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Form 2 Mathematics Mixed Revision Questions With Answers Set 1
Find non-zero scalars k and h such that ka + hb = c, given that:
a=
(
3
2
)
, b=
(
4
6
)
and c=
(
2
2
)
.
(2m 17s)
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1.
Evaluate:
√
(
0.64
×
(
1.69
)
1
2
(
0.04
)
1
2
×
38.44
)
2.
Given that the triangles ABC and PQR in the figure below are similar, find: (a) the size of ?PRQ. (b) the lengths of RQ and PR.
3.
Evaluate
a
b
-
b
a
b
b
×
a
a
if a = 2 and b = -2.
4.
Given that y =
a
x
n
, find the value of y when
a
=
5
3
, x = 2 and n = -4 .
5.
A straight line
l
1
has a gradient
-
1
2
and passes through the point P (-1, -3). Another straight line
l
2
passes through the points Q(1, -3) and R(4, 5). Find: (a) the equation of
l
1
. (b) the gradient of
l
2
. (c) the equation of
l
2
. (d) the co-ordinates of the point of intersection of
l
1
and
l
2
. (e) the equation of a line through R parallel to
l
1
.
6.
In the figure below, rectangles ABCD and PQRS are similar. Find the area of PQRS.
7.
In the figure below, AB//DC,
∠
DAB =
∠
DBC =
90
0
, CD = 7 cm and BC =4 cm. Calculate the length of AB correct to four significant figures.
8.
On a map with a scale of 1: 50 000, a coffee plantation covers an area of
20
c
m
2
. Find the area of the plantation in hectares.
9.
Determine the equation of the straight line with: (a) gradient
-
4
3
and y-intercept -2. (b) gradient
-
2
5
and x-intercept
1
4
10.
In the figure OA = BQ = 10 cm. AP = 5cm. Given that OPQ is a straight line calculate: (a) OB (b) AB (c) PQ
11.
The gradient of a straight line
l
1
passing through the points P (3, 4) and Q (a, b) is
-
3
2
. A line
l
2
is perpendicular to
l
1
and passes through the points Q and R (2, —1). Determine the values of a and b.
12.
Evaluate
{
(
3.2
)
1
2
-
4
}
-
2
, giving your answer in standard form.
13.
The scale of a map is 1:125 000. Find the actual distance in kilometres represented by 16.8 cm on the map.
14.
A line L is perpendicular to y = 3x. If L passes through point (0, 4), find: (a) the equation of L. (b) the point Q where L intersects the line y = 3x.
15.
If A=
3
√
t
(
W
-
B
)
r
, find A if t = 0.0034, W = 4.634, B = 2.342 and r = 3.006.
16.
The width of rectangle is 10 cm and its area is
120
c
m
2
. Calculate the width of a similar rectangle whose area is
480
c
m
2
.
17.
Two perpendicular lines intersect at (3, 9). If one of them passes through
(
2
,
9
1
3
)
, find the equation of each line.
18.
The volume of a sphere is given by
V
=
4
π
r
3
3
. Find the value of r if v=311 and
π
=
3.142
.
19.
The vertices of a triangle are P (4, -3), Q(2, 1) and R(- 8, 2). Find the gradients of PQ, QR and RP. Arrange the gradients in an increasing order.
20.
A square whose vertices are P (1, 1), Q(2, 1), R(2, 2) and S(1, 2) is given an enlargement with centre at (0, 0). Find the images of the vertices if the scale factor is 2.
21.
The ratio of the radii of two spheres is 2:3. Calculate the volume of the first sphere if the volume of the second is
20
c
m
3
.
22.
Evaluate without using tables
3
√
0.729
×
409.6
0.1728
23.
A particle moves from rest and attains a velocity of 10 m/s after two seconds. It then moves with this velocity for four seconds. It finally decelerates uniformly and comes to rest after another six seconds. (a) Draw a velocity-time graph for the motion of this particle. (b) From your graph find: (i) the acceleration during the first two seconds. (ii) the uniform deceleration during the last
24.
The mean of the numbers m, 8m + 1, 17 and 20 is 14. Calculate: (a) the value of m. (b) the mode.
25.
Ten casual labourers were hired by a garment factory for one week and were paid in shillings, according to productivity of each, as below: 615, 633, 720, 509, 633, 710, 614, 630, 633, 720 Find the mean, mode and median for the data.
26.
In a bakery, the mass of 20 loaves taken at random is 10.03 kg. If the mean mass of the first 13 loaves is 505 g, find the mean mass of the other 7 loaves.
27.
The total marks scored in a test by 6 pupils was 420. If the mean mark for the first 5pupils was 68, find the marks scored by the sixth pupil.
28.
The table below shows a record of the number of goals scored in various matches by a football team: Calculate the mean number of: (a) goals per match. (b) matches per goal.
29.
Maina divided his farm among his four sons. The first son got twice as much as the second. The third got 2 hectares more than the first, while the fourth got 2 hectares less than the second. If on a pie chart, the share of the fourth son is represented by a sector of 48°, find the size of the shamba.
30.
The heights of 40 pupils in a class were measured to the nearest centimetre and recorded as below: (a) Use class interval of 5 to group the data starting with 145 — 149. (b) Estimate the mean and the median height of the class.
31.
The mean of a set of n numbers is 28. If an extra number, 18, is included in the set, the mean becomes 26. Calculate the value of n.
32.
The mean of 100 numbers is 13. Each of the numbers is multiplied by 3 and 5 subtracted from the result, calculate the new mean.
33.
Solve the equation
1
5
x
2
+
3
x
=
-
10
34.
The co-ordinates of points A, B and C are (0, -4), (2, -1) and (4, 2) respectively. (a) Deduce the position vectors of A, B and C. (b) Find the lengths of AB and AC. (c) Show that the points A, B and C are collinear.
35.
A pentagon is described by the vectors a =
(
2
5
)
, b =
(
1
3
)
, c =
(
-
2
1
)
, d =
(
-
3
11
)
and e =
(
2
-
1
)
in that order. What is the relationship between the vectors c and e?
36.
Solve the equation
x
2
=
4
(
1
2
x
+
2
)
37.
Simplify: a)
49
a
2
-
9
b
2
14
a
+
7
a
b
+
6
b
+
3
b
2
b)
(
4
x
+
2
m
)
2
(
-
4
m
2
+
16
x
2
)
(
4
m
2
+
16
x
2
+
16
m
x
)
(
-
2
m
+
4
x
)
38.
Use vectors to show that the points A (6, 1), B(3, 4) and C(1, 6) lie on a straight line. If 0 is the origin and E and F are the midpoints of OC and OA respectively, find the co-ordinates of E and F. Show that AC//EF.
39.
The column vectors of two sides of a triangle PQR are given by PQ =
(
4
2
)
and QR =
(
3
-
3
)
. Write down the column vectors RP and PR.
40.
Find possible values of x if
9
x
2
=
27
2
x
+
12
.
41.
The point P (-2, 5) is mapped onto P' (1, 9) by a translation T1. If P' is mapped onto P'' by a translation T2 given by
(
-
4
-
1
)
find the co-ordinates of P''.
42.
If a=
(
4
-
3
)
and b=
(
14
1
)
, find: (a) 2a - 3b. (b)
1
2
a - b. (c) Scalars x and y if xa - yb =
(
16
11
)
43.
Factorise the expressions: (a)
(
2
x
+
2
y
)
2
-
(
x
-
y
)
2
(b)
(
2
x
+
y
)
2
-
(
x
-
y
)
2
44.
Solve the simultaneous equations; 2a + 3b =
(
4
6
)
-a + b=
(
3
2
)
45.
The dimensions of a rectangle in centimetres are 2n - 3 by n + 1 and the area is
817
c
m
2
. Determine the dimensions of the rectangle.
46.
In the figure below,
∠
SPQ = 40° and
∠
TUS = 70°. Calculate
∠
QRS.
47.
In quadrilateral ABCD,
∠
ADC = 115°,
∠
ACB = 40°,
∠
DBC = 35° and
∠
BAC = 75°. Show that A, B, C, D are concyclic. Hence, find: (a)
∠
DAC. (b)
∠
DCA. (c)
∠
BDC.
48.
In the figure below, 0 is the centre of circle BCDE,
∠
BAE = 50° and
∠
BCE = 30°. Find
∠
BDC
49.
In the figure below, ADE and ABF are straight lines.
∠
DEC = 37° and
∠
BFC = 43°. Calculate
∠
BCF.
50.
In the figure below,
∠
QMN = 40°,
∠
QNR = 30°, MP = NR and MQR is a straight line passing through the centre of the circle. Find
∠
NRM and
∠
QNP.
51.
In the figure below, 0 is the centre of circle ABC,
∠
OAC = 55° and
∠
OBC = 25°. Calculate the reflex
∠
AOB.
52.
The region determined by the inequalities x + y > 0, y = x and 2y - x < 6 is reflected in the x-axis. Find the three inequalities that satisfy the image.
53.
In the figure below, calculate the value of y.
54.
Represent the simultaneous inequalities
y
3
+
3
8
≥
y
4
and
y
2
-
6
<
y
5
on a number line.
55.
Represent on a graph the region which satisfies the four inequalities:
y
≥
-
1
,
y
≤
x
-
1
,
y
≤
-
x
+
9
and
y
+
12
≥
2
x
. Determine the area of the region.
56.
The unshaded region in the figure below is reflected in the x-axis. Write down the inequalities which satisfy the new region. Find the area of the region.
57.
Represent the following inequalities on a graph paper by shading out the unwanted region:
y
≤
2
x
+
1
,
4
y
+
x
≤
22
,
2
y
≥
x
-
5
If the straight line through the points (5, 2) and (6, 4) forms the fourth boundary of the enclosed region, write down the fourth inequality by the region (points on the line are in the region).
58.
Find seven integral co-ordinates that satisfy all these inequalities:
y
+
3
x
≥
5
3
y
+
x
≤
15
y
-
x
≥
1
59.
Write down the inequalities satisfied by the illustrated region in the figure below:
60.
Find the area of the trapezoidal region bounded by the inequalities:
y
-
2
x
≤
1
3
y
-
x
>
-
2
2
y
+
x
<
7
2
y
+
x
≥
2
61.
A car which accelerates uniformly from rest attains a velocity of 30 m/s after 15 seconds. It travels at a constant velocity for the next 30 seconds before decelerating to stop after another 10 seconds. Use a suitable scale to draw the graph of velocity against time and use it to determine: (a) the total distance covered by the car in the first 55 seconds. (b) the retardation in the last ten
62.
Evaluate
4
√
3.45
+
2.62
786
×
0.0007
63.
If a =
(
2
3
)
and b =
(
4
5
)
, find:
(
a
)
2
a
+
1
2
b
(
b
)
7
8
(
3
a
+
1
3
b
)
(
c
)
3
a
-
5
b
64.
A right pyramid has a square base of sides 12 cm and slant height of 20 cm. Calculate: (a) its total surface area. (b) its volume.
65.
Solve the following inequalities and represent the solutions on a number line: (a) 3x+7<-5 (b)
t
3
≤
5
6
t
-
4
(c)
5
(
2
r
-
1
)
≤
7
(
r
+
1
)
(d)
6
(
x
-
4
)
-
4
(
x
-
1
)
≥
52
66.
The vertices of a quadrilateral are A(5, 1) B(4, 4), C(1, 5) and D (2, 2) (a) show that: (i) AB is parallel to CD. (ii) AD is parallel to CB. (iii) AC is perpendicular to DB.
67.
Simplify:
(
x
+
y
)
2
+
(
y
+
z
)
2
+
(
z
+
x
)
2
-
(
x
+
y
+
z
)
2
68.
Two translation
T
1
and
T
2
are represented by the vectors
(
3
-
1
)
and
(
3
-
2
)
respectively. If L is the point (4, 5), find the image of L under: (a)
T
1
(b)
T
2
(c)
T
1
followed by
T
2
69.
Write down the equations of the following lines, given the gradients and a point on the line: (a) (0, 0),
-
1
2
(b) (3, 0), 0 (c) (-1, 0), -1 (d) (3, 2),-2.
70.
The sum of two numbers exceeds a third number by four. If the sum of the three numbers is at least 20 and at most 28, find any three integral values satisfying the inequality.
71.
Below are records of daily bookings in Karibu Hotel during the month of June 2003: Find the mean number of people booked in the hotel each day.
72.
Factorise each of the following expressions: (a)
4
x
2
-
6
x
y
+
4
x
z
-
6
y
z
(b)
3
p
2
–
8
p
+
4
(c)
2
s
2
+
3
s
t
-
2
t
2
(d)
64
k
2
-
49
n
2
73.
The volume of a right pyramid with a square base is
256
c
m
3
. If its height is 16 cm, calculate: (a) the base area (b) the side of the square base
74.
Draw the following figures and indicate, where possible, their lines of symmetry: (a) Regular hexagon (b) Rhombus (c) Square (d) Regular pentagon (e) Circle (f) Equilateral triangle
75.
The figure below shows a cone with the vertex at A and diameter BC. The cone is cut off along DE: Find the: (a) base radius of the smaller cone. (b) volume of the frustum.
76.
The total surface area of two metallic solid sphere is
116
π
c
m
2
and their radii differ by 3 cm. Find the radius of each sphere. If the two spheres are melted and the material moulded into two spheres of equal radii, find the new total surface area.
77.
The data below is a record of time in seconds taken in a 100-metre race by 30 athletes: (a) Group the data using class intervals of 0.4 starting from 10.4 - 10.7. (b) Find the mean time taken in the race. (c) Estimate the median of the data.
78.
Find the vertices of a triangle defined by the intersection of the lines: y = x,
y
-
1
x
-
5
=-3 and 3y – 6 = 2 - x.
79.
The vertices of a kite are A (0, 0), B(5, 2), C(9, 9) and D (2, 5). Find the equation of the line of symmetry of the kite. If ABCD is reflected in the line x=0 find: (a) the co-ordinates of the vertices of the image. (b) the equation of the line of symmetry of the image.
80.
Points A (-6, 10), B(-7, 8), C (-6, 6), D(- 4, 6), E(-3, 8) and F(- 4, 10) are the vertices of a regular hexagon. The hexagon is reflected in the line y = x + 4, followed by a reflection in the line y=-x. Find the co-ordinates of the vertices of the final image of the hexagon.
81.
Solve the equation:
(
p
+
1
)
2
+
3
p
-
1
=
0
82.
Write down the gradients of the following lines without drawing them: (a) 3x + 2y - 6 = 0 (b) 4x = 6 - 2y (c)
x
3
+
y
2
= 1 (d) x = - 4 (e) y = 7
83.
If
2
x
+
y
=
16
and
4
2
x
-
y
=
1
4
find the values of x and y.
84.
Points P (2, 3), Q(4, 5) and R (7, 6) are vertices of triangle PQR. Find the co-ordinates of N, M and D, the midpoints of PQ, QR and PR respectively.
85.
Find non-zero scalars k and h such that ka + hb = c, given that: a=
(
3
2
)
, b=
(
4
6
)
and c=
(
2
2
)
.
86.
Calculate the surface area and volume of a sphere of radius 14 cm.
87.
Solve the simultaneous inequalities:
7
x
-
8
≤
342
and
4
x
-
1
≥
135
.
88.
The masses of 50 babies born in a certain hospital were recorded as below: (a) Find the mean mass of the babies (b) Estimate the median mass.
89.
AC is a diameter of a circle ABCD. AB = (x - 2) cm, BC = (x + 5) cm and the area of ABC is 30
c
m
2
. Find the radius of the circle. If ACD is isosceles, find its area.
90.
The internal and external diameters of a spherical shell are 12 cm and 18 cm respectively. Calculate the volume of the material of the shell.
91.
Use vectors to show that a triangle with vertices P(2, 3), Q(6, 4) and R (10, 5) is isosceles.
92.
The number of vehicles passing through a check-point were recorded for 20 days as follows: Find the mean number of vehicles which passed through this check-point per day.
93.
The ratio of the base areas of two cones is 9:16. (a) Find the ratio of their volumes. (b) If the larger cone has a volume of 125
c
m
3
, find the volume of the smaller cone.
94.
Use gradient to show that a triangle whose vertices are A (1, 3), B(4, -6) and C(4. 3) is right-angled. Find the area of this triangle.
95.
Given that
3
5
x
-
2
y
=
243
and
3
2
x
-
y
=
3
, calculate the values of x and y.
96.
Factorise
9
x
2
—
16
y
2
+
(
3
x
-
4
y
)
2
97.
A polygon with vertices A(6,2), B(6, 4.5), C(5, 4), D (4, 4.5), E(4, 2) and F(5, 2.5) is enlarged by scale factors (a) 2 (b) -1 with centre (5, 0). Find the co-ordinates of its image in each case.
98.
A triangle with the vertices at A (5, - 2), B(3, - 4) and C (7, - 4) is mapped onto a triangle with the vertices at A'(5, - 12), B'(7, - 10) and C'(3, - 10) by an enlargement, Find the linear scale factor and the centre of the enlargement.
99.
Illustrate the region defined by the following inequalities on graph paper: (a)
y
≤
2
x
+
1
,
y
≥
4
-
x
and
x
≤
3
.
100.
The depths of two similar buckets are 28 cm and 21 cm respectively. If the larger bucket holds 3.1 litres, find the capacity of the smaller one in litres.
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