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Form 4 Mathematics Paper 2 Topical Questions and Answers
Given that
√
6
2
-
√
3
=
h
√
6
+
k
√
2
,Find the values of the rational numbers h and k.
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1.
The probability that three girls Fatuma, Jane and Adede will pass this year’s exam is
2
3
,
2
5
and
3
4
respectively. Using a tree diagram, determine the probability that a).All three will fail b).All three will pass c).Only two will pass
2.
Five bags of coffee and three bags of tea have a total mass of 716 kg. Three bags of coffee and five bags of tea weigh 216 kg less. Determine the total mass of one bag of coffee and seven bags of tea.
3.
The seventh and eighth terms of an AP are 42 and 45 respectively. Find the sum of the third and fifth terms of the AP.
4.
Given that log x = m,log y = n and log z = p,express in terms of m,n and p.
log
(
x
2
y
4
(
√
z
)
3
)
5.
Solve the following quadratic equation using the completing square method.
5
x
2
–
19
x
+
12
6.
The price of two grades of tea T1 and T2 ,is sh 180 and sh 120 per kg respectively. Kiprono mixed T1 and T2 in a ratio such that by selling the mixture at sh 175 per kg, he made a profit of 25%. Determine the ratio T1 :T2 in the mixture.
7.
Solve the following simultaneous equations
x
2
-
x
y
=
2
x+y = 3
8.
The volume V of a cylinder varies jointly with its height h and the square of its radius r. Determine the percentage increase in the volume of the cylinder if its radius increases in the ratio 3:2 and its height decreases by 30%.
9.
Given that vectors a =
(
1
-
2
3
)
,
b
=
(
-
2
3
-
1
)
and
c
=
(
-
3
-
1
-
4
)
and that p = 2a – b + c,find to one decimal place the length of vector p.
10.
Two towns R and S are situated on a river bank 64 km apart. A boat can travel at a speed of 36km/h in still water. The boat travelled from town R downstream to town S and back to town R in 3 hours 36 minutes. Determine the speed of the river current.
11.
A tea broker bought Kenyan tea at sh 150 per kg,Ugandan tea at sh 120 per kg and Zairean tea at sh 82.50 per kg. He mixed them in the ratio 2:3:4 by mass to make a blend which he sold at a profit of 50%.Determine the price at which he sold one kg of the blend.
12.
Given that 2n, 3n and 4n +2 are consecutive of a geometric progression determine a).the common ratio b).the value of n
13.
Given that
3
x
2
-
11
x
+
m
is a perfect square, find the value of m.
14.
The cash price for an article is sh 15000.The article can be bought by hire purchase which requires the customer to pay one-third of the HP value as deposit followed by equal monthly instalments of sh 1625 each.The HP value is 30% more than the cash price how many instalments will be required.
15.
The equation of a circle is
x
2
+
y
2
-
4
x
+
8
y
-
29
=
0
. Find the coordinates of the centre of the circle and its radius.
16.
Without using a calculator or mathematical tables, evaluate
5
log
10
2
-
1
2
log
10
16
+
3
log
10
5
17.
Solve the equation
a
a
+
3
-
2
a
-
3
=
1
18.
The velocity V metres per second of a particle in motion is given by
V
=
4
t
2
-
2
t
+
7
Where t is time in seconds. Determine the total distance moved by the particle between the second and third seconds of its motion.
19.
Three quantities F,L and Q are connected by the equation F=L +Q. The quantity L varies as r and Q varies as the square of r. Given that F =54 when r = 3 and F =140 when r =5,find the value of F when r = 6.
20.
Make D the subject of the formula
T
=
k
2
f
(
s
2
-
D
2
)
21.
Simplify the following surd by writing in the form
a
+
√
b
.
1
4
-
√
15
22.
Solve the following quadratic equation by completing the square
5
x
2
-
11
x
-
12
=
0
23.
Three quantities X, Y and Z are such that X varies jointly with Y and the square of Z. Given that X =160 when Y =10 and Z = 2, Find a).The equation connecting X,Y and Z. b).The value of X when Y=6 and Z= 3
24.
The velocity V metres per second of a moving object is given by
V
=
4
t
3
-
3
t
2
+
10
t
-
3
Where t is time in seconds after the object passes a fixed point O. Determine the total distance moved by the object in the first 2 seconds.
25.
The expression
25
x
2
-
80
x
+
k
is a perfect square. Find the value of k.
26.
Make d the subject of the formula
P
=
√
d
2
+
x
d
2
x
27.
The present value of a matatu is sh 764 500.For the last 4 years the value has been depreciating at a uniform rate of 4% per annum. Calculate to the nearest one thousand shillings, the value of the matatu at the start of the 4 year period.
28.
The equation of a curve is
y
=
2
x
2
+
4
x
+
5
. Determine the coordinates of the turning point of the curve
29.
The matrix
T
=
(
k
-
3
4
k
k
+
2
)
has no inverse. a).Find two possible values of k b).Write down two possible matrices for T.
30.
The sum of the first n terms of the series 3.5 +10.5 +17.5 +24.5+.. is 224.Find the value of n.
31.
Use the matrix method to solve the simultaneous equations 6a -2b =16 4a – 3b = 14
32.
The position vectors of C is
(
-
1
3
-
2
)
and vector CD is
(
4
-
5
-
2
)
.Determine the coordinates of D.
33.
The vectors x =
(
-
2
4
-
8
)
,y =
(
1
-
2
1
)
and z =
(
6
-
3
3
)
are such that
p
=
1
2
x
+
y
-
2
3
z
a).Express p as the column vector. b).Hence calculate /p/ to 2 decimal places.
34.
The equation of a circle is
x
2
-
4
x
+
y
2
+
6
y
=
3
. Determine the centre and the radius of the circle.
35.
A transformation is represented by the matrix T =
(
2
1
-
1
3
)
A triangle ABC whose area is
13.5
c
m
2
is mapped onto triangle A’B’C’ by transformation T. What is the area of triangle A’B’C’?.
36.
Find the value of x which satisfy the equation
2
x
-
1
6
=
x
+
1
3
x
37.
Two bags A and B each contain a mixture of red balls and green balls. Bag A contains red balls and green balls in the ratio 8:7.Bag B contains a total of 18 balls out of which 6 are green. A student picked a bag at random and from it picked a ball at random. What is the probability that the ball picked was red.
38.
Without using a calculator or mathematical tables, evaluate the expression.
4
log
10
5
-
1
2
log
10
16
+
6
log
10
2
39.
Two grades of coffee A and R cost sh 200 per kg and sh 120 per kg respectively. Wangare mixed the two grades and sold the mixture at sh 195 per kg.If in so doing she made a profit of 30%,find the ratio in which she mixed them.
40.
Given the matrices A =
(
2
1
3
-
4
)
and B =
(
p
q
q
-
1
-
q
)
and C =
(
4
5
-
5
13
)
are such that AB = C.Find the values of p and q.
41.
Two quantities E and x are such that E partly varies as x and partly varies as the square root of x. When x=4,E =50 and when x = 9,E =105.Determine a).the equation which relates E and x b).The value of E when x = 16.
42.
The curve which passes through the point (0, 8) has a gradient function
d
y
d
x
=
2
x
-
6
Find the equation of the curve and hence determine the coordinates of its turning point.
43.
Sorghum costs sh 90 per kg and millet sh 40 per kg. A trader mixed sorghum and millet and sold the mixture at sh 84 per kg.If in so doing he made a profit of 40%,find the ratio in which he mixed them.
44.
Given the matrices P =
(
3
x
-
2
x
)
Q =
(
y
2
-
3
1
)
,R =
(
-
3
8
-
8
-
2
)
and that PQ = R, Find the values of x and y.
45.
Given that
√
6
2
-
√
3
=
h
√
6
+
k
√
2
,Find the values of the rational numbers h and k.
46.
The position vector of B is 4i +j -3k and vector AB= 6i -3j +2k.Express the position vector of A in terms of i,j and k and hence state the coordinates of A.
47.
The acceleration a m/
s
2
of a moving particle is given by a = 2t -3 where t is time seconds. Determine a).An expression for the velocity V m/s of the particle given that the initial velocity is 5m/s. b).The total distance moved by the particle in the first 3 seconds of motion.
48.
Given the equation below express in fraction form sin x
6
cos
2
x
–
sin
x
-
4
=
0
×
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