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Sample KCSE Preparation Mathematics Paper 2 Questions and Answers
Given that
25
x
2
-
20
x
+
k
is a perfect square, find the value of k.
(2m 7s)
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1.
Given that
2
-
√
3
2
+
√
3
=
a
+
b
√
c
find the values of a, b and c
2.
The position vector of P is
(
4
-
3
2
)
and PQ =
(
5
7
-
4
)
.Determine the coordinates of Q
3.
The acceleration a =m/
s
2
of a projectile in spaces is given by equation
a
=
3
t
2
–
2
t
+
5
.Determine the velocity of the projectile after 2 seconds.
4.
A cold water tap running alone fills a bath in 10 minutes while a hot water tap fills it in 15 minutes. The drainage pipe can empty the full bath in 8 minutes. The cold and hot water taps are left raining for 4 minutes and then the drainage pipe is opened. Determine how many more minutes it will take to fill the bath.
5.
a).Expand and simplify
(
1
-
4
x
)
6
up to the term in
x
3
b).Use the expansion to calculate, to 4 decimal places, the value of
(
0.96
)
6
6.
Solve for x in
5
log
10
x
+
log
10
5
=
1
+
2
log
10
4
7.
The equation of a curve is
y
=
2
x
2
–
3
x
+
5
.Find the equation of the tangent to the curve at the point (2, 7).
8.
A biscuit factory starts producing biscuits at the rate of 50000 per hour. This rate of production decreases by 10% every hour calculate: a).the total number of biscuits produced in the first 3 hours. b).the number of biscuits produced during the fourth hour.
9.
Find the value of y in the equation below
8
y
+
2
3
y
+
3
=
131
10.
Simplify as far as possible
x
-
3
x
+
3
-
4
x
-
12
x
2
-
9
11.
A particle is fired vertically upwards so that its height h (metres) after time t (seconds) is given by
h
=
20
t
–
t
2
.Determine the maximum height to which the particle rises.
12.
At the end of 3 year period a flour mill was valued at sh.548720.The flour mill had been depreciating in value at a constant rate of 5% per annum throughout this period. What was the value of the flour mill at the beginning of this period?
13.
Without using mathematical tables or a calculator, evaluate the expression
3
log
10
5
-
1
2
log
10
2500
+
2
log
10
20
14.
The mass of a cylindrical metal pipe varies jointly as its length and the square of its radius. A pipe whose length is 5m and radius 4 cm has a mass of 6.4kg.Determine the mass of a similar metal pipe of length 8 m and radius 5 cm.
15.
The position vector of R is 2i +3j -4 and vector RS = 3i-2j + k. Express the vector S in terms of i ,j and k and hence state its coordinates.
16.
Find the value of x in the equation.
15
2
x
-
4
=
3
2
x
-
4
17.
Solve the following equation
2
x
+
1
=
x
+
2
3
18.
Evaluate
4
3
2
×
8
2
3
÷
(
16
)
1
2
19.
Solve the following quadratic equation by completing the square. Give your answer to one decimal place
2
x
2
-
7
x
+
2
=
0
20.
The equation of a circle is
x
2
+
6
x
+
y
2
-
8
y
-
24
=
0
.Find the coordinates of the centre of the circle and its radius.
21.
The total cost C shillings of manufacturing tv sets is partly constant and partly varies as the square of the number of the tv sets n manufactured. The total cost of manufacturing 50 tv sets is sh 500000 and the total cost of manufacturing 80 tv sets is sh 968000.Find a).an equation relating C and n. b).the total cost of manufacturing 60 tv sets.
22.
The points A’(-1,2),B’(2,-1) and C’(-3,-2) are the images of A,B and C respectively under a transformation whose matrix is
M
=
(
2
3
1
2
)
Find the inverse of M and hence find the coordinates of A,B and C.
23.
Two contractors A and B applied for a tender to build estate for the Nairobi City Council. Contractor A can complete the job in 120days while B would take 180days.It was decided that both contractors work together. Determine how long it took to complete the job.
24.
Brass is made by mixing zinc,tin and copper in the ratio 1:3:4 by mass. Copper costs sh 61000 per tonne,tin sh 108000 per tonne and zinc sh 80000 per tonne. Calculate of making one kilogram of the brass.
25.
Make M the subject of the formula
Q
=
d
2
π
√
F
-
h
M
M
26.
Given that
8
4
-
2
√
3
=
a
+
b
√
3
.Find the values of a and b.
27.
A solid lead cuboid measuring 8cm long, 5cm wide and 3cm high is melted down and recast into a solid sphere. In the process 5% by volume of the metal is lost. Calculate to the nearest centimeter the radius of the sphere.
28.
Solve for x in the equation
4
x
+
1
×
(
1
32
)
2
-
x
=
(
16
)
x
-
1
2
29.
Solve for y in the logarithmic equation Log (y-2) +log(y-4) = log 24
30.
Find the equation of the circle centre (2,3) and radius 5 units.
31.
The sum of the first five terms of an arithmetic progression (AP) is 160.The sum of the first nine terms of the same AP is 396.Determine the sum of the first seven terms of the AP.
32.
Given that
L
=
m
r
√
p
2
+
s
m
,find the value of L when m =6.25,r =2.5,p= 15 and s =175
33.
Pure Arabica coffee costs sh 400 per kg while pure robusta coffee costs sh 190 per kg.A trader bought both types of coffee and mixed them to make a blend which he sold at sh 375 per kg. In so doing he made a profit of 50%. Find the ratio of Arabica coffee to Robusta coffee in the mixture.
34.
Two matrices P and Q are such that
P
2
=
Q
-
2
P
.Given that
P
=
(
-
2
1
3
-
1
)
, find matrix Q and hence determine its inverse.
35.
The major arc of a circle whose radius is 12cm has a length of 44cm.Taking
π
=
22
7
,calculate the since of the reflex angle the arc subtends at the centre of the circle.
36.
A boat which can travel at a speed of 33km/h in still water takes 8 hours 48minutes to travel from a point A to a point B upstream and back to point A. The river current is flowing at a speed of 3km/h. What is the distance between A and B?
37.
Four years ago Korir bought a tractor at sh1.5million. The value of the tractor depreciates at the rate of 12% of its value at the beginning of each year. Korir now sells his tractor to a friend for sh 1.2 million. Calculate how much more money Korir received than the tractor is worth.
38.
The marked price of a radio cassette player is sh 12000.On cash payment the customer is given a 10% discount on the marked price.The radio cassette player can also be bought on hire purchase where the customer is required to pay 35% of the HP price as a deposit followed by 9 equal monthly instalments. If the HP price is 25% higher than the cash price, calculate the amount of each instalment.
39.
An equilateral triangle PQR has an area of
97.43
c
m
2
.Calculate the length of each side of the triangle giving your answer to the nearest whole number.
40.
Triangle PQR is mapped onto P’Q’R’ by a transformation whose matrix is
P
=
(
3
4
2
1
)
a).Find the determinant of P b).Given that the area of the triangle P’Q’R’ is
52.5
c
m
2
,find the area of PQR.
41.
The ratio of girls to boys in form 4 class is 2:3.Last term one-quarter of the girls and one-third of the boys joined the debating club. This number formed one-fifth of all the members of the club. If there are 60 members in the debating club, find the number of students in the form 4 class.
42.
Given that vector a = (n-3)i+(2n +4)j +(2n- 5)k and that the length /a/ = 7,find the two possible values of n.
43.
Three quantities P,Q and R are such that P varies jointly with the Q and the square of R. Find the percentage decrease in P if Q is increased by 50% and R is decreased by 20%.
44.
A town Q is 1 800 nautical miles east of town
P
(
60
0
N
,
25
0
W
)
.Find the longitude of town Q.
45.
Given that
25
x
2
-
20
x
+
k
is a perfect square, find the value of k.
46.
Solve the quadratic equation
5
x
2
-
2
x
-
3
=
0
47.
Given that matrices
P
=
(
2
-
1
4
3
)
,
Q
=
(
1
-
2
-
2
5
)
and that R = PQ, find the inverse of R.
48.
Solve for x in the equation
9
x
+
1
–
54
=
3
2
x
+
1
×
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