LFM Pure and Mechanics

taiwan-gsat 2025 Q8 5 marks Algebraic Simplification and Expression Manipulation View

Consider points $P ( x , y )$ on the coordinate plane satisfying the equation $\frac { 2 ^ { x ^ { 2 } } } { 8 } = \frac { 4 ^ { x } } { 2 ^ { y ^ { 2 } } }$. Select the correct options.
(1) When $x = 3$, there are 2 distinct solutions satisfying this equation
(2) If point $( a , b )$ satisfies this equation, then point $( - a , - b )$ also satisfies this equation
(3) All possible points $P ( x , y )$ form a circle
(4) Point $P ( x , y )$ may lie on the line $x + y = 4$
(5) For all possible points $P ( x , y )$, the maximum value of $x - y$ is $1 + 2 \sqrt { 2 }$

taiwan-gsat 2025 Q18 3 marks Applied/Contextual Exponential Modeling View

It is known that UVI values have an exponential relationship with altitude: for every 300-meter increase in altitude, the UVI value increases by 4\% of the value before the increase. At ground level, the ultraviolet radiation received from the sun is 400 joules per square meter. At a mountain 4500 meters above ground level, the UVI value of the ultraviolet radiation received is which of the following options? (Single choice question, 3 points)
(1) $4 ( 1 + 0.04 \times 15 )$
(2) $4 \left( 1 + 0.04 ^ { 15 } \right)$
(3) $4 ( 1 + 0.04 ) ^ { 15 }$
(4) $4 \times 100 ( 1 + 0.04 ) ^ { 15 }$
(5) $4 \times 100 \left( 1 + 0.04 ^ { 45 } \right)$

tmua None Q11 Exponential Equation Solving View

11. The sum of the roots of the equation $2 ^ { 2 x } - 8 \times 2 ^ { x } + 15 = 0$ is
A 3
B 8
C $\quad 2 \log _ { 10 } 2$
D $\quad \log _ { 10 } \left( \frac { 15 } { 4 } \right)$
E $\quad \frac { \log _ { 10 } 15 } { \log _ { 10 } 2 }$

tmua 2017 Q7 1 marks True/False or Multiple-Statement Verification View

The graphs of two functions are shown here:

$y = a ^ { x }$ is shown with a solid line, where $a$ is a positive real number
$y = f ( x )$ is shown with a dashed line

Which of the following statements $( \mathbf { 1 } , \mathbf { 2 } , \mathbf { 3 } , \mathbf { 4 } )$ could be true?
$1 f ( x ) = b ^ { x }$ for some $b > a$
$2 f ( x ) = b ^ { x }$ for some $b < a$
$3 f ( x ) = a ^ { k x }$ for some $k > 1$
$4 f ( x ) = a ^ { k x }$ for some $k < 1$
A $\mathbf { 1 }$ only
B 2 only
C 3 only
D 4 only
E 1 and 3 only
F 1 and 4 only
G 2 and 3 only
H 2 and 4 only

tmua 2017 Q14 1 marks Exponential Equation Solving View

The solution of the simultaneous equations
$$\begin{array} { r } 2 ^ { x } + 3 \times 2 ^ { y } = 3 \\ 2 ^ { 2 x } - 9 \times 2 ^ { 2 y } = 6 \end{array}$$
is $x = p , y = q$.
Find the value of $p - q$
A $\frac { 5 } { 12 }$
B $\frac { 7 } { 3 }$
C $\log _ { 2 } \frac { 5 } { 12 }$
D $\log _ { 2 } \frac { 7 } { 3 }$
E $\log _ { 2 } 9$
F $\quad \log _ { 2 } 15$

tmua 2018 Q11 1 marks True/False or Multiple-Statement Verification View

Consider the equation $2 ^ { x } = m x + c$, where $m$ and $c$ are real constants.
Which of the following statements is/are true?
I The equation has a negative real solution only if $c > 1$.
II The equation has two distinct real solutions if $c > 1$.
III The equation has two distinct positive real solutions if and only if $c \leq 1$.

tmua 2020 Q5 1 marks MCQ on Function Properties View

Which one of the following shows the graph of
$$y = \frac { 2 ^ { x } } { 1 + 2 ^ { x } }$$
(Dotted lines indicate asymptotes.)

tmua 2020 Q6 1 marks MCQ on Function Properties View

Find the maximum value of the function
$$\mathrm{f}(x) = \frac{1}{5^{2x} - 4(5^x) + 7}$$
A $\frac{1}{7}$
B $\frac{1}{4}$
C $\frac{1}{3}$
D $3$
E $4$
F $7$

tmua 2021 Q4 1 marks MCQ on Function Properties View

Find the minimum value of the function
$$2 ^ { 2 x } - 2 ^ { x + 3 } + 4$$
A - 16 B - 12 C - 8 D 0 E 4 F $\quad 20$

tmua 2021 Q14 1 marks Exponential Equation Solving View

Consider the following simultaneous equations, where $p$ is a real number:
$$\begin{array} { r } p 2 ^ { x } + \log _ { 2 } y = 2 \\ 2 ^ { x } + \log _ { 2 } y = 1 \end{array}$$
What is the complete range of $p$ for which these simultaneous equations have a real solution $( x , y )$ ?
A $p < 1$
B $p \neq 1$
C $p > 1$
D $p < 1$ or $p > 2$
E $\quad p \neq 1$ and $p < 2$ F $p > 1$ and $p < 2$ G $p > 2$ H All real values of $p$

tmua 2022 Q18 1 marks True/False or Multiple-Statement Verification View

P, Q, R and S show the graphs of
$$y = ( \cos x ) ^ { \cos x } , y = ( \sin x ) ^ { \sin x } , y = ( \cos x ) ^ { \sin x } \text { and } y = ( \sin x ) ^ { \cos x }$$
for $0 < x < \frac { \pi } { 2 }$ in some order.
Which row in the following table correctly identifies the graphs?

	$y = ( \cos x ) ^ { \cos x }$	$y = ( \sin x ) ^ { \sin x }$	$y = ( \cos x ) ^ { \sin x }$	$y = ( \sin x ) ^ { \cos x }$
A	P	Q	R	S
B	P	Q	S	R
C	Q	P	R	S
D	Q	P	S	R
E	R	S	P	Q
F	R	S	Q	P
G	S	R	P	Q
H	S	R	Q	P

tmua 2023 Q3 1 marks True/False or Multiple-Statement Verification View

Consider the claim: For all positive real numbers $x$ and $y$,
$$\sqrt { x ^ { y } } = x ^ { \sqrt { y } }$$
Which of the following is/are a counterexample to the claim? I $x = 1 , y = 16$ II $x = 2 , y = 8$ III $x = 3 , y = 4$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

tmua 2023 Q7 1 marks Exponential Equation Solving View

$\mathrm { P } ( x )$ and $Q ( x )$ are defined as follows:
$$\begin{aligned} & \mathrm { P } ( x ) = 2 ^ { x } + 4 \\ & \mathrm { Q } ( x ) = 2 ^ { ( 2 x - 2 ) } - 2 ^ { ( x + 2 ) } + 16 \end{aligned}$$
Find the largest value of $x$ such that $\mathrm { P } ( x )$ and $Q ( x )$ are in the ratio $4 : 1$, respectively.

tmua 2023 Q15 1 marks Parameter Determination from Conditions View

The difference between the maximum and minimum values of the function $f ( x ) = a ^ { \cos x }$, where $a > 0$ and $x$ is real, is 3 .
Find the sum of the possible values of $a$.

turkey-yks 2011 Q5 Simplify or Evaluate a Logarithmic Expression View

$12^{a} = 2$
$$6^{b} = 3$$
Given that, what is the value of the expression $\mathbf{12}^{\boldsymbol{(}\mathbf{1} - \mathbf{a}\mathbf{)2b}}$?
A) 15 B) 16 C) 9 D) 8 E) 4

turkey-yks 2011 Q9 Solve a Logarithmic Equation View

$$\frac{2^{x^{2} - y^{2}}}{4^{x^{2} + xy}} = \frac{1}{2}$$
Given that, what is the value of the expression $(x + y)^{2}$?
A) 2 B) 4 C) 1 D) $\frac{1}{2}$ E) $\frac{1}{4}$

turkey-yks 2011 Q23 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) $2 \ln 4$

turkey-yks 2012 Q8 Exponential Equation Solving View

Let x be a real number such that
$$( \sqrt { 7 } + \sqrt { 3 } ) ^ { x } = 4$$
Given this, which of the following is the expression $( \sqrt { 7 } - \sqrt { 3 } ) ^ { x }$ equal to?
A) $2 ^ { - x }$
B) $2 ^ { - x + 1 }$
C) $4 ^ { x }$
D) $4 ^ { x - 1 }$
E) $4 ^ { x + 1 }$

turkey-yks 2013 Q39 Modular Arithmetic Computation View

$$\lim _ { x \rightarrow \infty } \frac { e ^ { - 3 x } + e ^ { 2 x } } { \ln x + 3 e ^ { 2 x } }$$
What is the value of this limit?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 1 } { 3 }$
D) 0
E) 1

turkey-yks 2014 Q4 Solve a Logarithmic Equation View

$\mathbf { x }$ is a real number and
$$\left( \frac { 1 } { 6 } \right) ^ { x } = \left( \frac { 4 } { 3 } \right) ^ { x + 1 }$$
Given this, what is $8 ^ { \mathbf { X } }$?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 1 } { 8 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 2 } { 9 }$

turkey-yks 2014 Q40 True/False or Multiple-Statement Verification View

A function f is defined on the set of real numbers as
$$f ( x ) = 1 + e ^ { - x }$$
Accordingly, I. The range of function f is $( 1 , \infty )$. II. Function f is decreasing on its domain. III. The line $y = 0$ is a horizontal asymptote of function f. Which of the following statements are true?
A) Only II
B) Only III
C) I and II

turkey-yks 2017 Q8 Exponential Equation Solving View

$$\begin{aligned} & 4 ^ { x } + 4 ^ { y } = 10 \\ & 4 ^ { x } - 4 ^ { y } = 8 \end{aligned}$$
Accordingly, what is the value of the expression $\mathbf { 2 } ^ { \mathbf { x } + \mathbf { y } }$?
A) 2 B) 3 C) 4 D) 5 E) 6

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49