TMUA Pure Mathematics 4

Question 1

Find ∫(3x² + 2x − 1) dx.

Accepted Answer

x³ + x² − x + C

Answer

Integrating term-by-term: 3x²→x³, 2x→x², −1→−x, plus constant C.

Question 2

The line y = 2x + c is tangent to the circle x² + y² = 5. Find |c|.

Accepted Answer

5

Answer

Distance from origin to the line 2x − y + c = 0 must equal √5: |c|/√5 = √5, so |c| = 5.

Question 3

If p(x) = x³ − 2x² − 5x + 6 and (x − 1) is a factor, fully factorise p(x).

Accepted Answer

(x−1)(x−3)(x+2)

Answer

Dividing by (x−1) gives x² − x − 6 = (x−3)(x+2), so p(x) = (x−1)(x−3)(x+2).

Question 4

What is the range of f(x) = −x² + 4 for x ∈ ℝ?

Accepted Answer

f(x) ≤ 4

Answer

The parabola opens downward with vertex at (0, 4), so the maximum value is 4 and f(x) ≤ 4.

Question 5

Simplify (√5 + 2)(√5 − 2).

Accepted Answer

1

Answer

Using the difference of two squares: (√5)² − 2² = 5 − 4 = 1.

(TMUA) Test of Mathematics for University Admission Practice Test