { 7 } { 25 }$

Q29 Sine and Cosine Rules Find an angle using the cosine rule View

The triangle ABC is drawn on unit squares as shown above. What is the tangent of angle $B$?
A) $\frac { 25 } { 4 }$
B) $\frac { 34 } { 5 }$
C) $\frac { 40 } { 9 }$
D) 4
E) 5

Q30 Function Transformations View

The graph of the function $f$ is given below.
Given that $\mathbf { g } ( \mathbf { x } ) = \mathbf { 3 } - \mathbf { f } ( \mathbf { x } - \mathbf { 2 } )$, what is the sum $\mathbf { g } ( - \mathbf { 2 } ) + \mathbf { g } ( \mathbf { 5 } )$?
A) - 3
B) - 1
C) 1
D) 2
E) 3

Q31 Inequalities Optimization Subject to an Algebraic Constraint View

For points $(x, y)$ on the boundary of the bounded region between the parabola $y = x ^ { 2 }$ and the line $y = 2 - x$, what is the maximum value of the expression $\mathbf { x } ^ { \mathbf { 2 } } + \mathbf { y } ^ { \mathbf { 2 } }$?
A) 25
B) 20
C) 17
D) 13
E) 10

Q32 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

The piecewise function $f : R \rightarrow R$ is defined as $f ( x ) = \left\{ \begin{array} { c l } 3 x + 1 , & x \text { is rational } \\ x ^ { 2 } , & x \text { is irrational } \end{array} \right.$
Accordingly, which of the following is $( f \circ f ) \left( \frac { \sqrt { 2 } } { 2 } \right)$?
A) $3 \sqrt { 2 } + 2$
B) $\sqrt { 2 } + 2$
C) $\frac { 1 } { 4 }$
D) $\frac { 5 } { 2 }$
E) $\frac { 7 } { 2 }$

Q33 Sequences and series, recurrence and convergence Direct term computation from recurrence View

The function f satisfies the equation
$$f ( n ) = 2 \cdot f ( n - 1 ) + 1$$
for integers $n \geq 1$. Given that $f ( 0 ) = 1$, what is $f ( 2 )$?
A) 8
B) 7
C) 6
D) 5
E) 4

Q34 Sequences and series, recurrence and convergence Direct term computation from recurrence View

The sequence $(a _ { k })$ is defined as
$$\begin{aligned} & a _ { 1 } = 40 \\ & a _ { k + 1 } = a _ { k } - k \quad ( k = 1,2,3 , \ldots ) \end{aligned}$$
Accordingly, what is the term $\mathrm { a } _ { 8 }$?
A) 4
B) 7
C) 12
D) 15
E) 19

Q35 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

An equilateral triangle ABC with side length 1 unit has points D and E marked on sides AB and AC respectively, where these sides are divided into three equal parts. Let K be the midpoint of the line segment DE. A new equilateral triangle is drawn with one vertex at K and the opposite side on BC, and the same process is applied to the newly drawn equilateral triangles.
What is the sum of the areas of all nested triangular regions drawn in this manner, in square units?
A) $\frac { \sqrt { 3 } } { 3 }$
B) $\frac { 3 \sqrt { 3 } } { 4 }$
C) $\frac { 8 \sqrt { 3 } } { 9 }$
D) $\frac { 5 \sqrt { 3 } } { 16 }$
E) $\frac { 9 \sqrt { 3 } } { 32 }$

Q36 Number Theory Divisibility and Divisor Analysis View

$$\prod _ { n = 1 } ^ { 7 } ( 3 n + 2 )$$
If this number is divisible by $10 ^ { \mathbf { m } }$, what is the maximum integer value that m can take?
A) 2
B) 3
C) 4
D) 5
E) 6

Q37 Small angle approximation View

$$\lim _ { x \rightarrow 0 } \frac { x + \arcsin x } { \sin 2 x }$$
What is the value of this limit?
A) 0
B) 1
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 1 } { 6 }$

Q38 Small angle approximation View

$$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + 2 x + 1 } - \sqrt { x ^ { 2 } + 1 } \right)$$
What is the value of this limit?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 5 } { 2 }$
D) 1
E) 2

Q39 Chain Rule Chain Rule with Composition of Explicit Functions View

$$f ( x ) = \sin ^ { 2 } \left( 3 x ^ { 2 } + 2 x + 1 \right)$$
Given this, what is the value of $f ^ { \prime } ( 0 )$?
A) $2 \cos 2$
B) $2 \cos 3$
C) $6 \sin 1$
D) $4 \sin 2$
E) $2 \sin 2$

Q40 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

$$\begin{aligned} & f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 3 \\ & f ( 0 ) = 2 \end{aligned}$$
Given this, what is the value of $\mathbf { f } ( - \mathbf { 1 } )$?
A) - 2
B) - 1
C) 0
D) 1
E) 2

Q41 Small angle approximation View

$$\begin{aligned} & f ( x ) = 2 x - 1 \\ & g ( x ) = \frac { x } { 2 } - \frac { 1 } { x } \end{aligned}$$
Given this, what is the value of $\lim _ { x \rightarrow 2 } \frac { f ( g ( x ) ) } { x - 2 }$?
A) 0
B) 1
C) 3
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

Q42 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View

At what point does the tangent line to the curve $\mathbf { y } = \sin ( \pi \mathrm { x } ) + \mathrm { e } ^ { \mathrm { x } }$ at $\mathrm { x } = 1$ intersect the y-axis?
A) $- \pi$
B) - 1
C) 0
D) $e - 1$
E) $\pi$

Q43 Applied differentiation MCQ on derivative and graph interpretation View

Below is the graph of the derivative of a function f defined on the interval $[ - 5,5 ]$.
According to this graph, I. The function f is decreasing for $x > 0$. II. $f ( - 2 ) > f ( 0 ) > f ( 2 )$. III. The function f has local extrema at $x = - 2$ and $x = 2$. Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) I and III
E) I, II and III

Q44 Stationary points and optimisation Geometric or applied optimisation problem View

A line $d$ with negative slope passing through the point $(1,2)$ forms a triangular region with the coordinate axes. What is the minimum area of this triangular region in square units?
A) 2
B) 3
C) 4
D) $\frac { 9 } { 2 }$
E) $\frac { 7 } { 2 }$

Q45 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

The slope of the tangent line to the graph of a function f at $\mathrm { x } = \mathrm { a }$ is $1$, and the slope of the tangent line at $x = b$ is $\sqrt { 3 }$. Given that the second derivative function $\mathbf { f } ^ { \prime \prime } ( \mathbf { x } )$ is continuous on the interval $[ \mathbf { a } , \mathbf { b } ]$, what is the value of
$$\int _ { b } ^ { a } f ^ { \prime } ( x ) \cdot f ^ { \prime \prime } ( x ) d x$$
?
A) - 1
B) 1
C) 2
D) $\frac { 1 } { 3 }$
E) $\frac { 2 } { 3 }$

Q46 Areas Between Curves Find Parameter Given Area Condition View

In the graph below, the line $y = k$ is drawn such that the areas of regions A and B are equal.
Accordingly, what is the value of k?
A) 2
B) 3
C) 4
D) $\frac { 9 } { 4 }$
E) $\frac { 11 } { 2 }$

Q47 Integration by Parts Reduction Formula or Recurrence via Integration by Parts View

$$\int _ { 1 } ^ { e } \ln ^ { 3 } x \, d x = 6 - 2 e$$
Given this, what is the value of the integral $\int _ { 1 } ^ { e } \ln ^ { 4 } x \, d x$?
A) $7 e - 16$
B) $8 e - 18$
C) $9 e - 24$
D) $10 e - 26$
E) $11 e - 28$

Q48 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

In the integral $\int \frac { \ln \sqrt { x } } { \sqrt { x } } d x$, if the substitution $u = \sqrt { x }$ is made, which of the following integrals is obtained?
A) $\int \ln u \, d u$
B) $\int 2 \ln u \, d u$
C) $\int \frac { \ln u } { u } d u$
D) $\int \frac { \ln u } { 2 u } d u$
E) $\int u \ln u \, d u$

Q49 Matrices Matrix Algebra and Product Properties View

$$A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right], \quad B = \left[ \begin{array} { l l } 1 & 0 \\ \cdots & \cdots \end{array} \right]$$

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2011 lys1-math