TMUA Problem Solving 5

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?

Accepted Answer

6

Answer

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?

Accepted Answer

12 m

Answer

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?

Accepted Answer

12

Answer

(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?

Accepted Answer

1/4

Answer

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?

Accepted Answer

−2 < k < 6

Answer

For no real roots, discriminant < 0: k²−4(k+3)<0 → k²−4k−12<0 → (k−6)(k+2)<0 → −2<k<6.

(TMUA) Test of Mathematics for University Admission Practice Test