Summer Tasks July 2016 (YEAR 12 – 13)
SUBJECT:
Summer Task Title / Instructions:
Summer Task Title / Instructions:
Complete all four sections. You must answer all questions to the best of your ability.
Do your work on separate paper with full working shown.
Do not leave questions unattempted: make use of the support available to you as detailed below.
This task must be submitted to your maths teacher during your first mathematics lesson in
September.
Support:
You can email [email protected] any questions relating to the task or the A-level
course.
As a school, we subscribe to www.mymaths.co.uk . This website provides excellent A Level
resources. If you would like to be able to use these resources, please email the above address for
a username and password.
www.examsolutions.net is also a useful website which provides video tutorials on each topic.
Please submit the task to your teacher on the first lesson in September.
Section 1: Functions
1)
Figure 2
Figure 2 shows part of the curve with equation y = f(x).
The curve passes through the points P(−1.5, 0)and Q(0, 5) as shown.
On separate diagrams, sketch the curve with equation
(a) y = f(x)
(2)
(b) y = f(x)
(2)
(c) y = 2f(3x)
(3)
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets
the axes.
2) The function f is defined by
f : x |→ |2x − 5|,
x  ℝ.
(a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph
cuts or meets the axes.
(2)
(b) Solve f(x) =15 + x.
(3)
The function g is defined by
g : x |→ x2 – 4x + 1,
x  ℝ,
0 ≤ x ≤ 5.
(c) Find fg(2).
(2)
(d) Find the range of g.
(3)
3)
Figure 1
Figure 1 shows part of the curve with equation y = f(x) , x  ℝ.
The curve passes through the points Q(0, 2) and P(−3, 0) as shown.
(a) Find the value of ff (−3).
(2)
On separate diagrams, sketch the curve with equation
(b) y = f −1(x),
(2)
(c) y = f(x) − 2,
(2)
(d) y = 2f  12 x  .
(3)
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets
the axes.
4) The function f is defined by
f: x 
(a) Show that f(x) =
2( x  1)
1
–
, x > 3.
x3
x  2x  3
2
1
, x > 3.
x 1
(4)
(b) Find the range of f.
(2)
(c) Find f –1 (x). State the domain of this inverse function.
(3)
The function g is defined by
g: x  2x2 – 3, x  ℝ.
(d) Solve fg(x) =
1
.
8
(3)
5.
Section 2: Logs and exponentials
6.
7.
8.
9.
Section 3: Numerical methods
f(x) = x3 + 3x2 + 4x – 12
10)
(a) Show that the equation f(x) = 0 can be written as
x=
 4(3  x) 

 ,
 (3  x) 
x  –3.
(3)
The equation x3 + 3x2 + 4x – 12 = 0 has a single root which is between 1 and 2.
(b) Use the iteration formula
xn + 1 =
 4(3  x n ) 

 ,
(
3

x
)
n


n  0,
with x 0 = 1 to find, to 2 decimal places, the value of x1 , x 2 and x3 .
(3)
The root of f(x) = 0 is  .
(c) By choosing a suitable interval, prove that  = 1.272 to 3 decimal places.
(3)
f(x) = x3 + 2x2 − 3x – 11
11) (a)
Show that f(x) = 0 can be rearranged as
(2)
x=
 3x  11 

,
 x2 
x  –2.
The equation f(x) = 0 has one positive root α.
The iterative formula xn + 1 =
 3x n  11 

 is used to find an approximation to α .
 xn  2 
(b) Taking x1 = 0, find, to 3 decimal places, the values of x 2 , x3 and x 4 .
(3)
(c) Show that α = 2.057 correct to 3 decimal places.
(3)
12)
f(x) = 4 cosec x − 4x +1,
where x is in radians.
(a) Show that there is a root α of f(x) = 0 in the interval [1.2, 1.3].
(2)
(b) Show that the equation f(x) = 0 can be written in the form
x=
1
1
+
4
sin x
(2)
(c) Use the iterative formula
xn+ 1 =
1
1
+ ,
sin x n
4
x0 = 1.25,
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.
(3)
(d) By considering the change of sign of f(x) in a suitable interval, verify that α = 1.291 correct to 3
decimal places.
(2)
Section 4: Mechanics
12) A car is moving on a straight horizontal road. At time t = 0, the car is moving with speed 20 m s–1
and is at the point A. The car maintains the speed of 20 m s–1 for 25 s. The car then moves with
constant deceleration 0.4 m s–2, reducing its speed from 20 m s–1 to 8 m s–1. The car then moves
with constant speed 8 m s–1 for 60 s. The car then moves with constant acceleration until it is
moving with speed 20 m s–1 at the point B.
(a) Sketch a speed-time graph to represent the motion of the car from A to B.
(3)
(b) Find the time for which the car is decelerating.
(2)
Given that the distance from A to B is 1960 m,
(c) find the time taken for the car to move from A to B.
(8)
13) A particle P is projected vertically upwards from a point A with speed u m s–1. The point A
is 17.5 m above horizontal ground. The particle P moves freely under gravity until it reaches
the ground with speed 28 m s–1.
(a) Show that u = 21.
(3)
At time t seconds after projection, P is 19 m above A.
(b) Find the possible values of t.
(5)
The ground is soft and, after P reaches the ground, P sinks vertically downwards into the ground
before coming to rest. The mass of P is 4 kg and the ground is assumed to exert a constant
resistive force of magnitude 5000 N on P.
(c) Find the vertical distance that P sinks into the ground before coming to rest.
(4)
14) A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting
from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s–1. She
maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the
finishing line with a speed of V m s–1.
(a) Sketch a speed-time graph for the motion of the girl during the whole race.
(2)
(b) Find the distance run by the girl in the first 64 s of the race.
(3)
(c) Find the value of V.
(5)
(d) Find the deceleration of the girl in the final 20 s of her race.
(2)
15) A ball is projected vertically upwards with a speed of 14.7 m s −1 from a point which is 49 m
above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
(a) the greatest height, above the ground, reached by the ball,
(4)
(b) the speed with which the ball first strikes the ground,
(3)
(c) the total time from when the ball is projected to when it first strikes the ground.
(3)