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Date	May 2019	Marks available	4	Reference code	19M.1.SL.TZ1.S_10
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	S_10	Adapted from	N/A

Question

Consider $f(x) = \sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right)$ and $g(x) = \sqrt x$ for $x$ ≥ 0. The first time the graphs of $f$ and $g$ intersect is at $x = 0$ .

The set of all non-zero values that satisfy $f(x) = g(x)$ can be described as an arithmetic sequence, ${u_n} = a + bn$ where $n$ ≥ 1.

Find the two smallest non-zero values of $x$ for which $f(x) = g(x)$ .

[5]

At point P, the graphs of $f$ and $g$ intersect for the 21st time. Find the coordinates of P.

[4]

The following diagram shows part of the graph of $g$ reflected in the $x$ -axis. It also shows part of the graph of $f$ and the point P.

Find an expression for the area of the shaded region. Do not calculate the value of the expression.

[4]

Markscheme

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

correct working (A1)

eg ${\text{sin}}\left( {\frac{\pi }{4}x} \right) = 1$ , $\sqrt x \left( {1 - {\text{sin}}\left( {\frac{\pi }{4}x} \right)} \right) = 0$

${\text{sin}}\left( {\frac{\pi }{2}} \right) = 1$ (seen anywhere) (A1)

correct working (ignore additional values) (A1)

eg $\frac{\pi }{4}x = \frac{\pi }{2}$ , $\frac{\pi }{4}x = \frac{\pi }{2} + 2\pi$

$x$ = 2, 10 A1A1 N1N1

[5 marks]

valid approach (M1)

eg first intersection at $x = 0$ , $n = 20$

correct working A1

eg $- 6 + 8 \times 20$ , $2 + \left( {20 - 1} \right) \times 8$ , ${u_{20}} = 154$

P(154, $\sqrt {154}$ ) (accept $x = 154$ and $y = \sqrt {154}$ ) A1A1 N3

[4 marks]

valid attempt to find upper boundary (M1)

eg half way between ${u_{20}}$ and ${u_{21}}$ , ${u_{20}} + \frac{d}{2}$ , 154 + 4, $- 2 + 8n$ , at least two values of new sequence {6, 14, ...}

upper boundary at $x = 158$ (seen anywhere) (A1)

correct integral expression (accept missing ${{\text{d}}x}$ ) A1A1 N4

eg $\int_0^{158} {\left( {\sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right) + \sqrt x } \right)} {\text{d}}x$ , $\int_0^{158} {\left( {g + f} \right)} \left. {{\text{d}}x} \right)$ , $\int_0^{158} {\sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right)} {\text{d}}x - \int_0^{158} { - \sqrt x \,{\text{d}}x}$

Note: Award A1 for two correct limits and A1 for correct integrand. The A1 for correct integrand may be awarded independently of all the other marks.

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

Show 82 related questions

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22M.1.AHL.TZ1.10b.ii:
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19M.2.SL.TZ1.T_5a:
Calculate, in CAD, the total amount John pays for the bicycle.
18M.2.AHL.TZ1.H_1b:
Calculate the number of positive terms in the sequence.
18N.1.SL.TZ0.S_3a:
Find u₈.
22M.3.AHL.TZ1.1e:
By using a suitable table of values or otherwise, determine the smallest positive integer, greater than $1$ , that is both a triangular number and a pentagonal number.
18M.2.SL.TZ1.S_7b:
Hence find the value of n such that $\sum\limits_{k = 1}^n {{x_k} = 861}$ .
18M.2.SL.TZ1.S_7a:
Given that x_k_{+ 1} = x_k + a, find a.
17M.1.AHL.TZ2.H_3b:
the value of $r$ ;
17N.1.SL.TZ0.S_2a:
Find the common difference.
19N.1.SL.TZ0.S_1a:
Find the common difference.
22M.3.AHL.TZ1.1d:
Hence show that $P_{5} (n) = \frac{n (3 n - 1)}{2}$ for $n \in ℤ^{+}$ .
22M.1.SL.TZ1.8b.ii:
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22M.1.AHL.TZ1.10b.i:
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22M.1.AHL.TZ1.10b.iii:
The sum of the first $n$ terms of the series is $\ln (\frac{1}{x^{3}})$ .

Find the value of $n$ .
22M.1.SL.TZ2.2b:
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22M.1.SL.TZ2.2c:
Find the common difference, $d$ .
18M.2.AHL.TZ1.H_7b:
Find the approximate number of fish in the lake at the start of 2042.
17N.1.SL.TZ0.S_2b:
Find the tenth term.
19N.1.SL.TZ0.S_1c:
Find the sum of the first $20$ terms.
19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
EXN.2.SL.TZ0.7c.i:
Find the value of $N$ .
EXN.2.SL.TZ0.7b:
Given that $J_{n}$ follows a geometric sequence, state the value of the common ratio, $r$ .
EXN.2.SL.TZ0.7d:
Find Jane’s total earnings at the start of her $10$ th year of employment. Give your answer correct to the nearest dollar.
20N.2.SL.TZ0.T_5g:
Show that the tank will never be completely filled using this pump.
17M.1.AHL.TZ1.H_7a:
find the value of $d$ .
21M.1.SL.TZ1.3:
Consider an arithmetic sequence where $u_{8} = S_{8} = 8$ . Find the value of the first term, $u_{1}$ , and the value of the common difference, $d$ .
17M.1.SL.TZ2.S_1a:
Find the common difference.
17M.1.SL.TZ2.S_1b:
Find the tenth term.
20N.2.SL.TZ0.T_5b:
Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.
19M.1.SL.TZ1.S_10d:
The following diagram shows part of the graph of $g$ reflected in the $x$ -axis. It also shows part of the graph of $f$ and the point P.

Find an expression for the area of the shaded region. Do not calculate the value of the expression.
19M.1.SL.TZ1.S_10a:
Find the two smallest non-zero values of $x$ for which $f(x) = g(x)$ .
19M.2.SL.TZ2.S_10b:
Find the exact value of ${S_k}$ .
19M.2.SL.TZ2.S_10c:
Show that $F = 3240$ .
EXN.1.SL.TZ0.4b:
Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number.
EXN.2.SL.TZ0.7a:
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21M.2.SL.TZ2.3a:
Given that the $k$ th term of the sequence is zero, find the value of $k$ .
18M.1.SL.TZ1.T_7b:
On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.
18M.2.AHL.TZ1.H_7a:
Show that there will be approximately 2645 fish in the lake at the start of 2020.
EXN.2.SL.TZ0.7c.ii:
For the value of $N$ found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.
19N.1.SL.TZ0.S_1b:
Find the first term.
21M.2.SL.TZ2.3b:
Let $S_{n}$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_{n}$ .
EXN.1.SL.TZ0.4a:
Show that $u_{1} = 4$ .
18N.2.AHL.TZ0.H_1a:
Find the common ratio of this sequence.
22M.1.SL.TZ2.2a:
State the value of the first term, $u_{1}$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
17N.2.SL.TZ0.T_2b:
Find the value of $k$ .
18M.1.SL.TZ1.T_7a.ii:
For that day find the weight of Sergei’s first lift.
20N.2.SL.TZ0.T_5c:
Write down the common difference, $d$ .
18M.2.SL.TZ2.T_4b:
Calculate the café’s total profit for the first 12 weeks.
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
19M.2.SL.TZ1.S_7a:
Write down the first three non-zero terms of ${w_n}$ .
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
17M.1.AHL.TZ1.H_7b:
determine the value of $\sum\limits_{r = 1}^N {{u_r}}$ .
19M.2.SL.TZ2.S_10d:
An infinite geometric series is given as ${S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots$ , $a \in {\mathbb{Z}^ + }$ .

Find the largest value of $a$ such that ${S_\infty } < F$ .
16N.1.SL.TZ0.T_13a:
Find the year in which the comet was seen from Earth for the fifth time.
17M.1.SL.TZ2.S_1c:
Find the sum of the first ten terms of the sequence.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
18M.1.SL.TZ1.T_7a.i:
For that day find how much weight was added after each lift.
16N.1.SL.TZ0.T_13b:
Determine how many times the comet has been seen from Earth up to the year 2014.
17M.1.SL.TZ2.T_5a:
Diagram $n$ is formed with 52 sticks. Find the value of $n$ .
17N.2.SL.TZ0.T_2e:
Calculate the total distance Carlos runs in the first year.
20N.2.SL.TZ0.T_5e.ii:
Write down the number of hours that the pump was pumping fuel into the tank.
17M.1.SL.TZ1.T_5b:
The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.
20N.2.SL.TZ0.T_5d:
Find the amount of fuel pumped into the tank in the $13th$ hour.
20N.2.SL.TZ0.T_5f:
Find the total amount of fuel pumped into the tank in the first $8$ hours.
19M.2.SL.TZ1.T_5e:
John purchased the bicycle in 2008.

Justify why John should not insure his bicycle in 2019.
17M.1.SL.TZ2.T_5b:
Find the total number of sticks used by Tomás for all 24 diagrams.
20N.2.SL.TZ0.T_5e.i:
Find the value of $n$ such that $u_{n} = 0$ .
20N.2.SL.TZ0.T_5a:
Find $h$ , the height of the tank.
21N.1.SL.TZ0.8a:
Show that $a = 8$ .
21N.1.SL.TZ0.8d.ii:
Find the value of $p$ and the value of $q$ .
21N.1.SL.TZ0.8d.i:
Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.
19M.2.SL.TZ1.T_5b:
Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
19M.2.SL.TZ1.T_5c:
Calculate, in years, when the bicycle value will be less than 50 USD.
19M.2.SL.TZ1.T_5d:
Find the total amount John has paid to insure his bicycle for the first 5 years.
17N.2.SL.TZ0.T_2a.i:
Write down the distance Rosa runs in the third training session;
17N.2.SL.TZ0.T_2a.ii:
Write down the distance Rosa runs in the $n$ th training session.
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
17N.2.SL.TZ0.T_2d:
Find the distance Carlos runs in the fifth month of training.
18M.2.SL.TZ2.T_4d:
Calculate the tea-shop’s total profit for the first 12 weeks.

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