TMUA Pure Mathematics 3

Question 1

Given that 4^(x+1) = 8^x, find x.

Accepted Answer

2

Answer

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).

Accepted Answer

−1/4

Answer

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.

Accepted Answer

7 + 4√3

Answer

(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°)?

Accepted Answer

2

Answer

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)).

Accepted Answer

4x − 3

Answer

g(g(x)) = g(2x − 1) = 2(2x − 1) − 1 = 4x − 2 − 1 = 4x − 3.

(TMUA) Test of Mathematics for University Admission Practice Test