Download PDF opens your browser print dialog - choose “Save as PDF” as the destination to save a clean A4 solution sheet. The page below is exactly what prints.
S4 Mathematics Final Examination
Detailed Solutions - Questions 16 & 17
Wednesday, June 24, 2026
Question 16
A circle passes through the origin O(0,0) and meets the y-axis and x-axis at A(0,12) and B(5,0) respectively. A straight line L parallel to AB cuts the circle at C and D in quadrant I.
(a)Equation of AB
Slope mAB=5−00−12=−512; with the intercept form:
5x+12y=1⟹12x+5y−60=0
(b)Centre G and radius r
As ∠AOB=90∘, AB is a diameter (angle in a semicircle), so G is its midpoint:
G=(2.5,6),r=2152+122=6.5
(c)Shortest distance between AB and CD when CD = 5
d2=r2−(2CD)2=6.52−2.52=36⟹d=6
(d)Point M on CD closest to G
Step d=6 from G along the unit normal 131(12,5):
M=G+136(12,5)=(26209,13108)≈(8.04,8.31)
Diameter AB, parallel chord CD, and closest point M at distance d = 6 from G.
Question 17
f(x)=k1[x2+(2k+8)x+(25k−20)]
(a)(i)Show f passes through F(2, 29)
f(2)=k1[4+2(2k+8)+25k−20]=k29k=29✓
(a)(ii)Vertex of f(x)
Vertex=(−k−4,17−k−k36)
(b)(i)Vertex U of g(x) = f(-x) + 8
U=(k+4,25−k−k36)
(b)(ii)Values of k giving minimum value 12
25−k−k36=12⇒k2−13k+36=0:
(k−4)(k−9)=0⟹k=4ork=9
(b)(iii)Is the angle claim true?
For k=4, ∠FOV≈29.7∘. Since 29.7∘<32∘, the claim is FALSE.