Reproduced in its entirety is a question set I designed in Year 3 as midterm homework for my maths module on polynomial functions.
JOURNAL – QUADRATIC INEQUALITIES:
Omission of essential working will result in loss of marks. Each question is worth 5 marks.
Tai Toh Chian and Suma Tee Chee Poh want to buy a million-dollar bungalow. They don’t have enough money individually, so they decide to share. Tai Toh Chian suggests they split 50:50 – each person pays $500,000. However, Suma Tee comes out with another idea. He wants to pay according to this expression:
-100,000z2 + 200,000z + 50,000 dollars
He lets Toh Chian choose any value for z. Will Tai Toh Chian benefit from accepting this offer? Prove it (i.e. if he gains, prove that -100,000z2 + 200,000z + 50,000 can be greater than 500,000 and state the range of values for z when this is true. If he loses, prove that -100,000z2 + 200,000z + 50,000 is always less than 500,000 for all real z). Finally, find the values of z for which Chee Poh pays over $30,000.
After consulting Mr Carl Khoo Lay Ter, Tai Toh Chian decides to reject the offer. Chee Poh grudgingly pays half and the house is being built. Mr Carl Khoo Lay Ter notices that the doorway from the hall to the kitchen arches from the top of the left doorway pillar to the top of the right pillar according to the expression
-2a2 + 4a + 2 1/10 metres above the ground
The two pillars are of the same height. a is the distance from the left doorway pillar. Calculate the values for a for the doorway, and express this as a solution set.
Tai Toh Chian runs into some financial business problems and is forced to sell his share in the house. Jenner Oh See Ti, who is another quadratic freak, offers to buy up his share in the form 100,000b2 – 700,000b + 600,000 dollars. Toh Chian is allowed to pick any value for b from 0 to 7. Find the values for x where Toh Chian will make a profit. In other words, find x for which 100,000b2 – 700,000b + 600,000 > 500,000.
Tai Toh Chian sells his share of the house, again after consulting Carl Khoo Lay Ter. Carl, as a consultant, is tired of giving free consultations and starts charging Tai Toh Chian. His price is 5c2 + 10c + 20 dollars for all the consultations up to the cth. Toh Chian has $400,000 to spare after his house sale. What is the c that he can afford? Express this as a solution set.
Tai Toh Chian uses the money he gained from the house sale to help his business and consults Carl Khoo Lay Ter for the 32nd time, spending $5,460. The consultant suggests expanding his food business and thus Toh Chian decides to sell burgers. Unfortunately, since he is running the burger kiosk himself, his burgers are only slightly more edible than his plates. One reason is that his cooking skills are terrible, and the second reason is that he does not want to hire a cook and that he uses utterly disgusting combinations of flavours. Jenner Oh See Ti and Suma Tee Chee Poh are his first customers. Jenner Oh bites into the soyabean-strawberry jam-bacon burger and flings it into a rubbish chute exactly 8 metres in front of him according to the parabola
y = -x2 + 8x
where his hand is the origin and the chute is his hand level, y = 0. At exactly the same moment, Suma Tee, who is beside Jenner Oh, chews an asparagus-durian-chilli burger, cracking his front tooth. He hurls the burger in the same direction as the rubbish chute, but according to the parabola
y = -1/50x2 + x
into a forest, hitting a termite nest also at y = 0, 50 metres away. Air resistance is not considered because the burgers are about as solid as artillery shells and thus it is negligible. Find the values for x where the height (y-value) of the soyabean-strawberry jam-bacon burger is less than the height of the asparagus-durian-chilli burger. Note that 0 ≤ x ≤ 8 as we discount the burger when it lands in the chute.
(Hint: Try sketching the two graphs of the burger flight paths. This may help sum up the data above.)
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