Spesh U3/4 Complex Vectors Trial 5 PDF

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Trial 5 Specialist Mathematics : Analysis Task 1

Name:

You may use the VC VCAA AA For Formula mula Sheet. T Technology echnology Free. Time : 2 periods 1. q is a vector such that |q | = 2 diagr diagram. am. Dr Draw aw q on the diagr diagram. am.

45 marks

| p|

and q . p = 0, where p is the vector shown in the

p

[2] 2. Show that a = 8i + 5j + 2k, b = 4i  3j + k and c = 2i  j + 2k are linearly independent vectors.

[4]

3. A c a O P b d B PA and PB are tangents to a circle, centr centre e O. Complete the proof that the llength ength of the two tangents are equal. [Y You ou will need to use the ffact act that a tangen tangentt makes a 900 angle with the rradius adius ]. To prove Proof :

|c |

=

|d |

.

PO = c + ………….. |PO |2 = PO . PO =( = 2 = |c | + PO = d + ………….. |PO |2 =( = 2 = |b | + 2 |c | + |a |2 ∴ But ∴ ∴

[6]

|a |

=

|b |

).(

|a |2

)

since

).(

)

2

|d | =

|b |2

because

2

+

|d |

3. The diagr diagram am shows a str straight aight shoreline w with ith a town at O. Let i and j be unit vector vectorss in the east and north directio directions ns respectively respectively.. Displacements are measur measured ed in kilometres.

At 12:00 midd midday ay ay,, a cargo ship is at C ( 2,  6 ) and a sailing ship is at S ( 8,  4 ). Find CS in terms of i and j a and nd hence determine the distanc distance e between the cargo ship and the sailing ship at 12:00 mi midday dday dday.. Give your answer to the nea nearest rest tenth of a kilometre.

Let P ( 6m,  2m ) denote a point on the shoreline, where m > 0. Express the scalar product OP . PS in terms of m.

Hence, or otherwise, find the coordin coordinates ates of the closest point on the shoreli shoreline ne to the sailing ship at 12:00 midday midday..

To the nearest tenth of a kil kilometre, ometre, how close is the sailing shi ship p to the shoreline at 12: 12:00 00 midday ?

Let t be the time in hours aft after er 12:00 midday midday.. The position vectors of the cargo sship hip and the sailing ship are re respectively spectively r c = (1 (15t 5t + 2)i  (6+5t)j and r s = (12t +8)i  (3 cos t + 8t +1)j. Show that after two hours, at exactly 2: 2:00 00 pm, the cargo ship is di directly rectly south of the sailing ship. [ 2 + 2 + 3 + 1 + 2 = 10 ]

5. Linh starts at point O and w walks alks on level gr ground ound 200 metres iin n a north westerly direction to P P.. She then w walks alks 50 metres due north to Q Q,, which is at the botto bottom m of a building. Linh climbs to T T,, the top of the building, which is 30 metres vertica vertically lly above Q Q.. Express each of the ffollowing ollowing in terms of i, j and k i) OP

ii) OQ

[3]

6.

Find the exact vvalue alue of 7π 7π cos −sin 12 12

( ) ( )

[4]

iii) O OT T

7. Simpl Simplify ify cos 4 ( 2 x ) −sin 4 ( 2 x )

sin

[4] 8. State the implied domain and rrange ange of these functions 2

−1

g ( x ) =sin (sin (x−π) )

[4]

x sin¿ )) −1 f ( x ) =cos ¿

( x4 ) cos ( x4 )

9. a) Sketch llabelling abelling all intercepts, turnin turning g points and asymptotes

( π4 )

f ( x ) =cosec x− [4]

9. b) Sk Sketch etch labelling all asymptotes and showing all turni turning ng points and intercepts, but not labelling

1 2 g ( x ) = + x −2 x [4]...