TMUA Pure Mathematics 3 — Questions and Answers
Question 1: Given that 4^(x+1) = 8^x, find x.
- 2 (Correct answer)
- 4
- 1
- 3
Correct answer: 2
Rewrite as 2^(2x+2) = 2^(3x), so 2x + 2 = 3x giving x = 2.
Question 2: Find the gradient of the normal to y = x² at the point (2, 4).
- −1/4 (Correct answer)
- 4
- 1/4
- −4
Correct answer: −1/4
dy/dx = 2x, so the tangent gradient at x = 2 is 4; the normal gradient is −1/4.
Question 3: Expand and simplify (2 + √3)².
- 7 + 4√3 (Correct answer)
- 7 + 2√3
- 4 + 3
- 7
Correct answer: 7 + 4√3
(2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3.
Question 4: How many solutions does the equation sin(x) = 1/2 have in [0°, 360°)?
- 2 (Correct answer)
- 1
- 3
- 4
Correct answer: 2
sin(x) = 1/2 gives x = 30° and x = 150° in one full period.
Question 5: A function is defined by g(x) = 2x − 1. Find g(g(x)).
- 4x − 3 (Correct answer)
- 4x − 1
- 2x − 3
- 4x + 1
Correct answer: 4x − 3
g(g(x)) = g(2x − 1) = 2(2x − 1) − 1 = 4x − 2 − 1 = 4x − 3.
Question 6: What is the coefficient of x³ in the expansion of (1 + 2x)⁶?
- 160 (Correct answer)
- 40
- 120
- 60
Correct answer: 160
C(6,3)·(2x)³ = 20·8x³ = 160x³, so the coefficient is 160.
Question 7: Prove by contradiction: if n² is even, then n is even. Which step is essential?
- Assuming n is odd and showing n² is odd (Correct answer)
- Assuming n is even and showing n² is odd
- Assuming n² is odd and showing n is even
- Assuming n is odd and showing n² is even
Correct answer: Assuming n is odd and showing n² is odd
For contradiction, assume n is odd (n = 2k+1), then n² = 4k²+4k+1 is odd — contradicting n² being even.
TMUA Pure Mathematics 3
Given that 4^(x+1) = 8^x, find x.