Here’s the detailed solution for the questions on the sheet:

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1. Multiple-choice questions:

(1) $\frac{a + b}{5} = \frac{a - b}{3}$, find $\frac{a}{b}$.

$\text{Equate the fractions: } \frac{a+b}{5} = \frac{a-b}{3}$

Multiply through by $15$:

$3(a + b) = 5(a - b)$

Simplify:

$3a + 3b = 5a - 5b \quad \Rightarrow \quad 2a = 8b \quad \Rightarrow \quad \frac{a}{b} = 4$

Answer: (b) $4$.

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(2) Given $(x^3, y + 3) = (1, \sqrt{4})$, find $x - y$.

$x^3 = 1 \quad \Rightarrow \quad x = 1$

$y + 3 = \sqrt{4} \quad \Rightarrow \quad y + 3 = 2 \quad \Rightarrow \quad y = -1$

$x - y = 1 - (-1) = 2$

Answer: (a) $2$.

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(3) If $x \propto y$, then $x = \dots$ (where $m$ is a constant $\neq 0$).

By the definition of proportionality:

$x = m \cdot y$

Answer: (b) $m \cdot y$.

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(4) If $X = [-4, 2]$ and $Y = [-2, 5]$, then $(-3, -3) \in \dots$.

$(-3, -3)$ satisfies $x \cdot y$, where $x \in X$ and $y \in Y$:

$x \cdot y = (-3)(-3) = 9$

Thus, it satisfies $x^2$.

Answer: (c) $x^2$.

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(5) $a : b = a^2 : \dots$.

From the ratio $\frac{a}{b} = \frac{a^2}{k}$:

$k = ab$

Answer: (b) $ab$.

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2. Graphical representation of $f(x) = 2x - x^2$, $x \in [-1, 3]$:

Rewrite $f(x) = -x^2 + 2x$ (a downward parabola).

• Complete the square: $f(x) = -(x^2 - 2x) = -(x^2 - 2x + 1 - 1) = -(x - 1)^2 + 1$

Vertex: $(1, 1)$. Maximum value is $1$.

Axis of symmetry: $x = 1$.

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3. Solve for $f(k) + f(m)$:

The given function is:

$f(x) = (m - 3)x^2 + (2 - k)x + 2k + 2m$

Evaluate $f(k)$ and $f(m)$:

1. Substitute $x = k$ into $f(x)$: $f(k) = (m - 3)k^2 + (2 - k)k + 2k + 2m$

2. Substitute $x = m$ into $f(x)$: $f(m) = (m - 3)m^2 + (2 - k)m + 2k + 2m$

Combine $f(k) + f(m)$: [ f(k) + f(m) = [(m - 3)(k^2 + m^2)] + [(2 - k)k + (2 - k)m] + 2k + 2m + 2k + 2m ]

Simplify for further calculations.

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Do you want a deeper explanation for any specific question? Let me know! Here are some related questions to try:

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Related questions:

1. How do you graph a parabola step-by-step?
2. Can you explain the proportionality rule with examples?
3. What is the significance of axis of symmetry in a quadratic function?
4. How do you determine the vertex of a parabola algebraically?
5. How do you solve quadratic functions using substitution?

Tip: Always recheck equations while solving to avoid small calculation errors!