LFM Stats And Pure

mat None Q2 Polynomial identity or factoring to simplify a given expression View

2. For ALL APPLICANTS.

Suppose that the equation
$$x ^ { 4 } + A x ^ { 2 } + B = \left( x ^ { 2 } + a x + b \right) \left( x ^ { 2 } - a x + b \right)$$
holds for all values of $x$.
(i) Find $A$ and $B$ in terms of $a$ and $b$.
(ii) Use this information to find a factorization of the expression
$$x ^ { 4 } - 20 x ^ { 2 } + 16$$
as a product of two quadratics in $x$.
(iii) Show that the four solutions of the equation
$$x ^ { 4 } - 20 x ^ { 2 } + 16 = 0$$
can be written as $\pm \sqrt { 7 } \pm \sqrt { 3 }$.

mat 1996 Q2 Counting solutions or configurations satisfying a quadratic system View

2. (a) Factorise the expression $x ^ { 2 } + x - 6$.
(b) For which values of the real constant $a$ does the equation
$$x ^ { 2 } + x - a = 0$$
have at least one real solution? Write down these solutions in terms of $a$.
(c) Show that, for any value of the real constant $b$, the equation
$$x ^ { 3 } - ( b + 1 ) x + b = 0$$
has $x = 1$ as a solution. Find all values of $b$ for which this equation has exactly two distinct solutions.

mat 2025 Q26Y(i)(b) 3 marks Simplifying a rational or algebraic expression and solving View

Find a linear polynomial $f ( x )$ such that $f ( x ) \cdot ( 1 + 2 x ) = ( 3 + 4 x )$.

taiwan-gsat 2022 Q12 5 marks Finding roots or coefficients of a quadratic using Vieta's relations View

Let $a, b, c$ be nonzero real numbers, and the two roots of the quadratic equation $ax^2 + bx + c = 0$ both lie between 1 and 3. Select the equation whose two roots must lie between 4 and 5.
(1) $a(x-2)^2 + b(x-2) + c = 0$
(2) $a(x+2)^2 + b(x+2) + c = 0$
(3) $a(2x-7)^2 + b(2x-7) + c = 0$
(4) $a\left(\frac{x+7}{2}\right)^2 + b\left(\frac{x+7}{2}\right) + c = 0$
(5) $a(3x-11)^2 + b(3x-11) + c = 0$

taiwan-gsat 2024 Q13 5 marks Finding a ratio or relationship between variables from an equation View

A sales station sells three types of mobile phones: A, B, and C. The profit per unit is 100 yuan for type A, 400 yuan for type B, and 240 yuan for type C. Last year, $A, B, C$ units of types A, B, C were sold respectively, with an average profit of 260 yuan per unit. It is also known that the average profit for selling types A and B together ($A + B$ units) is 280 yuan per unit. The ratio of the quantities of the three types of mobile phones sold last year is $A : B : C =$ (13－1)：(13－2)：(13－3) (expressed as a ratio of integers in lowest terms)

taiwan-gsat 2025 Q15 4 marks Vertex and parameter conditions for a quadratic graph View

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Prove that when $-1 \leq x \leq 1$, $f(x) \geq 0$ always holds. (Non-multiple choice question, 4 points)

tmua None Q3 View

3. Consider the following attempt to solve an equation. The steps have been numbered for reference. [Figure]
Which one of the following statements is true?
A Both - 4 and - 1 are solutions of the equation.
B Neither - 4 nor - 1 are solutions of the equation.
C One solution is correct and the incorrect solution arises as a result of step (1).
D One solution is correct and the incorrect solution arises as a result of step (2).
E One solution is correct and the incorrect solution arises as a result of step (3).

tmua 2020 Q9 1 marks Finding roots or coefficients of a quadratic using Vieta's relations View

The quadratic expression $x^2 - 14x + 9$ factorises as $(x - \alpha)(x - \beta)$, where $\alpha$ and $\beta$ are positive real numbers.
Which quadratic expression can be factorised as $(x - \sqrt{\alpha})(x - \sqrt{\beta})$?
A $x^2 - \sqrt{10}x + 3$
B $x^2 - \sqrt{14}x + 3$
C $x^2 - \sqrt{20}x + 3$
D $x^2 - 178x + 81$
E $x^2 - 176x + 81$
F $x^2 + 196x + 81$

tmua 2020 Q19 1 marks Optimization or extremal value of an expression via completing the square View

Find the lowest positive integer for which $x^2 - 52x - 52$ is positive.
A $26$
B $27$
C $51$
D $52$
E $53$
F $54$

turkey-yks 2010 Q1 Finding roots or coefficients of a quadratic using Vieta's relations View

$$(3x-1)(x+1)+(3x-1)(x-2)=0$$
What is the sum of the real numbers $x$ that satisfy the equation?
A) $\frac{2}{3}$
B) $\frac{3}{4}$
C) $\frac{3}{5}$
D) $\frac{5}{6}$
E) $\frac{7}{6}$

turkey-yks 2011 Q3 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) $t ^ { 2 } + 3$

turkey-yks 2011 Q5 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

turkey-yks 2011 Q10 Qualitative Analysis of DE Solutions View

$\frac{1}{x + 1} + x - 1 = \frac{1}{x^{2}}$
Given that, which of the following is the expression $x^{3} - 1$ equal to?
A) $\frac{2}{x - 1}$ B) $\frac{1}{x}$ C) $\frac{x - 1}{x}$ D) $-x$ E) $\frac{1}{x + 1}$

turkey-yks 2012 Q9 Solving an equation via substitution to reduce to quadratic form View

$$x \cdot \left( \sqrt { \frac { 1 } { x } - \frac { 1 } { x ^ { 2 } } } \right) = \frac { 1 } { 2 }$$
Given that, what is x?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 4 }$
C) $\frac { 9 } { 4 }$
D) $\frac { 6 } { 5 }$
E) $\frac { 7 } { 5 }$

turkey-yks 2012 Q13 Exponential Equation Solving View

The operation $\Delta$ is defined on the set of real numbers for all real numbers a and b as
$$a \Delta b = a ^ { 2 } + 2 ^ { b }$$
Given that $2 \Delta ( 1 \Delta x ) = 12$, what is x?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$
D) 1
E) 2

turkey-yks 2012 Q26 Quadratic equation with parametric or self-referential conditions View

$$x ^ { 2 } - ( \sin a ) x - \frac { 1 } { 4 } \left( \cos ^ { 2 } a \right) = 0$$
One root of the equation is $\frac { 2 } { 3 }$. Accordingly, what is $\sin a$?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { \sqrt { 2 } } { 3 }$
C) $\frac { \sqrt { 2 } } { 6 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 3 }$

turkey-yks 2013 Q5 Evaluating an algebraic expression given a constraint View

Let $\mathbf { a }$ and $\mathbf { b }$ be real numbers such that
$$\begin{aligned} & a ^ { 2 } - a = b ^ { 2 } - b \\ & a \cdot b = - 1 \end{aligned}$$
Given this, what is the sum $a ^ { 2 } + b ^ { 2 }$?
A) 6
B) 5
C) 4
D) 3
E) 2

turkey-yks 2014 Q11 Geometric or real-world application leading to a quadratic equation View

$\frac { \mathrm { a } } { 5 } , \frac { \mathrm {~b} } { \mathrm { a } }$ and $\frac { \mathrm { a } } { 3 }$ are three consecutive integers arranged from smallest to largest.
Given this, what is the sum $\mathrm { a } + \mathrm { b }$?
A) 60
B) 70
C) 75
D) 80
E) 90

turkey-yks 2014 Q22 Quadratic equation with parametric or self-referential conditions View

Let k be a positive real number. If one root of the equation
$$3 x ^ { 2 } + k x - 2 = 0$$
is k, what is the other root?
A) $\frac { \sqrt { 2 } } { 3 }$
B) $\frac { 2 \sqrt { 3 } } { 3 }$
C) $\frac { - 2 \sqrt { 2 } } { 3 }$
D) $\frac { - \sqrt { 2 } } { 6 }$
E) $\frac { - \sqrt { 3 } } { 6 }$

turkey-yks 2015 Q7 Linear Diophantine Equations View

$$\frac { x + \frac { 1 } { x + 2 } } { 1 - \frac { 1 } { x + 2 } } = \frac { 1 } { 4 }$$
What is the value of $x$ that satisfies this equality?
A) $\frac { - 3 } { 2 }$
B) $\frac { - 3 } { 4 }$
C) $\frac { - 1 } { 4 }$
D) $\frac { - 5 } { 4 }$
E) $\frac { - 3 } { 8 }$

turkey-yks 2015 Q8 Finite Geometric Sum and Term Relationships View

Let x be a positive integer such that
$$\frac { 10 x } { x + 3 }$$
is equal to the square of an integer. What is the sum of the values that x can take?
A) 26
B) 27
C) 29
D) 31
E) 32

turkey-yks 2015 Q15 Qualitative Analysis of DE Solutions View

A function f defined on the set of natural numbers is defined for every n as
$$f ( n ) = \begin{cases} 5 n + 40 , & 0 \leq n < 10 \\ f ( n - 10 ) , & n \geq 10 \end{cases}$$
Example: $f ( 23 ) = f ( 13 ) = f ( 3 ) = 5 \cdot 3 + 40 = 55$
Accordingly, what is the sum of the two-digit numbers AB that satisfy the equation $f ( A B ) = A B$?
A) 75 B) 80 C) 90 D) 100 E) 105

turkey-yks 2016 Q8 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

turkey-yks 2016 Q23 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

turkey-yks 2017 Q5 Geometric or real-world application leading to a quadratic equation View

The numbers $\frac { x } { y }$, $x - y$, and $x$ are three consecutive even integers arranged from smallest to largest.\ Accordingly, what is the sum $\mathrm{x} + \mathrm{y}$?\ A) 8\ B) 10\ C) 12\ D) 14\ E) 16

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

2. For ALL APPLICANTS.

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12