Exam Practice Worksheet

Circular Measure • SPM Paper 1 & 2

SPM KSSM Diagnostic Drill

Form 5 Chapter 1: Circular Measure (Sukatan Membulat)

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]

Diagram below shows a sector $POQ$ with center $O$ and radius $8\text{ cm}$. The length of arc $PQ$ is $12\text{ cm}$.

(a) Find the angle $\theta$ subtended at center $O$ in radians. [2 marks]

(b) Calculate the area of sector $POQ$. [2 marks]

Soalan 2 [4 Marks]

A sector of a circle has a fixed perimeter of $36\text{ cm}$. Find the radius $r$, in $\text{cm}$, and central angle $\theta$, in radians, such that the area of the sector is maximum.

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

Diagram below shows two concentric circular sectors $AOB$ and $COD$ with common center $O$. The radius of the larger sector $OB = 15\text{ cm}$ and $OD = 9\text{ cm}$. The subtended angle $\angle AOB$ is $0.8\text{ radians}$.

O D B C A 0.8 rad

(a) Calculate the perimeter, in $\text{cm}$, of the shaded region $ABDC$. [4 marks]

(b) Calculate the area, in $\text{cm}^2$, of the shaded region $ABDC$. [4 marks]

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

Answer all questions
Soalan 4 (Paper 2 Standard) [10 Marks]

Diagram shows a circle with center $O$ and radius $6\text{ cm}$. $PA$ and $PB$ are tangents to the circle at points $A$ and $B$ respectively, and the length of $OP = 10\text{ cm}$.

(a) Show that $\angle AOB = 1.855\text{ radians}$ (to 3 decimal places). [3 marks]

(b) Calculate the length of the tangent $PA$. [2 marks]

(c) Calculate the perimeter, in $\text{cm}$, of the shaded region bounded by tangents $PA, PB$ and minor arc $AB$. [2 marks]

(d) Calculate the area, in $\text{cm}^2$, of the shaded region. [3 marks]

Soalan 5 (KBAT Architectural Gothic Arch Tracery) [8 Marks]
A gothic cathedral window arch is designed using two intersecting circular sectors with centers at the baseline endpoints $A$ and $B$, where $AB = 1.2\text{ meters}$. The arch is bounded by circular arc $BC$ with center $A$ and radius $1.2\text{ m}$, and circular arc $AC$ with center $B$ and radius $1.2\text{ m}$.

(a) State the angle $\angle CAB$ in radians. [1 mark]

(b) Calculate the total perimeter of the gothic window arch $ABCA$. [3 marks]

(c) Calculate the surface area, in $\text{m}^2$, of stained glass required to glaze the entire gothic arch window $ABCA$. [4 marks]

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

A sector of a circle with radius $r\text{ cm}$ has an arc length equal to the perimeter of an equilateral triangle of side length $\frac{1}{2}r$.

(a) Calculate the angle subtended by the arc at the center in radians. [3 marks]

(b) Calculate the ratio of the area of the sector to the area of the equilateral triangle. [3 marks]