2nd Review Exercise for Semester 2 Exam Grade 10 Math Extended
1. Let f(𝑥) = 𝑥 ! + 5𝑥 " + 𝑎𝑥 # + b𝑥 + 2, for 𝑥 ∈ ℝ, where 𝑎, 𝑏 are constants. The
remainder when f(𝑥) is divided by (𝑥– 1) is 6, and the remainder when f(𝑥) is divided by
(x + 2) is –6. Find the value of 𝑎 and the value of 𝑏.
2. A rota?ng sprinkler waters part of a lawn, shown as the shaded area in the diagram below.
The sprinkler covers a distance of 13 m, reaching a maximum distance of 17 m and
covering an angle of 130°.
(a) Calculate the length of the outer arc the sprinkler can reach, the arc from point A to
point B.
(b) Find the area the sprinkler can cover. Give your answer to the nearest square metre.
3. The side lengths of a quadrilateral ABCD are AB = 6 mm, BC = 5mm, CD = 8 mm, and DA =
4C is equal to 90°.
9 mm. The angle AB
(a)
(b)
(c)
(d)
Calculate the length of AC.
4 C.
Calculate the size of angle AD
Calculate the area of triangle ACD.
Calculate the area of quadrilateral ABCD.
A point P is chosen on the side of CD in a such way that the area of triangle ACP is equal to
the area of the triangle ABC.
(e) Calculate the length of CP.
(f) Calculate the length of AP.
4. A company produces and sells electric cars. The company’s profit, P, in thousands of
dollars, changes based on the number of cars, x, they produce per month.
The rate of change of their profit from producing 𝑥 electric cars is modelled by
$%
=– 1.6𝑥 + 48, 𝑥 ≥ 0.
$&
The company makes a profit of 260 (thousand dollars) when they produce 15 electric cars.
(a) Find an expression for P in terms of 𝑥.
The company regularly increases the number of car it produces.
(b) Describe how their profit changes if they increase produc?on to over 30 cars per
month and up to 50 cars per month. Jus?fy your answer.
5. A military helicopter flies from an airbase at A to a baYlefield at B and then to a hospital
at H. The routes taken by the helicopter are given by the vectors
– 140
=====⃗
AB = ?
A
70
165
and =====⃗
BH = ?
A.
– 20
Distances are measured in kilometres.
=====⃗.
(a) Determine the vector AH
A[er evacua?ng wounded soldiers from the baYlefield to the front-line hospital, the
helicopter returns back to the airbase.
(b) Write down the vector that describes this return flight.
(c) Find the direct distance from the hospital to the airbase.
'
*
6. Consider the curve 𝑦 = (–& + &–*. Find the 𝑥-coordinates of the points on the curve where
the gradient is zero.
7. The cross-sec?onal view of a tunnel is shown on the axes below. The line [AB] represents
a ver?cal wall located at the le[ side of the tunnel. The height, in metres, of the tunnel
above the horizontal ground is modelled by 𝑦 =– 0.1𝑥 " + 0.8𝑥 # , 2 ≤ 𝑥 ≤ 8, rela?ve to
an origin O.
Point A has coordinates (2, 0), point B has coordinates (2, 2.4), and point C has coordinates
(8, 0).
$+
(a) (i) Find $&.
(ii) Hence find the maximum height of the tunnel.
When 𝑥 = 4 the height of the tunnel is 6.4 m and when 𝑥 = 6 the height of the tunnel is
7.2 m. These points are shown as D and E on the diagram, respec?vely.
(b) Use the trapezoidal rule, with three intervals, to es?mate the cross-sec?onal area of
the tunnel.
(c) (i) Write down the integral which can be used to find the cross-sec?onal area of the
tunnel.
(ii) Hence find the cross-sec?onal area of the tunnel.
8. The Texas Star is a Ferris wheel at the state fair in Dallas. The Ferris wheel has a diameter
of 61.8 m. To begin the ride, a passenger gets into a chair at the lowest point on the wheel,
which is 2.7 m above the ground, as shown in the following diagram. A ride consists of
mul?ple revolu?ons, and the Ferris wheel makes 1.5 revolu?ons per minute.
The height of a chair above the ground, ℎ, measured in metres, during a ride on the Ferris
wheel can be modelled by the func?on ℎ(𝑡) =– 𝑎cos(𝑏𝑡) + 𝑑, where 𝑡 is the ?me, in
seconds, since a passenger began their ride.
(a) Calculate the value of
(i)
𝑎;
(ii)
𝑏;
(iii)
𝑑.
A ride on the Ferris wheel lasts for 12 minutes in total.
(b) Calculate the number of revolu?ons of the Ferris wheel per ride.
(c) For exactly one ride on the Ferris wheel, suggest
(i)
an appropriate domain for ℎ(𝑡);
(ii)
an appropriate range for ℎ(𝑡).
Big Tex is a 16.7 metre-tall cowboy statue that stands on the horizontal ground next to
the Ferris wheel.
(d) By considering the graph of ℎ(𝑡), determine the length of ?me during one revolu?on
of the Ferris wheel for which the chair is higher than the cowboy statue.
There is a plan to relocate the Texas Star Ferris wheel onto a taller plaeorm which will
increase the maximum height of the Ferris wheel to 65.2 m. This will change the value of
one parameter, 𝑎, 𝑏 or 𝑑, found in part (a).
(e) (i)
(ii)
Iden?fy which parameter will change.
Find the new value of the parameter iden?fied in part (e)(i).
9. Air is being pumped into a spherical balloon so that its volume is increasing at a constant
rate of 15 cm3min–1. Find the rate at which the surface area of the balloon is increasing
when its radius hits 10cm.
The surface area S and the volume V of a sphere of radius r are given by 𝑆 = 4𝜋𝑟 # and
!
𝑉 = " 𝜋𝑟 " .
10. Jack makes an open container in the shape of a cuboid with square base, as shown in the
following diagram.
The container has base length 𝑥 m and height 𝑦 m. The volume is 32 m3.
Let 𝐴(𝑥) be the outside surface area of the container.
*#,
(a) Show that 𝐴(𝑥) = & + 𝑥 # .
(b) Find 𝐴'(𝑥).
(c) Given that the outside surface area is a minimum, find the base length of the
container.
(d) Jack coats the outside of the container with waterproof resin. A can of resin covers a
surface area of 5 m2 and costs $15. Find the total cost of the cans needed to coat the
container.