Date | November 2019 | Marks available | 2 | Reference code | 19N.2.SL.TZ0.T_3 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find and Hence | 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 level, at the bottom of the glass face, has triangular panes. The level has triangular panes, and the level has triangular panes. Each additional level has more triangular panes than the level below it.
Maegan has 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 level.
Show that the total number of triangular panes, , in the first levels is given by:
.
Hence, find the total number of panes in a glass face with levels.
Find the maximum number of complete levels that Maegan can build.
Each triangular pane has an area of .
Find the total area of the decorative glass face, if the maximum number of complete levels were built. Express your area to the nearest .
Markscheme
(M1)(A1)
Note: Award (M1) for substituted arithmetic sequence formula, (A1) for correct substitutions.
(A1)(G3)
[3 marks]
(M1)(A1)
Note: Award (M1) for substituted arithmetic sequence formula, (A1) for correct substitutions.
OR (M1)
Note: Award (M1) for evidence of expansion and simplification, or division by leading to the final answer.
(AG)
Note: The final line must be seen, with no incorrect working, for the final (M1) to be awarded.
[3 marks]
(M1)
Note: Award (M1) for correctly substituted formula for .
(A1)
Note: The use of “hence” in the question paper means that the formula (from part (b)) must be used.
[2 marks]
OR (or equivalent) (M1)
Note: Award (M1) for equating to or for equating the correctly substituted sum of arithmetic sequence formula to .
OR
a sketch of the graphs and intersecting (M1)
Note: Award (M1) for a sketch of a quadratic and a horizontal line with at least one point of intersection.
OR
a sketch of intersecting the -axis (M1)
Note: Award (M1) for a sketch of with at least one -intercept.
OR (A1)
Note: Award (A1) for or seen. Can be implied by a correct final answer.
(A1)(ft)(G2)
Note: Do not accept . Award a maximum of (M1)(A1)(A0) if two final answers are given. Follow though from their unrounded answer.
OR
and (A2)
Note: Award (A2) for both “crossover” values seen. Do not split this (A2) mark.
(A1)(G2)
[3 marks]
(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 .
OR
(A1)(ft)(M1)
Note: Award (A1)(ft) for their seen. Award (M1) for multiplying their number of panes by . Follow through from part (d).
(A1)(ft)(G2)
(A1)(ft)(G3)
[4 marks]