IB Questionbank

User interface language: English | Español

Date	May 2022	Marks available	3	Reference code	22M.1.SL.TZ1.7
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Hence or otherwise and Find	Question number	7	Adapted from	N/A

Question

A function, $f$ , has its derivative given by $f' (x) = 3 x^{2} - 12 x + p$ , where $p \in ℝ$ . The following diagram shows part of the graph of $f'$ .

The graph of $f'$ has an axis of symmetry $x = q$ .

The vertex of the graph of $f'$ lies on the $x$ -axis.

The graph of $f$ has a point of inflexion at $x = a$ .

Find the value of $q$ .

[2]

Write down the value of the discriminant of $f'$ .

[1]

b.i.

Hence or otherwise, find the value of $p$ .

[3]

b.ii.

Find the value of the gradient of the graph of $f'$ at $x = 0$ .

[3]

Sketch the graph of $f ″$ , the second derivative of $f$ . Indicate clearly the $x$ -intercept and the $y$ -intercept.

[2]

Write down the value of $a$ .

[1]

e.i.

Find the values of $x$ for which the graph of $f$ is concave-down. Justify your answer.

[2]

e.ii.

Markscheme

EITHER

attempt to use $x = - \frac{b}{2 a}$ (M1)

$q = - \frac{- 12}{2 \times 3}$

attempt to complete the square (M1)

$3 {(x - 2)}^{2} - 12 + p$

attempt to differentiate and equate to $0$ (M1)

$f'' (x) = 6 x - 12 = 0$

THEN

$q = 2$ A1

[2 marks]

discriminant $= 0$ A1

[1 mark]

b.i.

EITHER

attempt to substitute into $b^{2} - 4 a c$ (M1)

${(- 12)}^{2} - 4 \times 3 \times p = 0$ A1

$f' (2) = 0$ (M1)

$- 12 + p = 0$ A1

THEN

$p = 12$ A1

[3 marks]

b.ii.

$f'' (x) = 6 x - 12$ A1

attempt to find $f'' (0)$ (M1)

$= 6 \times 0 - 12$

gradient $= - 12$ A1

[3 marks]

A1A1

Note: Award A1 for line with positive gradient, A1 for correct intercepts.

[2 marks]

$a = 2$ A1

[1 mark]

e.i.

$x < 2$ A1

$f'' (x) < 0$ (for $x < 2$ ) OR the $f''$ is below the $x$ -axis (for $x < 2$ )

OR $f''$ (sign diagram must be labelled $f''$ ) R1

[2 marks]

e.ii.

Examiners report

Candidates did score well on this question. As always, candidates are encouraged to read the questions carefully for key words such as 'value' as opposed to 'expression'. So, if asked for the value of the discriminant, their answer should be a number and not an expression found from $b^{2} - 4 a c$ . As such the value of the discriminant in (b)(i) was often seen in (b)(ii). Please ask students to use a straight edge when sketching a straight line! Overall, the reasoning mark for determining where the graph of f is concave-down, was an improvement on previous years. Sign diagrams were typically well labelled, and the description contained clarity regarding which function was being referred to.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

Topic 2—Functions » SL 2.6—Quadratic function

Show 39 related questions

22M.1.SL.TZ1.7b.i:
Write down the value of the discriminant of $f'$ .
22M.1.SL.TZ2.7b.i:
Write down the values of $h$ and $k$ .
22M.1.SL.TZ2.7a:
Write down the equation of the axis of symmetry.
21N.1.SL.TZ0.7b:
Sketch a graph of $v$ against $t$ , clearly showing any points of intersection with the axes.
18M.2.AHL.TZ2.H_10c:
Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.
18M.2.AHL.TZ2.H_10a.iii:
Explain why $f$ has no inverse on the given domain.
21N.1.SL.TZ0.1a.i:
find the $x$ -coordinates of the $x$ -intercepts.
18M.2.AHL.TZ2.H_10a.ii:
With reference to your graph, explain why $f$ is a function on the given domain.
18M.2.AHL.TZ2.H_10a.iv:
Explain why $f$ is not a function for $- \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}$ .
18M.2.AHL.TZ2.H_10a.i:
Sketch the graph of $y = f\left( x \right)$ for $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .
21M.1.SL.TZ1.7d:
Hence find the values of $x$ where the graph of $f$ is both negative and increasing.
18M.2.AHL.TZ2.H_10b:
Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$ .
22M.1.SL.TZ1.7c:
Find the value of the gradient of the graph of $f'$ at $x = 0$ .
EXN.1.SL.TZ0.3:
The following table shows the probability distribution of a discrete random variable $X$ where $x = 1, 2, 3, 4$ .

Find the value of $k$ , justifying your answer.
22M.1.SL.TZ2.7b.ii:
Point $Q$ has coordinates $(5, 12)$ . Find the value of $a$ .
22M.1.AHL.TZ2.11c:
Given that $(h \circ g) (a) = \frac{π}{4}$ , find the value of $a$ .

Give your answer in the form $p + \frac{q}{2} \sqrt{r}$ , where $p, q, r \in ℤ^{+}$ .
20N.1.SL.TZ0.T_11b:
Complete the table below placing a tick (✔) to show whether the unknown parameters $a$ and $b$ are positive, zero or negative. The row for $c$ has been completed as an example.
21M.1.SL.TZ1.7b:
Find the value of $p$ and the value of $q$ .
16N.1.SL.TZ0.S_1b:
(i) Write down the value of $h$ .

(ii) Find the value of $k$ .
21M.2.SL.TZ2.3b:
Let $S_{n}$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_{n}$ .
21M.2.SL.TZ2.5a:
Find the range of $f$ .
21M.2.SL.TZ2.5b:
Given that $(g \circ f) (x) \leq 0$ for all $x \in ℝ$ , determine the set of possible values for $c$ .
19N.1.SL.TZ0.S_3b:
The graph of $f\left( x \right) = {x^2}$ is transformed to obtain the graph of $g$ .

Describe this transformation.
21M.1.SL.TZ1.7a:
Show that $m = 4$ .
18M.2.AHL.TZ2.H_10d.ii:
Show that $\alpha$ + β < −2.
21N.1.SL.TZ0.1a.ii:
find the coordinates of the vertex.
21N.1.SL.TZ0.1b:
The function $f$ can be written in the form $f (x) = - 2 {(x - h)}^{2} + k$ .

Write down the value of $h$ and the value of $k$ .
21N.1.SL.TZ0.7c:
Find the total distance travelled by $P$ .
21N.1.SL.TZ0.7a.ii:
Show that the distance of $P$ from $O$ at this time is $\frac{88}{27}$ metres.
22M.1.SL.TZ1.7a:
Find the value of $q$ .
17M.1.SL.TZ1.S_9a:
Find the value of $p$ .
17M.1.SL.TZ1.S_9b:
Find the value of $a$ .
19N.1.SL.TZ0.S_3a:
Find the value of $b$ .
17M.1.AHL.TZ1.H_11a.i:
Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$ .
21M.1.SL.TZ1.7c:
Find the value of $h$ and the value of $k$ .
18M.2.AHL.TZ2.H_10d.i:
Find $\alpha$ and β in terms of $k$ .
20N.1.SL.TZ0.T_11a:
Write down the other solution of $f (x) = k$ .
20N.1.SL.TZ0.T_11c:
State the values of $x$ for which $f (x)$ is decreasing.
21N.1.SL.TZ0.7a.i:
Find the value of $t$ when $P$ reaches its maximum velocity.

Hide related questions

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options