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Date	November 2019	Marks available	4	Reference code	19N.2.SL.TZ0.T_3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Express and Find	Question number	T_3	Adapted from	N/A

Question

Maegan designs a decorative glass face for a new Fine Arts Centre. The glass face is made up of small triangular panes. The first three levels of the glass face are illustrated in the following diagram.

The $1 st$ level, at the bottom of the glass face, has $5$ triangular panes. The $2 nd$ level has $7$ triangular panes, and the $3 rd$ level has $9$ triangular panes. Each additional level has $2$ more triangular panes than the level below it.

Maegan has $1000$ triangular panes to build the decorative glass face and does not want it to have any incomplete levels.

Find the number of triangular panes in the $12 th$ level.

[3]

Show that the total number of triangular panes, $S_{n}$ , in the first $n$ levels is given by:

$S_{n} = n^{2} + 4 n$ .

[3]

Hence, find the total number of panes in a glass face with $18$ levels.

[2]

Find the maximum number of complete levels that Maegan can build.

[3]

Each triangular pane has an area of $1.84 m^{2}$ .

Find the total area of the decorative glass face, if the maximum number of complete levels were built. Express your area to the nearest $m^{2}$ .

[4]

Markscheme

$u_{12} = 5 + (12 - 1) \times (2)$ (M1)(A1)

Note: Award (M1) for substituted arithmetic sequence formula, (A1) for correct substitutions.

$27$ (A1)(G3)

[3 marks]

$S_{n} = \frac{n}{2} (2 \times 5 + (n - 1) (2))$ (M1)(A1)

Note: Award (M1) for substituted arithmetic sequence formula, (A1) for correct substitutions.

$S_{n} = \frac{n}{2} (8 + 2 n)$ OR $S_{n} = n (5 + n - 1)$ (M1)

Note: Award (M1) for evidence of expansion and simplification, or division by $2$ leading to the final answer.

$S_{n} = n^{2} + 4 n$ (AG)

Note: The final line must be seen, with no incorrect working, for the final (M1) to be awarded.

[3 marks]

$(S_{18} =) 18^{2} + 4 \times 18$ (M1)

Note: Award (M1) for correctly substituted formula for $S_{n}$ .

$(S_{18} =) 396$ (A1)

Note: The use of “hence” in the question paper means that the $S_{n}$ formula (from part (b)) must be used.

[2 marks]

$1000 = n^{2} + 4 n$ OR $1000 = \frac{n}{2} (10 + (n - 1) 2)$ (or equivalent) (M1)

Note: Award (M1) for equating $S_{n}$ to $1000$ or for equating the correctly substituted sum of arithmetic sequence formula to $1000$ .

a sketch of the graphs $S_{n} = n^{2} + 4 n$ and $S_{n} = 1000$ intersecting (M1)

Note: Award (M1) for a sketch of a quadratic and a horizontal line with at least one point of intersection.

a sketch of $n^{2} + 4 n - 1000$ intersecting the $x$ -axis (M1)

Note: Award (M1) for a sketch of $n^{2} + 4 n - 1000$ with at least one $x$ -intercept.

$(n =) 29.6859 \dots$ OR $- 2 + 2 \sqrt{251}$ (A1)

Note: Award (A1) for $29.6859 \dots$ or $- 2 + 2 \sqrt{251}$ seen. Can be implied by a correct final answer.

$(n =) 29$ (A1)(ft)(G2)

Note: Do not accept $30$ . Award a maximum of (M1)(A1)(A0) if two final answers are given. Follow though from their unrounded answer.

$S_{30} = 1020$ and $S_{29} = 957$ (A2)

Note: Award (A2) for both “crossover” values seen. Do not split this (A2) mark.

$(n =) 29$ (A1)(G2)

[3 marks]

$(A =) (29^{2} + 4 \times 29) \times (1.84)$ (M1)(M1)

Note: Award (M1) for their correct substitution to find the total number of triangular panes. Award (M1) for multiplying their number of panes by $1.84$ .

$(A =) 957 \times 1.84$ (A1)(ft)(M1)

Note: Award (A1)(ft) for their $957$ seen. Award (M1) for multiplying their number of panes by $1.84$ . Follow through from part (d).

$(A =) 1760.88 (m^{2})$ (A1)(ft)(G2)

$(A =) 1761 (m^{2})$ (A1)(ft)(G3)

[4 marks]

Examiners report

[N/A]

Syllabus sections

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

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Find the amount of fuel pumped into the tank in the $13th$ hour.
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Find the number of triangular panes in the $12 th$ level.
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Find the total cost of buying 2 tickets in each of the first 16 rows.
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Find the distance from the base of this ladder to the top rung.
19N.2.SL.TZ0.T_3d:
Find the maximum number of complete levels that Maegan can build.
18M.1.SL.TZ2.S_7b:
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21M.2.SL.TZ2.3a.ii:
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21M.2.SL.TZ2.3a.i:
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Find the value of $n$ such that $u_{n} = 0$ .
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17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
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Find an expression for the width of $U_{n}$ in centimetres.
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Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
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A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
19M.2.SL.TZ2.S_10a:
Find the value of $k$ .
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
Daniella ran on day $20$ of her fitness programme.
EXN.3.AHL.TZ0.2b:
Given the width of a pixel is approximately $0.025 cm$ , find the number of squares in the final image.
21N.2.SL.TZ0.2a:
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21N.2.SL.TZ0.2b.ii:
Find an expression for $u_{n}$ .
21N.2.SL.TZ0.2b.iii:
Find the number of student applications the university expects to receive when $n = 11$ . Express your answer to the nearest integer.
21N.2.SL.TZ0.2c:
Write down an expression for $v_{n}$ .
21N.2.SL.TZ0.2d:
Calculate the total amount of acceptance fees paid to the university in the first $10$ years.
21N.2.SL.TZ0.2e:
Find $k$ .

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