## Questions

1. In the figure alongside OA = a , OB = b. T lies on AN such that AN : TN = 13:6. M lies on AB such that AM:MB=1:3 and N lies on OB such that OB:BN = 7:-5.
1. Express in terms of a and b in the simplest form.
1. AN
2. AT
3. AM
2. Show that O, T and M are collinear and state the ratio of OT: TM
2. A point (-3, 4) divides AB internally in the ratio 3:5. Find the coordinates of point A given that point B is (6, -5)
3. Given that O is the origin, OA = 3i + 2j – 4k and OB = 6i + 11j + 2k. If x divides AB in the ratio 1:2, find the modulus of OX to 2d.p
4. In the figure OABC is a trapezium in which 3 AB = 2OC. S divides OC in the ratio 2:1
and AS produced meets BC produced at T. Given that OC = 3c and OA = a
1. Express AS and BC in terms of a and c
2. Given further that AT = hAS and BT = kBC where h and k are constants
1. Express AT in two ways in terms a, c , h and k
3. The obtuse angle between the lines PQ
4. Hence find the ratio BT: BC

5. In the figure above, OPQ is a triangle in which OS = ¾ OP and PR: RQ = 2 : 1. Lines OR and SQ meet at T.
1. Given that OP = P and OQ = q, express the following vectors in term of p and q
1. PQ
2. OR
3. SQ
2. You area further given that ST = mSQ and OT = nOR. Determine the values of m and n.

1.
1.
1. AN = OA + ON
= -a + 2/7b
= 2/7b - a
2. AT = 7/13AN
7/13 (-a + 2/7b)
2/13b - 7/13a
3. AM = 1/4AB
= 1/4 (AO + OB)
= 1/4 (b - a)
2. OT = OA + AT
= a + (2/13b - 7/13a)
= 2/13(3a + b)
OM = OA + AM
= a + (-1/4a + 1/4b)
=3/4a + 1/4 b
=1/4(3a + b)
OT = 2/13 (3a + b)
OM     1/4 ( 3a + b)
OT = 8/13 OM
Or OM = 13/8OT
Since OT = 8/13OM
Then OT : TM = 8/13 : 5/13
= 8 : 5

2. 3. OX = 2/3(3i + 2j – 4k) + 1/3(6i + 11j + 2k)
= 2i + 4j8/3k + 2i + 11/3j + 2/3
= 4i + 5j -2k
10x1 = √(16 + 25 + 4)
= 6.71units
1. AS = AO + OS
= -a + 2(3c)
= 2 ca…………
BC = BA + AC
= a - b + AC
But AC = AO + OC = -a + 3c
= 3ca……….
AB + 2/3OC = 2/3 3 c = 2 c
BA = 2 c…….
BC = -12c +3ca = c -a.
2.
1. AT = hAS = h(2ca)
= 2hc -ha
AT
= AB + BT = 2c + K ( c -a)
= 2c + Kc – Ka
= ( 2 + k)c – Ka
2. 2 + K = 2h
(i) K = h
(ii) 2 + h = 2h
2 = 2h - h
2 = h, K = 2
3. BT : BC
BT = 2 BC
4.
1.
1. PQ = PO + OQ
= P + q or qp
2. OR = OP + PR
= P + 2/3PQ
= P + 2/3(qp)
= P + 2/3q - 2p
= 1/3p + 2/3q
3. SQ = SO + OQ
= -3/4OP + OQ
= -3/4p + q or q3/4p
2. Express OT in two different ways:
Given OT = nOR
= n (1/3P + 2/3q)
= n/3p + 2n/3q
From ΔOST,
OT = OS + ST
= 3/4OP + MSQ
= 3/4P + M (-3/4P +q)
= (3/4 - 3/4m) p + mq
n/3p + 2nq = (3/4 - 3/4m) p + mq
Compare the coefficients of p and q
n/3 = 3/4 - 3/4m
4n = 9 – 9m
4n + 9m = 9 ………………..eq (1)
2/3n = m
m = 2/3n …………….eq. (2)
Substitutes form in equation (1)
4n + 9 (2/3n) = 9
4n + 6n = 9
10n = 9
n = 9/10
Substitute for n in equation (2)
m = 2/3 x 9/10 = 3/5

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