Download IB Math Studies 2 Review for Geometry and Trig Name 1. Find the

IB Math Studies 2
Review for Geometry and Trig
1.
Find the volume of the following prism.
Name ________________________
Diagram not to scale
8 cm
42°
5.7 cm
(Total 4 marks)
2.
2
Amir needs to construct an isosceles triangle ABC whose area is 100 cm .
The equal sides, AB and BC, are 20 cm long.
(a) Angle AB̂C is acute. Show that the angle AB̂C is 30.
(2)
(b)
Find the length of AC.
(3)
(Total 5 marks)
3.
In triangle ABC, AB = 3.9 cm, BC = 4.8 cm and angle AB̂C = 82.
B
Diagram not to scale
82°
3.9 cm
4.8 cm
A
C
(a)
Calculate the length of AC.
(3)
(b)
Calculate the size of angle AĈB.
(3)
(Total 6 marks)
4.
The figure shows a triangular area in a park surrounded by the paths AB, BC and CA, where
AB = 400 m, AB̂C = 50 and BĈA = 30.
diagram not to scale
A
400 m
30º
(a)
C
B
Find the length of AC using the above information.
Diana goes along these three paths in the park at an average speed of 1.8 m s–1.
(b) Given that BC = 788m, calculate how many minutes she takes to walk once around the
park.
(Total 6 marks)
1
IB Math Studies 2
Review for Geometry and Trig
Name ________________________
5.
Triangle ABC is drawn such that angle AB̂C is 90, angle AĈB is 60 and AB is 7.3 cm.
(a) (i)
Sketch a diagram to illustrate this information. Label the points A, B, C.
Show the angles 90, 60 and the length 7.3 cm on your diagram.
(ii) Find the length of BC.
(3)
Point D is on the straight line AC extended and is such that angle CD̂B is 20.
(b) (i)
Show the point D and the angle 20 on your diagram.
(ii) Find the size of angle CB̂D .
(3)
(Total 6 marks)
6.
P (4, 1) and Q (0, –5) are points on the coordinate plane.
(a) Determine the
(i)
coordinates of M, the midpoint of P and Q;
(ii) gradient of the line drawn through P and Q;
(iii) gradient of the line drawn through M, perpendicular to PQ.
The perpendicular line drawn through M meets the y-axis at R (0, k).
(b) Find k.
7.
On a map three schools A, B and C are situated as shown in the diagram.
Schools A and B are 625 metres apart.
Angle AB̂C = 102 and BC = 986 metres.
C
(Total 6 marks)
986 m
102 B
A
(a)
625 m
Find the distance between A and C.
(3)
(b)
Find the size of angle BÂC .
(3)
(Total 6 marks)
8.
Points P(0,–4), Q (0, 16) are shown on the diagram.
y
Q
8
0
2
4
6
8
10
12
14
16
18
x
P
2
IB Math Studies 2
Review for Geometry and Trig
(a) Plot the point R (11,16).
(b) Calculate angle QP̂R.
M is a point on the line PR. M is 9 units from P.
(c) Calculate the area of triangle PQM.
Name ________________________
(Total 6 marks)
9.
The mid-point, M, of the line joining A(s, 8) to B(−2, t) has coordinates M(2, 3).
(a) Calculate the values of s and t.
(2)
(b)
Find the equation of the straight line perpendicular to AB, passing through
the point M.
(4)
(Total 6 marks)
10.
The diagram shows a circle of radius R and centre O. A triangle AOB is drawn inside the circle.
The vertices of the triangle are at the centre, O, and at two points A and B on the circumference.
Angle AÔB is 110 degrees.
A
B
R
110°
R
O
(a)
(b)
(c)
Given that the area of the circle is 36 cm2, calculate the length of the radius R.
Calculate the length AB.
Write down the side length L of a square which has the same area as the given circle.
(Total 6 marks)
11.
Triangle ABC is such that AC is 7 cm, angle AB̂C is 65 and angle AĈB is 30.
(a) Sketch the triangle writing in the side length and angles.
(1)
(b)
Calculate the length of AB.
(c)
Find the area of triangle ABC.
(2)
(3)
(Total 6 marks)
12.
(a)
Write down the gradient of the line y = 3x + 4.
(b)
Find the gradient of the line which is perpendicular to the line y = 3x + 4.
(c)
Find the equation of the line which is perpendicular to y = 3x + 4 and which passes
through the point (6, 7).
(1)
(1)
(2)
(d)
Find the coordinates of the point of intersection of these two lines.
(2)
(Total 6 marks)
3
IB Math Studies 2
Review for Geometry and Trig
Name ________________________
13. The diagram shows a point P, 12.3 m from the base of a building of height h m. The angle
measured to the top of the building from point P is 63°.
hm
P
63°
12.3
(a) Calculate the height h of the building.
Consider the formula h = 4.9t2, where h is the height of the building and t is the time in seconds
to fall to the ground from the top of the building.
(b) Calculate how long it would take for a stone to fall from the top of the building to the
ground.
(Total 6 marks)
14.
Consider the line l: 2x + y + 4 = 0.
(a) Write down the gradient of l.
(b) Find the gradient of a line perpendicular to l.
(c) Find the equation of a line perpendicular to l, passing through the point (5, 3). Give your
answer in the form ax + by + d = 0, where a, b, d  .
15.
Three points A (1, 3), B (4, 10) and C (7, –1) are joined to form a triangle. The mid-point of AB
is D and the mid-point of AC is E.
(a) Plot the points A, B, C, on the grid.
(Total 6 marks)
(b)
Find the distance DE.
(Total 6 marks)
16.
(a)
A farmer wants to construct a new fence across a field. The plan is shown below. The
4
IB Math Studies 2
Review for Geometry and Trig
new fence is indicated by a dotted line.
Name ________________________
Diagram not to scale
75°
40°
410 m
Calculate the length of the fence.
(5)
(b)
The fence creates two sections of land. Find the area of the smaller section of land ABC,
given the additional information shown below.
A
B
Diagram not to scale
245 m
24
C
(3)
(Total 8 marks)
17.
Jenny has a circular cylinder with a lid. The cylinder has height 39 cm and diameter 65 mm.
(a) Calculate the volume of the cylinder in cm3. Give your answer correct to two decimal
places.
(3)
The cylinder is used for storing tennis balls.
Each ball has a radius of 3.25 cm.
(b) Calculate how many balls Jenny can fit in the cylinder if it is filled to the top.
(1)
(c)
(i)
(ii)
Jenny fills the cylinder with the number of balls found in part (b) and puts the lid
on. Calculate the volume of air inside the cylinder in the spaces between the tennis
balls.
Convert your answer to (c) (i) into cubic metres.
(4)
(Total 8 marks)
18.
D
A
4.5 cm
3 cm
25°
C
3 cm
In the diagram, AB = BC = 3 cm, DC = 4.5 cm, angle AB̂C = 90 and angle AĈD = 25.
(a) Calculate the length of AC.
(b) Calculate the area of triangle ACD.
(c) Calculate the area of quadrilateral ABCD.
(Total 8 Marks)
B
19.
5
IB Math Studies 2
Review for Geometry and Trig
Name ________________________
y
3
2
1
–5 –4 –3 –2 –1
–1
0
1
2
3
4
5
x
–2
–3
–4
–5
–6
(a)
(b)
On the grid above, draw a straight line with a gradient of –3 that passes through the point
(–2, 0).
Find the equation of this line.
(Total 8 marks)
20.
Sylvia is making a square-based pyramid. Each triangle has a base of length 12 cm and a height
of 10 cm.
O
(not to scale)
M
(a)
Show that the height of the pyramid is 8 cm.
(2)
M is the midpoint of the base of one of the triangles and O is the apex of the pyramid.
(b) Find the angle that the line MO makes with the base of the pyramid.
(3)
(c)
Calculate the volume of the pyramid.
(d)
Daniel wants to make a rectangular prism with the same volume as that of Sylvia’s
pyramid. The base of his prism is to be a square of side 10 cm. Calculate the height of the
prism.
(2)
(2)
(Total 9 marks)
6
IB Math Studies 2
Review for Geometry and Trig
21.
Name ________________________
An old tower (BT) leans at 10 away from the vertical (represented by line TG).
The base of the tower is at B so that MB̂T = 100.
Leonardo stands at L on flat ground 120 m away from B in the direction of the lean.
He measures the angle between the ground and the top of the tower T to be BL̂T = 26.5.
T
not to scale
M
MB = 200
90°
100°
26.5°
B G
BL = 120
L
(a)
(i)
(ii)
Find the value of angle BT̂L .
Use triangle BTL to calculate the sloping distance BT from the base, B to the top,
T of the tower.
(b)
Calculate the vertical height TG of the top of the tower.
(c)
Leonardo now walks to point M, a distance 200 m from B on the opposite side of the
tower. Calculate the distance from M to the top of the tower at T.
(5)
(2)
(3)
(Total 10 marks)
The graph shows the cost, in dollars, of posting letters with different weights.
3.00
cost in dollars
22.
2.50
2.00
1.50
1.00
0.50
0
50
100
150
200
250
300
350
400
450
500
550
600
weight in grammes
(a) Write down the cost of posting a letter weighing 60 g.
(b) Write down the cost of posting a letter weighing 250 g.
Kathy pays 2.50 dollars to post a letter.
(c) Write down the range for the weight, w, of the letter.
(Total 8 marks)
7
IB Math Studies 2
Review for Geometry and Trig
Name ________________________
23. The diagram below shows a child’s toy which is made up of a circular hoop, centre O, radius 7
cm. The hoop is suspended in a horizontal plane by three equal strings XA, XB, and XC. Each
string is of length 25 cm. The points A, B and C are equally spaced round the circumference of
the hoop and X is vertically above the point O.
X
25 cm
B
C
7 cm
diagram not to scale
A
(a)
Calculate the length of XO.
(b)
Find the angle, in degrees, between any string and the horizontal plane.
(2)
(2)
(c)
Write down the size of angle AÔB.
(d)
Calculate the length of AB.
(e)
Find the angle between strings XA and XB.
(1)
(3)
(3)
(Total 11 marks)
24.
The figure below shows a rectangular prism with some side lengths and diagonal lengths
marked. AC = 10 cm, CH = 10 cm, EH = 8cm, AE 8 cm.
8 cm
E
H
(not to scale)
8 cm
D
A
10 cm
F
10 cm
(a)
G
B
C
Calculate the length of AH.
(2)
(b)
Find the size of angle AĈH.
(3)
2
(c)
Show that the total surface area of the rectangular prism is 320 cm .
(d)
A triangular prism is enclosed within the planes ABCD, CGHD and ABGH. Calculate the
volume of this prism.
(3)
(3)
(Total 11 marks)
8
IB Math Studies 2
Review for Geometry and Trig
25.
Name ________________________
The triangular faces of a square based pyramid, ABCDE, are all inclined at 70 to the base. The
edges of the base ABCD are all 10 cm and M is the centre. G is the mid-point of CD.
E
(Diagram not to scale)
B
C
M
A
G
(a)
D
Using the letters on the diagram draw a triangle showing the position of a 70 angle.
(b)
Show that the height of the pyramid is 13.7 cm, to 3 significant figures.
(c)
Calculate
(i)
the length of EG;
(ii) the size of angle DÊC .
(d)
Find the total surface area of the pyramid.
(e)
Find the volume of the pyramid.
(1)
(2)
(4)
(2)
(2)
(Total 11 marks)
26.
Three points are given A(0, 4), B(6, 0) and C(8, 3).
(a) Calculate the gradient (slope) of line AB.
(2)
(b)
Find the coordinates of the midpoint, M, of the line AC.
(c)
Calculate the length of line AC.
(d)
Find the equation of the line BM giving your answer in the form ax + by + d = 0 where a,
b and d  .
(e)
State whether the line AB is perpendicular to the line BC showing clearly your working
and reasoning.
(2)
(2)
(5)
(3)
(Total 14 marks)
27.
The diagram below shows a field ABCD with a fence BD crossing it. AB = 15m, AD = 20 m
and angle BÂD = 110°. BC = 22 m and angle BD̂C = 30°.
A
diagram not to scale
15
110°
20
B
30°
D
22
C
9
IB Math Studies 2
Review for Geometry and Trig
(a) Calculate the length of BD.
Name ________________________
(3)
(b)
Calculate the size of angle BĈD.
(3)
One student gave the answer to (a) “correct to 1 significant figure” and used this answer to calculate
the size of angle BĈD.
(c) Write down the length of BD correct to 1 significant figure.
(1)
(d)
Find the size of angle BĈD that the student calculated, giving your answer correct to 1
decimal place.
(2)
(e)
Hence find the percentage error in his answer for angle BĈD.
(3)
(Total 12 marks)
28.
On the coordinate axes below, D is a point on the y-axis and E is a point on the x-axis.
1
O is the origin. The equation of the line DE is y  x = 4.
2
y
(not to scale)
D
C
O
(a)
B
E
x
Write down the coordinates of point E.
(2)
C is a point on the line DE. B is a point on the x-axis such that BC is parallel to the y-axis. The
x-coordinate of C is t.
1
(b) Show that the y-coordinate of C is 4  t.
2
(2)
OBCD is a trapezium. The y-coordinate of point D is 4.
1
(c) Show that the area of OBCD is 4t  t 2 .
4
(3)
(d)
The area of OBCD is 9.75 square units. Write down a quadratic equation that expresses
this information.
(e)
(i)
(1)
(ii)
Using your graphic display calculator, or otherwise, find the two solutions to the
quadratic equation written in part (d).
Hence find the correct value for t. Give a reason for your answer.
(4)
(Total 12 marks)
10
IB Math Studies 2
Review for Geometry and Trig
29.
Name ________________________
The vertices of quadrilateral ABCD as shown in the diagram are A (–8, 8), B (8, 3), C (7,–1)
and D (–4, 1).
y
A
8
7
6
5
4
3
B
2
1
D
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
7
–1
8
x
C
–2
(a)
5
.
16
Calculate the gradient of the line DC.
(b)
State whether or not DC is parallel to AB and give a reason for your answer.
The gradient of the line AB is –
(2)
(2)
The equation of the line through A and C is 3x + 5y = 16.
(c) Find the equation of the line through B and D expressing your answer in the form
ax + by = c, where a, b and c  .
(5)
The lines AC and BD intersect at point T.
(d) Calculate the coordinates of T.
(4)
(Total 13 marks)
30.
ABCDV is a solid glass pyramid. The base of the pyramid is a square of side 3.2 cm. The
vertical height is 2.8 cm. The vertex V is directly above the centre O of the base.
V
D
C
O
A
(a)
B
Calculate the volume of the pyramid.
11
IB Math Studies 2
Review for Geometry and Trig
Name ________________________
(2)
(b)
The glass weighs 9.3 grams per cm3. Calculate the weight of the pyramid.
(c)
Show that the length of the sloping edge VC of the pyramid is 3.6 cm.
(2)
(4)
(d)
Calculate the angle at the vertex, BV̂C .
(e)
Calculate the total surface area of the pyramid.
(3)
(4)
(Total 15 marks)
31.
A closed box has a square base of side x and height h.
(a) Write down an expression for the volume, V, of the box.
(1)
(b)
Write down an expression for the total surface area, A, of the box.
(1)
3
The volume of the box is 1000 cm
(c) Express h in terms of x.
(2)
(d)
Hence show that A = 4000x–l + 2x2.
(2)
(e)
Find
dA
.
dx
(2)
(f)
Calculate the value of x that gives a minimum surface area.
(4)
(g)
Find the surface area for this value of x.
(3)
(Total 15 marks)
32.
(a)
Sketch the graph of the function f (x) =
2x  3
, for −10  x  10. Indicating clearly the
x 4
axis intercepts and any asymptotes.
(6)
(b)
Write down the equation of the vertical asymptote.
(c)
On the same diagram sketch the graph of g (x) = x + 0.5.
(d)
Using your graphical display calculator write down the coordinates of one of the points of
intersection on the graphs of f and g, giving your answer correct to five decimal places.
(e)
Write down the gradient of the line g (x) = x + 0.5.
(f)
The line L passes through the point with coordinates (−2, −3) and is perpendicular to the
line g (x). Find the equation of L.
(2)
(2)
(3)
(1)
(3)
(Total 17 marks)
12
IB Math Studies 2
Review for Geometry and Trig
33.
Name ________________________
The diagram below shows the graph of a line L passing through (1, 1) and (2, 3) and the graph P
of the function f (x) = x2 − 3x − 4
y
L
P
0
x
(a)
Find the gradient of the line L.
(b)
Differentiate f (x).
(c)
Find the coordinates of the point where the tangent to P is parallel to the line L.
(d)
Find the coordinates of the point where the tangent to P is perpendicular to
the line L.
(e)
Find
(i)
the gradient of the tangent to P at the point with coordinates (2, −6);
(ii) the equation of the tangent to P at this point.
(f)
State the equation of the axis of symmetry of P.
(e)
Find the coordinates of the vertex of P and state the gradient of the curve
at this point.
(2)
(2)
(3)
(4)
(3)
(1)
(3)
(Total 18 marks)
13
IB Math Studies 2
Review for Geometry and Trig
Name ________________________
34. A child’s toy is made by combining a hemisphere of radius 3 cm and a right circular cone of
slant height l as shown on the diagram below.
diagram not to scale
l
3 cm
(a)
Show that the volume of the hemisphere is 18 cm3.
(2)
(b)
The volume of the cone is two-thirds that of the hemisphere.
Show that the vertical height of the cone is 4 cm.
(4)
(c)
Calculate the slant height of the cone.
(2)
(d)
Calculate the angle between the slanting side of the cone and the flat surface of the
hemisphere.
(3)
(e)
The toy is made of wood of density 0.6 g per cm3. Calculate the weight of the toy.
(3)
(f)
Calculate the total surface area of the toy.
(5)
(Total 19 marks)
14