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Date	November 2019	Marks available	5	Reference code	19N.1.AHL.TZ0.H_3
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	H_3	Adapted from	N/A

Question

Three planes have equations:

$2x - y + z = 5$

$x + 3y - z = 4$ , where $a{\text{, }}b \in \mathbb{R}$ .

$3x - 5y + az = b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.

Markscheme

attempt to eliminate a variable (or attempt to find det $A$ ) M1

$\left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 1&3&{ - 1} \\ 3&{ - 5}&a \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 4 \\ b \end{array}} \right) \to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ - 14}&{a + 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 12} \end{array}} \right)$ (or det $A = 14\left( {a - 3} \right)$ )

(or two correct equations in two variables) A1

$\to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ 0}&{a - 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 6} \end{array}} \right)$ (or solving det $A = 0$ )

(or attempting to reduce to one variable, e.g. $\left( {a - 3} \right)z = b - 6$ ) M1

$a = 3{\text{, }}b \ne 6$ A1A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » AHL 1.16—Solution of systems of linear equations

22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .
18N.1.AHL.TZ0.H_4b:
Find the solution of the system of equations when $a = 2$ .
22M.1.AHL.TZ1.11b.ii:
Find a vector equation of $L$ , the line of intersection of $\prod_{1}$ and $\prod_{2}$ .
22M.1.AHL.TZ1.11c:
Find the distance between $L$ and $\prod_{3}$ .
17N.2.AHL.TZ0.H_1:
Boxes of mixed fruit are on sale at a local supermarket.

Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.

Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.

Box C contains 5 bananas and 4 kiwifruit and costs $3.00.

Find the cost of each type of fruit.
18N.1.AHL.TZ0.H_4a:
Find the value of $a$ for which the system of equations does not have a unique solution.
22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.

Topic 3— Geometry and trigonometry » AHL 3.18—Intersections of lines & planes

Topic 1—Number and algebra

Topic 3— Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

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