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 $\pm \sqrt { 7 } \pm \sqrt { 3 }$.

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.

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

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) $a(3x-11)^2 + b(3x-11) + c = 0$

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)

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)

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).

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 $x^2 + 196x + 81$

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

$$(3x-1)(x+1)+(3x-1)(x-2)=0$$
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) $\frac{7}{6}$

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) $t ^ { 2 } + 3$

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

$\frac{1}{x + 1} + x - 1 = \frac{1}{x^{2}}$
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) $\frac{1}{x + 1}$

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

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

$$x ^ { 2 } - ( \sin a ) x - \frac { 1 } { 4 } \left( \cos ^ { 2 } a \right) = 0$$
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) $\frac { 1 } { 3 }$

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

$\frac { \mathrm { a } } { 5 } , \frac { \mathrm {~b} } { \mathrm { a } }$ 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

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) $\frac { - \sqrt { 3 } } { 6 }$

$$\frac { x + \frac { 1 } { x + 2 } } { 1 - \frac { 1 } { x + 2 } } = \frac { 1 } { 4 }$$
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) $\frac { - 3 } { 8 }$

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

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

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

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

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