LFM Pure and Mechanics

taiwan-gsat 2023 Q9 5 marks Ordering and Comparing Surd or Numerical Values View

Given $a = 6$, $b = \frac{20}{3}$, $c = 2\sqrt{10}$, and $d$, where $d$ is a rational number. These four numbers are marked on a number line as $A(a)$, $B(b)$, $C(c)$, and $D(d)$. Select the correct options.
(1) $a + b + c + d$ must be a rational number (2) $abcd$ must be an irrational number (3) Point $D$ could possibly be at a distance of $2\sqrt{10} + 6$ from point $C$ (4) The midpoint of points $A$ and $B$ is to the right of point $C$ (5) Among all points on the number line at a distance less than 8 from point $B$, there are 14 positive integers and 1 negative integer

taiwan-gsat 2025 Q4 5 marks Exponential Inequalities and Counting Solutions View

On the coordinate plane, a point whose $x$-coordinate and $y$-coordinate are both integers is called a lattice point. How many lattice points are in the interior (not including the boundary) of the region bounded by the function graph $y = \log _ { 2 } x$, the $x$-axis, and the line $x = \frac { 61 } { 2 }$?
(1) 88
(2) 89
(3) 90
(4) 91
(5) 92

turkey-yks 2010 Q1 Numerical Arithmetic with Fractions and Decimals View

$$\frac { 0,2 - 0,025 } { 0,5 }$$
What is the result of this operation?
A) $\frac { 3 } { 5 }$
B) $\frac { 4 } { 5 }$
C) $\frac { 7 } { 20 }$
D) $\frac { 8 } { 25 }$
E) $\frac { 12 } { 25 }$

turkey-yks 2010 Q2 Numerical Arithmetic with Fractions and Decimals View

$$\frac { 5 \left( 2 - \frac { 3 } { 5 } \right) } { 2 \left( 3 - \frac { 5 } { 2 } \right) }$$
What is the result of this operation?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) 3
D) 5
E) 7

turkey-yks 2010 Q3 Simplifying Surd Expressions View

$$\frac { 6 } { \sqrt { 3 } } - \frac { 2 } { \sqrt { 3 } + 1 }$$
What is the result of this operation?
A) $\sqrt { 3 }$
B) $2 \sqrt { 3 }$
C) $\sqrt { 3 } - 1$
D) $\sqrt { 3 } + 1$
E) $2 \sqrt { 3 } - 1$

turkey-yks 2010 Q4 Algebraic Expansion and Simplification View

$$( a + 1 ) ^ { 2 } - ( a - 1 ) ^ { 2 }$$
Which of the following is this expression equal to?
A) a
B) $2 a$
C) $3 a$
D) 4 a
E) $5 a$

turkey-yks 2010 Q5 Evaluating Expressions Using Index Laws View

$10 ^ { -1 } + 10 ^ { -2 } + 10 ^ { -3 }$
What is the result of this operation?
A) 0,011
B) 0,101
C) 0,111
D) 0,123
E) 0,321

turkey-yks 2010 Q6 Solving Exponential or Index Equations View

$( 16 ) ^ { 3 n } = 8 ^ { 5 }$
Given this, what is n?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 3 } { 5 }$
D) $\frac { 5 } { 4 }$
E) $\frac { 5 } { 6 }$

turkey-yks 2010 Q7 Evaluating Expressions Using Index Laws View

$15 ^ { 13 } + 6 \cdot 15 ^ { 13 } + 8 \cdot 15 ^ { 13 }$
What is the result of this operation?
A) $15 ^ { 15 }$
B) $15 ^ { 14 }$
C) $14 \cdot 15 ^ { 13 }$
D) $10 \cdot 16 ^ { 13 }$
E) $16 ^ { 13 }$

turkey-yks 2010 Q14 Identifying Rational vs Irrational Numbers View

Which of the following is a rational number?
A) $\sqrt { 2 } + 1$
B) $2 \sqrt { 2 } - 1$
C) $\frac { 1 } { \sqrt { 2 } }$
D) $\frac { \sqrt { 2 } } { \sqrt { 2 } + 1 }$
E) $\frac { 2 \sqrt { 2 } - 2 } { 3 \sqrt { 2 } - 3 }$

turkey-yks 2011 Q1 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

turkey-yks 2011 Q3 Evaluating Expressions Using Index Laws View

$\frac { 4 ^ { \frac { 1 } { 2 } } + ( - 8 ) ^ { \frac { 1 } { 3 } } - 1 } { 2 ^ { - 1 } }$
What is the result of this operation?
A) 2 B) 6 C) $-1$ D) 0 E) $-2$

turkey-yks 2011 Q6 Ordering and Comparing Surd or Numerical Values View

$x = \sqrt[3]{4}$
$$y = \sqrt[4]{8}$$ $$z = \sqrt[5]{16}$$
Given that, which of the following orderings is correct?
A) $x < y < z$ B) $x < z < y$ C) $y < x < z$ D) $z < x < y$ E) $z < y < x$

turkey-yks 2011 Q8 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) $a ^ { 2 } + 1$

turkey-yks 2012 Q2 Numerical Arithmetic with Fractions and Decimals View

$$\frac { 16 ^ { 3 } } { 24 ^ { 3 } + 16 ^ { 3 } + 8 ^ { 3 } }$$
What is the result of this operation?
A) $\frac { 1 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 1 } { 5 }$
D) $\frac { 4 } { 7 }$
E) $\frac { 2 } { 9 }$

turkey-yks 2012 Q2 Evaluating Expressions Using Index Laws View

$$\frac { 6 ^ { -2 } - 4 \cdot 6 ^ { -3 } } { 3 ^ { -2 } - 2 \cdot 3 ^ { -3 } }$$
What is the result of this operation?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$

turkey-yks 2012 Q3 Simplifying Surd Expressions View

$$\begin{aligned} & a = \sqrt { 12 } - \sqrt { 8 } \\ & b = \sqrt { 27 } + \sqrt { 18 } \end{aligned}$$
Given that, what is the product a.b?
A) $4 \sqrt { 2 }$
B) $3 \sqrt { 3 }$
C) 4
D) 5
E) 6

turkey-yks 2012 Q4 Evaluating Expressions Using Index Laws View

Given that $x = \sqrt [ 4 ] { 5 }$,
$$\left( x ^ { 2 } - 2 \right) ^ { - 1 }$$
which of the following is this expression equal to?
A) $1 + \sqrt [ 4 ] { 5 }$
B) $2 + \sqrt [ 4 ] { 5 }$
C) $1 + \sqrt { 5 }$
D) $2 + \sqrt { 5 }$
E) $1 + 2 \sqrt { 5 }$

turkey-yks 2012 Q5 Simplifying Algebraic Expressions with Indices or Factoring View

$$\frac { x ( y + z ) + z ( y - x ) } { x ^ { 2 } + x y + x z + y z }$$
Which of the following is the simplified form of this expression?
A) $\frac { x } { x + y }$
B) $\frac { y } { x + y }$
C) $\frac { z } { x + z }$
D) $\frac { y } { x + z }$
E) $\frac { y } { y + z }$

turkey-yks 2012 Q6 Conjugate Surds and Sum Evaluation via Identities View

For positive real numbers x and y,
$$\begin{aligned} x \cdot y & = 5 \\ x ^ { 2 } + y ^ { 2 } & = 15 \end{aligned}$$
Given this, what is the value of the expression $x ^ { 3 } + y ^ { 3 }$?
A) 40
B) 45
C) 50
D) 60
E) 75

turkey-yks 2013 Q1 Solving Exponential or Index Equations View

$$\left( 1 - 3 ^ { - 1 } + a ^ { - 1 } \right) ^ { - 3 } = 8$$
Given this, what is a?
A) $- 6$
B) $- 4$
C) $\frac { - 2 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 6 }$

turkey-yks 2013 Q1 Solving Exponential or Index Equations View

$$\frac { 2 ^ { - 2 } } { 4 ^ { - 1 } + \frac { 1 } { m ^ { - 1 } } } = 13 ^ { - 1 }$$
Given this, what is m?
A) 3
B) 4
C) 5
D) 6
E) 7

turkey-yks 2013 Q2 Solving Equations Involving Surds View

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

turkey-yks 2013 Q2 Numerical Arithmetic with Fractions and Decimals View

$$2 \cdot ( 0,2 ) ^ { 3 } + ( 0,4 ) ^ { 3 }$$
What is the result of this operation?
A) 0,06
B) 0,08
C) 0,1
D) 0,12
E) 0,14

turkey-yks 2013 Q3 Ratio and Proportion Problems View

$x , y$ are positive real numbers and
$$\frac { 2 y } { x + \frac { 1 } { y } } - \frac { 3 x } { y + \frac { 1 } { x } } = \frac { 5 x ^ { 2 } } { x \cdot y + 1 }$$
Given this, what is the ratio $\frac { x } { y }$?
A) $\frac { 2 } { 5 }$
B) $\frac { 1 } { 5 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 2 }$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40