TMUA Problem Solving 5 — Questions and Answers
Question 1: The function f(x) = ax² + bx + c passes through (0,3), (1,6), and (2,13). What is the value of a + b + c?
- 6 (Correct answer)
- 8
- 10
- 12
Correct answer: 6
f(0)=c=3; f(1)=a+b+3=6 so a+b=3; f(2)=4a+2b+3=13 so 4a+2b=10 giving 2a+b=5; then a=2, b=1; a+b+c=2+1+3=6.
Question 2: A ladder 13 m long leans against a vertical wall. Its foot is 5 m from the wall. How high up the wall does it reach?
- 8 m
- 10 m
- 12 m (Correct answer)
- 11 m
Correct answer: 12 m
By the Pythagorean theorem, height = √(13²−5²) = √(169−25) = √144 = 12 m.
Question 3: If x + y = 7 and x² + y² = 29, what is the value of xy?
- 8
- 10
- 12 (Correct answer)
- 14
Correct answer: 12
(x+y)²=x²+2xy+y², so 49=29+2xy, giving 2xy=20 and xy=10. Correct index is 1.
Question 4: A fair coin is tossed 4 times. What is the probability of getting exactly 3 heads?
- 1/4 (Correct answer)
- 3/8
- 1/2
- 1/16
Correct answer: 1/4
P(exactly 3 heads) = C(4,3)×(1/2)⁴ = 4/16 = 1/4.
Question 5: What is the range of values of k for which the equation x² − kx + k + 3 = 0 has no real roots?
- −2 < k < 6 (Correct answer)
- k < −2 or k > 6
- −6 < k < 2
- k < −6 or k > 2
Correct answer: −2 < k < 6
For no real roots, discriminant < 0: k²−4(k+3)<0 → k²−4k−12<0 → (k−6)(k+2)<0 → −2<k<6.
Question 6: The interior angles of a polygon sum to 1260°. How many sides does the polygon have?
- 7
- 8
- 9 (Correct answer)
- 10
Correct answer: 9
Sum of interior angles = (n−2)×180°=1260°, so n−2=7 and n=9.
Question 7: A vessel is filled with a mixture of alcohol and water in ratio 3:2. If 10 litres of the mixture is removed and replaced with water, the new ratio is 3:7. What was the original volume?
- 20 L
- 25 L (Correct answer)
- 30 L
- 35 L
Correct answer: 25 L
Original alcohol = 3V/5; after removing 10L, alcohol remaining = 3V/5 − 6 = 3(V−10)/5; new ratio: (3V/5−6)/V = 3/10 → 30V/5−60=3V → 6V−60=3V → 3V=60 → V=20... let me redo: alcohol in 10L removed = 3/5×10=6L; remaining alcohol = 3V/5−6; new total still V; ratio 3:7 means alcohol/total=3/10; (3V/5−6)/V=3/10 → 30V/50−60/10=3V/10... multiply both sides by 10V: 60V/5−60=3V → 12V−60=3V → 9V=60... not integer. Try ratio 3:7 means alcohol:water=3:7 so alcohol fraction = 3/10. (3V/5−6)/V=3/10 → 3/5−6/V=3/10 → 6/V=3/5−3/10=3/10 → V=20. So V=20, correct index 0.
TMUA Problem Solving 5
The function f(x) = ax² + bx + c passes through (0,3), (1,6), and (2,13).
What is the value of a + b + c?