Pure 1 May June 2024 Guess Paper
CIE A Level Maths 9709
*This is not endorsed by Cambridge and is purely for practice purposes only.
Questions
1. A line with equation y = mx − 10 is a tangent to the curve with equation y = x2 − 8x + 15.
Find the possible values of the constant m, and the corresponding coordinates of the points at
which the line touches the curve. [6]
2. A function f is defined by f (x) = x2 − 3x + 7, for x ∈ R. A sequence of transformations is applied
in the following order to the graph of y = f (x) to give the graph of y = g(x).
Stretch parallel to the x-axis with scale factor 41
Reflection in the y-axis
Stretch parallel to the y-axis with scale factor 2
Find g(x), giving your answer in the form ax2 + bx + c, where a, b and c are constants to be found.
[4]
3. It is given that the coefficient of x3 in the expansion of
(5 + ax)4 (7 − ax)
is −80.
Find the value of the constant a. [5]
4. A large industrial water tank is such that, when the depth of the water in the tank is x metres, the
volume V m3 of water in the tank is given by V = 167 − 31 (4 − x)3 . Water is being pumped into
the tank at a constant rate of 2.4 m3 per hour.
Find the rate of increase of the depth of the water when the depth is 2 m, giving your answer in cm
per minute. [5]
5. (a) Show that the equation
3
3
−
=0
tan x sin x
may be expressed in the form a cos2 x + b cos x + c = 0, where a, b and c are integers to be
found. [3]
4 sin x −
(b) Hence solve the equation 4 sin x − tan3 x − sin3 x = 0 for 0◦ ≤ x ≤ 360◦ . [3]
6. The points A(6, 0), B(6, 6) and C(−2, 6) are on the circumference of a circle.
(a) Find an equation of the circle. [5]
(b) Find an equation of the tangent to the circle at B. [2]
1
7. asdf
A
r cm
O
θ rad
r cm
B
The diagram shows a sector OAB of a circle with centre O and radius r cm. Angle AOB = θ
radians. It is given that the length of the arc AB is 14.4cm and that area of the sector OAB is
129.6cm2 .
(a) Find the area of the shaded region. [5]
(b) Find the perimeter of the shaded region. [2]
8. The first term of a geometric progression is 64 and the fourth term is 27.
(a) Find the sum to infinity of the progression. [3]
The second term of the geometric progression is equal to the second term of an arithmetic
progression.
The third term of the geometric progression is equal to the fifth term of the same arithmetic
progression.
(b) Find the sum of the first 10 terms of the arithmetic progression. [6]
2
for x > 3.
9. The function f is defined by f (x) = 1 + x−3
(a) State the range of f . [1]
(b) Obtain an expression for f −1 (x) and state the domain of f −1 . [4]
The function g is defined by g(x) = 4x − 4 for x > 0.
(c) Obtain a simplified expression for gf (x). [2]
3
dy
. It is given that dx
= x− 2 + k, where
10. At the point (4, 36) on a curve, the gradient of the curve is 57
8
k is a constant.
(a) Show that k = 7. [1]
(b) Find the equation of the curve. [4]
(c) Find the coordinates of the stationary point. [3]
(d) Determine the nature of the stationary point. [2]
2
11. asdf
y
1
1
y = 2 x 2 + 3x− 2
A
B
1
y = 2x− 2 + 6
x
O
1
1
1
The diagram shows with equations y = 2x− 2 + 6 and y = 2 x 2 + 3x− 2 . The curves intersect at
points A and B.
(a) Find the coordinates of A and B. [4]
(b) Hence find the area of the region between the two curves. [5]
3
Answers
1. m = −18,
m = 2,
(−5, 80),
(5, 0)
2. 32x2 + 24x + 14
3. a = 2
= 1.00 cm per minute
4. dx
dt
5. (a) 4 cos2 x + 3 cos x − 1 = 0
(b) x = 75.5◦ , 180◦ , 284.5◦
6. (a) (x − 2)2 + (y − 3)2 = 25
(b) y = − 34 x + 14
7. (a) 13.4 cm2
(b) 28.4 cm
8. (a) S∞ = 256
(b) S10 = 340
9. (a) f (x) > 1
2
,
(b) f −1 (x) = 3 + x−1
x>1
8
(c) gf (x) = x−3
3
= (4)− 2 + k
10. (a) 57
8
57
= 18 + k
8
57
− 18 = k
8
k=7
1
(b) y = −2x− 2 + 7x + 9
(c) (0.273, 7.09)
(d) Maximum Point
11. (a) A(1, 8) B(4, 7)
(b) − 32
4