Pure Mathematics Flashcards

Find the sum of the infinite geometric series 3 + 1 + 1/3 + 1/9 + …

Answer: 9/2

The first term a = 3 and common ratio r = 1/3, so S = a/(1-r) = 3/(2/3) = 9/2.

If f(x) = x³ − 3x² + 4, find the x-coordinates of the stationary points.

Answer: x = 0 and x = 2

f'(x) = 3x² − 6x = 3x(x − 2) = 0 gives x = 0 and x = 2.

What is the value of log₂(8) − log₂(2)?

Answer: 2

log₂(8) = 3 and log₂(2) = 1, so the difference is 2.

Solve |2x − 3| = 7.

Answer: x = 5 or x = −2

Setting 2x − 3 = 7 gives x = 5; setting 2x − 3 = −7 gives x = −2.

The quadratic x² + bx + 16 = 0 has a repeated root. What is |b|?

Answer: 8

A repeated root requires discriminant b² − 4(16) = 0, so b² = 64 and |b| = 8.

What is the exact value of tan(135°)?

Answer: −1

tan(135°) = tan(180° − 45°) = −tan(45°) = −1.

If the binomial expansion of (1 + kx)⁵ has a second-term coefficient of 20, find k.

Answer: 4

The second term is 5k·x, so 5k = 20 gives k = 4.