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Date November 2013 Marks available 3 Reference code 13N.2.hl.TZ0.7
Level HL only Paper 2 Time zone TZ0
Command term Find Question number 7 Adapted from N/A

Question

The vectors a and b are such that  a \( = (3\cos \theta  + 6)\)i \( + 7\) j and b \( = (\cos \theta  - 2)\)i \( + (1 + \sin \theta )\)j.

Given that a and b are perpendicular,

show that \(3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0\);

[3]
a.

find the smallest possible positive value of \(\theta \).

[3]
b.

Markscheme

attempting to form \((3\cos \theta  + 6)(\cos \theta  - 2) + 7(1 + \sin \theta ) = 0\)     M1

\(3{\cos ^2}\theta  - 12 + 7\sin \theta  + 7 = 0\)     A1

\(3\left( {1 - {{\sin }^2}\theta } \right) + 7\sin \theta  - 5 = 0\)     M1

\(3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0\)     AG

[3 marks]

a.

attempting to solve algebraically (including substitution) or graphically for \(\sin \theta \)     (M1)

\(\sin \theta  = \frac{1}{3}\)     (A1)

\(\theta  = 0.340{\text{ }}( = 19.5^\circ )\)     A1

[3 marks]

b.

Examiners report

Part (a) was very well done. Most candidates were able to use the scalar product and \({\cos ^2}\theta  = 1 - {\sin ^2}\theta \) to show the required result.

a.

Part (b) was reasonably well done. A few candidates confused ‘smallest possible positive value’ with a minimum function value. Some candidates gave \(\theta  = 0.34\) as their final answer.

b.

Syllabus sections

Topic 2 - Core: Functions and equations » 2.6 » Solving quadratic equations using the quadratic formula.

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