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KCSE MATHEMATICS QUESTIONS AND SOLUTIONS


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Form 3
Form 2
Form 4
Form 1

QUESTION AND ANSWER ON VECTORS II MODEL06012023003

5/1/2023

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QUESTION AND ANSWER ON VECTORS II MODEL06012023017

5/1/2023

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Given that, the coordinates of two points P and Q are  (■(2,&3,&5))   and  (■(6,&k-1,&15)) respectively and that their position vectors are parallel, Calculate the value of  |PQ|.  (4marks)

11/12/2022

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​The diagram below shows a triangle OPQ in which M and N are points on OQ and PQ respectively such that OM = ⅔ OQ and PN = 1/4 PQ.  Lines PM and ON meets at X.

30/11/2021

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VECTORS (2) | FORM 3 LEVEL | PAPER 2 QUESTIONS | SECTION B

​The diagram below shows a triangle OPQ in which M and N are points on OQ and PQ respectively such that OM = ⅔ OQ and PN = ¼ PQ.  Lines PM and ON meets at X.

​The diagram below shows a triangle OPQ in which M and N are points on OQ and PQ respectively such that OM = ⅔ OQ and PN = ¼ PQ. Lines PM and ON meets at X.

(a) Given that OP = P and OQ = q express in term of P and q  the vectors.
(i) PQ

PQ=q-p
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​(ii) PM.                    (2mks)

​(iii) ON.                     (2mks)

​(b) You are further given that OX = KON and PX = hPM.
(i) Express OX in terms of P and q in two different ways.      (2mks)

(ii) Find the value of h and K.   (2mks)

(iii) Find the ratio PX : XM.               (1mk)

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The diagram below shows a square based pyramid V vertically above the middle of the base.  PQ = 10cm and VR = 13cm.  M is the midpoint of VR.

29/11/2021

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VECTORS II | FORM 3 LEVEL | PAPER 2 QUESTIONS | SECTION B

The diagram below shows a square based pyramid V vertically above the middle of the base.  PQ = 10cm and VR = 13cm.  M is the midpoint of VR.

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​Find
(a) (i) the length of vector PR.                 (2mks)

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​(ii) The height of the pyramid.                 (2mks)

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(b) (i) the angle between VR and the base PQRS. (2mks)

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(ii) The angle between MR and the base PQRS.   (2mks)

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(iii) The angle between the planes QVR and PQRS. (2mks)

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Given that position vectors of points A and B are

19/11/2021

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K.C.S.E Mathematics Q & A - MODEL 2019PP2QN23

24/9/2020

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Form 3 Mathematics


In the figure below, OA = a, OB = b and BX meets OY at C. OX:OA = 1:2 and BY:YA = 1:3.
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(a) Express in terms of a and b: 
(ii) OY; 
(iii) BX. 
(b) Given that OC = hOY and BC = kBX, determine the values of h and k.
answers
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K.C.S.E Mathematics Q & A - MODEL 2019PP2QN13

24/9/2020

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Form 3 Mathematics


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K.C.S.E Mathematics Q & A - MODEL 2019PP1QN11

22/9/2020

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Form 3 Mathematics


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K.C.S.E Mathematics Q & A - MODEL 2018PP2QN13

21/9/2020

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Form 3 Mathematics


In the figure below OP = p, OR = r, PQ:QR = 1:2 and PS = 3PR.
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Express QS in terms of p and r. 
answer
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K.C.S.E Mathematics Q & A - MODEL 2005.PP1.QN.18

19/9/2020

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Form 3 Mathematics



The points P, Q, R and S have position vectors 2p, 3p, r and 3r respectively, relative to an origin O. A point T divides PS internally in the ratio 1:6

(a) Find, in the simplest form, the vectors OT and QT in terms P and r ( 4 marks)
(b)

(i) Show that the points Q, T, and R lie on a straight line ( 3 marks)
(ii) Determine the ratio in which T divides QR ( 1 mark)

Answer
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K.C.S.E Mathematics Q & A - MODEL 2005.PP1.QN.13

17/9/2020

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Form 3 Mathematics



Point T is the midpoint of a straight line AB. Given the position vectors of A and T are i-j + k and 2i+ 1 ½ k respectively, find the position vector of B in terms of i, j \ and k. ( 3 marks)

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K.C.S.E Mathematics Q & A - MODEL 2017PP2QN16

17/9/2020

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Form 3 Mathematics


Given that OA = 3i+ 4j+ 7k, OB= 4i + 3j + 9k and OC = i + 6j + 3k. show that points A, B and C are collinear.
answer
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K.C.S.E Mathematics Q & A - MODEL 2005.PP1.QN.6

17/9/2020

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Form 3 Mathematics



A point R divides a line PQ internally in the ration 3:4. Another point S, divides the line PR externally in the ration 5:2. Given that PQ = 8cm, calculate the length of RS, correct to 2 decimal places.

Answer
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K.C.S.E Mathematics Q & A - MODEL 2016PP2QN11

15/9/2020

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Form 3 Mathematics


Given that qi + 1/3j + 2/3k is a unit vector, find q.
answer
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K.C.S.E Mathematics Q & A - MODEL 2016PP1QN20

14/9/2020

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Form 3 Mathematics


In the diagram below, the coordinates of points A and B are (1,6) and (15, 6) respectively.
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Point N is on OB and that 30N = 2OB. Line OA is produced to L such that OL = 3OA.
a)Find vector LN 
b) Given that a point M is on LN such that LM: MN = 3: 4 find the coordinate of M 
c) If line OM is produced to T such a that OM: MT= 6:1
i)Find the position vector of T 
ii) Show that points L, T and B are collinear 
ANSWERS
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K.C.S.E Mathematics Q & A - MODEL 2016PP1QN05

12/9/2020

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Form 3 Mathematics


Give that OA =2i a + 3 j and OB = 3i -2j, find the magnitude of AB to one decimal place
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K.C.S.E Mathematics Q & A - MODEL 2004.PP1.QN.21

12/9/2020

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Form 3 Mathematics



If A,B and C are the points P and Q are p and q respectively is another point with position vector r = 3q - ½p. Express in terms of p and q.
i) PR
ii) RQ hence show that P,Q and R are collinear.
iii) Determine the ratio PQ:QR.
Answer
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K.C.S.E Mathematics Q & A - MODEL 2015PP2QN16

10/9/2020

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Form 3 Mathematics


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K.C.S.E Mathematics Q & A - MODEL 2004.PP1.QN.4

9/9/2020

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Form 3 Mathematics



​Given that OA = 3i - 2j + and OB = 4i + j – 3k. Find the distance between points A and B to 2 decimal places.
Answer
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K.C.S.E Mathematics Q & A - MODEL 2014PP2QN22

8/9/2020

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Form 3 Mathematics


In the figure below OA= a and OB= b. M is the mid point of OA and AN: NB= 2.1
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a) Express in terms of a and b:
i) BA: 
ii) BN: 
iii) ON 
b) Given that BX= hBM AND OX = kON determine the values of h and k.
answers
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K.C.S.E Mathematics Q & A - MODEL 2003PP1QN021

4/9/2020

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Form 3 Mathematics



In the figure below, vector OP = P and OR =r. Vector OS = 2r and OQ = 3/2p.
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ANSWER
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K.C.S.E Mathematics Q & A - MODEL 2013PP1QN22

2/9/2020

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Form 3 Mathematics


In the figure below, OABC is a trapezium. AB parallel to OC and OC = 5AB. D is a point on OC such that OD: DC = 3:2
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a) Given that OA = p and AB = q, express in terms of p and q:
i. OB; 
ii. AD; 
iii. CB;
b) Lines OB and AD intersect at point X such that AX = kAD and OX = rOB, where
k and r are scalars. Determine the values k and r.
answers
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K.C.S.E Mathematics Q & A - MODEL 2012PP2QN19

1/9/2020

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Form 3 Mathematics


In triangle OPQ below, OP = p, OQ = q. Point M lies on OP such that OM : MP = 2 : 3 and point N lies on OQ such that ON : NQ = 5:1. Line PN
intersects line MQ at X.
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(a) Express in terms of p and q:
(i) PM
(ii) QM.
(c) Given that PX = kPN and QX = rQM, where k and r are scalars:
(i) write two different expressions for OX in terms of p, q, k and r; 
(ii) find the values of k and r; 
(iii) determine the ratio in which X divides line MQ.
answers
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K.C.S.E Mathematics Q & A - MODEL 2003PP1QN006

1/9/2020

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Form 3 Mathematics



Give that x = 2i +j – 2k,y = - 3i +4j –k and z = - 5i +3j +2k and that p = 3x – y +2z. Find the magnitude of vector p to 3 significant figures.
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