1 A curve for which y is inversely proportional to x is shown below.
y
Find the equation of the curve. [2]
2 The function f(x) = is defined on the
... [Show More] domain x H 0.
The function g(x) = 25 - x
2
is defined on the domain R.
(a) Write down an expression for fg(x). [1]
(b) (i) Find thedomain of fg(x). [3]
(ii) Find the range of fg(x). [2]
3 An infinite sequence a1
, a2
, a3
, f is defined by an
=
n
n
, for all positive integers n.
(a) Find the limit of the sequence. [1]
(b) Prove that this is an increasing sequence. [3]
© OCR 2022 H640/03 Jun22
5
4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation 2 cos2
x = 3 sin x for 0 G x G 2r. [5]
5 A curve is defined implicitly by the equation 2x
2
+ 3xy + y
2
+ 2 = 0.
(a) Show that
dy
=-
4x + 3y
. [3]
dx 3x + 2y
(b) In this question you must show detailed reasoning.
Find the coordinates of the stationary points of the curve. [4]
6 A hot drink is cooling. The temperature of the drink at time t minutes is T °C.
The rate of decrease in temperature of the drink is proportional to (T -20).
(a) Write down a differential equation to describe the temperature of the drink as a function of
time. [2]
(b) When t = 0, the temperature of the drink is 90 °C and the temperature is decreasing at a rate of
4.9 °C per minute.
Determine how long it takes for the drink to cool from 90 °C to 40 °C. [6]
© OCR 2022 H640/03 Jun22 Turn over
2 8
6
7 A student is trying to find the binomial expansion of
She gets the first three terms as 1 -
x
3
+
x
6
.
1 - x
3
.
She draws the graphs of the curves y = 1 - x
3
, y = 1 -
x
3
and y = 1 -
x
3
+
x
6
using software.
2 2 8
y
x
(a) Explain why 1 -
x
3
+
x
6
H 1 -
x
3
for all values of x. [1]
2 8 2
(b) Explain why the graphs suggest that the student has made a mistake in the binomial expansion.
[1]
(c) Find the first four terms in the binomial expansion of 1 - x
3
. [3]
(d) State the set of values of x for which the binomial expansion in part (c) is valid. [1]
(e) Sketch the curve y = 2.5 1 - x
3
on the grid in the Printed Answer Booklet. [2]
(f) In this question you must show detailed reasoning.
The end of a bus shelter is modelled by the area between the curve
x =-0.75, x = 0.75 and the x-axis. Lengths are in metres.
y = 2.5 1 - x
3
, the lines
Calculate, using your answer to part (c), an approximation for the area of the end of the bus
shelter as given by this model. [4]
© OCR 2022 H640/03 Jun22
3
y = 1 - x
3
2 y = 1 -
x
3
+
x
6
2 8
1
–2 –1 0 1 2 3
–1
y = 1 -
x
3
2
8 The curves y = h(x) and
7
y = h
-1
(x), where h(x) = x
3
-8, are shown below.
The curve
The curve
y = h(x) crosses the x-axis at B and the y-axis at A.
y = h
-1
(x) crosses the x-axis at D and the y-axis at C.
y
x
(a) Find an expression for h
-1
(x). [2]
(b) Determine the coordinates of A, B, C and D. [5]
(c) Determine the equation of the perpendicular bisector of AB. Give your answer in the form
y = mx + c, where m and c are constants to be determined. [4]
(d) Points A, B, C and D lie on a circle.
Determine the equation of the circle. Give your answer in the form (x -a)
2
+ (y -b)
2
= r
2
,
where a, b and r
2
are constants to be determined. [5]
© OCR 2022 H640/03 Jun22 Turn over
y = h
-1
(x) C
D B
O
y = h(x)
A
40500 -i(180 -i)
2
8
Answer all the questions.
Section B (15 marks)
The questions in this section refer to the article on the Insert. You should read the article before
attempting the questions.
9 Show that y = x has the same gradient as y = sin x when x = 0, as stated in line 5. [2]
10 In this question you must show detailed reasoning.
Fig. C2.2 indicates that the curve y =
4x (r - x)
-sin x has a stationary point near x = 3.
r
2
• Verify that the x-coordinate of this stationary point is between 2.6 and 2.7.
• Show that this stationary point is a maximum turning point. [5]
11 Show that, for the angle 45°, the formula sin i.
4i(180 -i)
given in line 28 gives the
same approximation for the sine of the angle as the formula sin x .
16x (r - x)
5r
2
-4x (r - x)
given in
line 23. [3]
12 (a) Show that cos x = sinax +
r
k. [2]
(b) Hence show that sin x .
16x (r - x)
gives the approximation cos x .
r
2
- 4x
2
, as stated
5r
2
- 4x (r - x) r
2
+ x
2
in line 31. [3]
END OF QUESTION PAPER [Show Less]