20N.1.SL.TZ0.S_7

pestleMathematicsAISLPaper 120N· ahl-5-10-second-derivatives-testing-for-max-and-minsource ↗

In this question, all lengths are in metres and time is in seconds.

Consider two particles, $P_{1}$ and $P_{2}$ , which start to move at the same time.

Particle $P_{1}$ moves in a straight line such that its displacement from a fixed-point is given by $s (t) = 10 - \frac{7}{4} t^{2}$ , for $t \geq 0$ .

Find an expression for the velocity of $P_{1}$ at time $t$ .

[2]

Particle $P_{2}$ also moves in a straight line. The position of $P_{2}$ is given by $r = (\begin{matrix} - 1 \\ 6 \end{matrix}) + t (\begin{matrix} 4 \\ - 3 \end{matrix})$ .

The speed of $P_{1}$ is greater than the speed of $P_{2}$ when $t > q$ .

Find the value of $q$ .

[5]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

recognizing velocity is derivative of displacement (M1)

eg $v = \frac{d s}{d t}, \frac{d}{d t} (10 - \frac{7}{4} t^{2})$

velocity $= - \frac{14}{4} t (= - \frac{7}{2} t)$ A1 N2

[2 marks]

valid approach to find speed of $P_{2}$ (M1)

eg $|(\begin{matrix} 4 \\ - 3 \end{matrix})|, \sqrt{4^{2} + {(- 3)}^{2}}$ , velocity $= \sqrt{4^{2} + {(- 3)}^{2}}$

correct speed (A1)

eg $5 {m s}^{- 1}$

recognizing relationship between speed and velocity (may be seen in inequality/equation) R1

eg $|- \frac{7}{2} t|$ , speed = | velocity | , graph of $P_{1}$ speed , $P_{1}$ speed $= \frac{7}{2} t, P_{2}$ velocity $= - 5$

correct inequality or equation that compares speed or velocity (accept any variable for $q$ ) A1

eg $|- \frac{7}{2} t| > 5, - \frac{7}{2} q < - 5, \frac{7}{2} q > 5, \frac{7}{2} q = 5$

$q = \frac{10}{7}$ (seconds) (accept $t > \frac{10}{7}$ , do not accept $t = \frac{10}{7}$ ) A1 N2

Note: Do not award the last two A1 marks without the R1.

[5 marks]

Examiners’ report

[N/A]