18N.1.AHL.TZ0.H_5

pestleMathematicsAIHLPaper 118N· ahl-3-13-scalar-and-vector-productssource ↗

The vectors a and b are defined by a = $\left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ t \end{array}} \right)$ , b = $\left( {\begin{array}{*{20}{c}} 0 \\ { - t} \\ {4t} \end{array}} \right)$ , where $t \in \mathbb{R}$ .

Find and simplify an expression for a • b in terms of $t$ .

[2]

Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .

[4]

Markscheme / solution

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

a • b = $\left( {1 \times 0} \right) + \left( {1 \times - t} \right) + \left( {t \times 4t} \right)$ (M1)

= $- t + 4{t^2}$ A1

[2 marks]

recognition that a • b = |a||b|cos θ (M1)

a • b < 0 or $- t + 4{t^2}$ < 0 or cos θ < 0 R1

Note: Allow ≤ for R1.

attempt to solve using sketch or sign diagram (M1)

$0 < t < \frac{1}{4}$ A1

[4 marks]

Examiners’ report

[N/A]