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Date	November 2020	Marks available	5	Reference code	20N.1.SL.TZ0.S_7
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find	Question number	S_7	Adapted from	N/A

Question

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

* 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]

Syllabus sections

Topic 5—Calculus » AHL 5.10—Second derivatives, testing for max and min

22M.1.AHL.TZ1.8a.ii:
Find $\frac{d^{2} y}{d x^{2}}$ .
22M.1.AHL.TZ1.8b:
The curve has a point of inflexion at $(a, b)$ .

Find the value of $a$ .
19M.1.AHL.TZ1.H_5:
A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ , the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .
16N.1.SL.TZ0.S_10b:
(i) Find the first three derivatives of $g(x)$ .

(ii) Given that ${g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})$ , find $p$ .
16N.1.SL.TZ0.S_10c:
(i) Find $h'(x)$ .

(ii) Hence, show that $h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}$ .
20N.2.SL.TZ0.S_10b:
Find the least value of $n$ .
20N.2.SL.TZ0.S_10a:
Show that $f'' (x) = \frac{24 - 6 x^{2}}{{(x^{2} + 4)}^{2}}$ .
20N.2.SL.TZ0.S_10c:
Find $\int \frac{6 x}{x^{2} + 4} d x$ .
20N.1.SL.TZ0.S_7a:
Find an expression for the velocity of $P_{1}$ at time $t$ .
20N.2.SL.TZ0.S_10d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 3$ . The area of $R$ is $19.6$ , correct to three significant figures.

Find $f (x)$ .
16N.1.SL.TZ0.S_10a:
(i) Find the first four derivatives of $f(x)$ .

(ii) Find ${f^{(19)}}(x)$ .

Topic 3—Geometry and trigonometry » AHL 3.12—Vector applications to kinematics

Topic 3—Geometry and trigonometry

Topic 5—Calculus

Question

Markscheme

Examiners report

Syllabus sections

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