Question 2(b) Achieved

An exponential integral, a negative-power differential equation, an upper limit that appears on both sides, a trig quotient whose top is a multiple of the derivative of its bottom, and a proportional-to-the-square model built from scratch.

Solve the differential equation dy/dx = (4x + 1)^−1/2, where x ≥ 0, given that when x = 6, y = 7.5.

The 2023 paper, page 6 ↗ Marking schedule, page 3 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1. 1 Integrate (4x + 1)^−1/2: add one to the power, divide by the new power and by the 4.

   y = (4x + 1)^1/2/(1/2 × 4) + c = (1/2)(4x + 1)^1/2 + c

2. 2 Use y = 7.5 when x = 6.

   4(6) + 1 = 25 and √25 = 5, so 7.5 = 2.5 + c and c = 5.

What the marker wanted

• Achieved The whole solution, with c = 5 and the integration shown.

In our words, from NZQA’s 2023 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.