Exam Practice Worksheet

Quadratic Functions • SPM Paper 1 & 2

SPM KSSM Diagnostic Drill

Form 4 Chapter 2: Quadratic Functions (Fungsi Kuadratik)

Total Marks: 40 Marks • Time Allowed: 50 Minutes

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A Bahagian A: SPM Paper 1 Format [16 Marks]

Answer all questions
Soalan 1 [4 Marks]

The quadratic equation $(p - 1)x^2 + 4x + 2p = 0$ has two equal real roots. Find the possible values of the constant $p$.

Soalan 2 [4 Marks]

Find the range of values of $x$ for which $3x(x - 2) \ge 2x + 9$.

Soalan 3 (Paper 1 Sec B) [8 Marks]

The curve $y = 2x^2 + px + q$ has a turning point at $(3, -5)$.

(a) Find the values of $p$ and $q$. [3 marks]

(b) State the range of values of $k$ such that the line $y = k$ intersects the curve at two distinct points. [2 marks]

(c) A line $y = 4x + c$ is a tangent to the curve. Find the value of $c$. [3 marks]

B Bahagian B: SPM Paper 2 Format & Real-World KBAT [24 Marks]

Answer all questions
Soalan 4 (Paper 2 Standard) [8 Marks]

The roots of the quadratic equation $x^2 - 4x + 2 = 0$ are $p$ and $q$.

(a) State the values of $p + q$ and $pq$. [2 marks]

(b) Show that $p^3 + q^3 = 40$. [3 marks]

(c) Hence, form a quadratic equation with integer coefficients whose roots are $\frac{p^2}{q}$ and $\frac{q^2}{p}$. [3 marks]

Soalan 5 (KBAT / HOTS Real-World Application) [8 Marks]
An agricultural smart greenhouse uses a parabolic glass roof truss to maximize solar capture. The profile of the parabolic roof truss is given by the function:
$h(x) = -\frac{1}{8}x^2 + x + 6, \quad 0 \le x \le 12$
where $x$ is the horizontal distance (in meters) from the left side wall ($x = 0$) to the right side wall ($x = 12$), and $h(x)$ is the height (in meters) of the truss above the ground.

(a) State the vertical height of the two side support walls above the ground. [1 mark]

(b) By expressing $h(x)$ in vertex form $a(x - h)^2 + k$, determine the maximum height reached by the roof truss and the horizontal position of the peak. [3 marks]

(c) A motorized internal solar thermal heating beam must be installed horizontally across the greenhouse at a height of $7.5\text{ meters}$ above the ground. Calculate the total length of this horizontal heating beam. [4 marks]

Soalan 6 (Non-Routine Synthesis) [8 Marks]

A family of curves has the equation $f(x) = x^2 - 2(k + 1)x + (k^2 + 5)$, where $k$ is a real constant.

(a) Show that the discriminant of $f(x)$ is given by $\Delta = 8k - 16$. [2 marks]

(b) Hence, find the range of values of $k$ such that the curve $y = f(x)$ lies entirely above the $x$-axis. [3 marks]

(c) Determine the coordinates of the vertex of the curve in terms of $k$, and find the Cartesian equation of the locus of this vertex as $k$ varies. [3 marks]