A-level · Algebra & functions

Simultaneous equations and inequalitiesExam-style questions

Two conditions at once, and conditions that pick out a range. What solving simultaneously means on a graph; a line meeting a curve, with the answers given as pairs; counting the meeting points with the discriminant; linear inequalities, and why dividing by a negative turns the sign round; quadratic inequalities from the critical values and a sketch; and the outside region, set notation and combining two conditions.

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15 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A circular patch of lichen is growing on a rock. A scientist measures the radius of the patch from time to time.

In a model, the radius of the patch, $r$ mm, $t$ years after the first measurement, is given by

$r = \sqrt{at + b} \qquad 0 \le t \le 40$

where $a$ and $b$ are constants.

The scientist's records show that

* when $t = 4$ , the radius of the patch was $9.30$ mm * when $t = 15$ , the radius of the patch was $15.8$ mm
(a) Use these two measurements to find a complete equation for the model, giving the values of $a$ and $b$ to 3 significant figures.

(4 marks)

When $t = 30$ , the scientist measured the radius of the patch as $21.4$ mm.
(b) Use this measurement to evaluate the model, giving a reason for your answer.

(2 marks)
Level 4 · 6 marksTry it
2
In this question you must show detailed reasoning.

A football club is planning a rectangular artificial-grass pitch of width $x$ m. The length of the pitch will exceed its width by $25$ m. League rules require the pitch to be at least $61$ m long, and to keep within budget the area of the pitch must be less than $2600$ m².

Form an inequality in $x$ for each of the two requirements. Solve these inequalities to determine the set of possible values of $x$ .

(6 marks)
Level 4 · 6 marksTry it