CAIE A-Level Mathematics 1 Pure Mathematics 1 Question Bank

CAIE A-Level Mathematics 1 Pure Mathematics 1 Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise Pure Mathematics 1 through quadratics, functions, coordinate geometry, circular measure, trigonometry, series, differentiation and integration with exact working.

Covered syllabus points

1.1 Quadratics1.2 Functions1.3 Coordinate geometry1.4 Circular measure1.5 Trigonometry1.6 Series1.7 Differentiation1.8 Integration

Question 1

Question 1(a)

(a)

Expand (2−12x)6\left(2-\frac{1}{2} x\right)^{6} in ascending powers of x up to and including the term in x3x^{3}.

[ 3 ]

Question 1(b)

(b)

Hence find the coefficient of x3x^{3} in the expansion of (3−x+2x3)(2−12x)6\left(3-x+2 x^{3}\right)\left(2-\frac{1}{2} x\right)^{6}.

[ 2 ]

Question 1

[Maximum number: 4]

Find the set of values of the constant k for which the quadratic equation

3kx2+(k+8)x+3=03 k x^{2}+(k+8) x+3=0

has two distinct real roots.

Question 2

[Maximum number: 6]

Let f(x)=4sin⁡23x\mathrm{f}(x)=4 \sin ^{2} 3 x.

Question 2(a)

(a)

Find the value of f′(14π)\mathrm{f}^{\prime}\left(\frac{1}{4} \pi\right).

[ 3 ]

Question 2(b)

(b)

Find ∫f(x)dx\int \mathrm{f}(x) \mathrm{d} x.

[ 3 ]

Question 3

Question 3(a)

(a)

Express 4x2+10x+64 x^{2}+10 x+6 in the form a(x+b)2+ca(x+b)^{2}+c, where a, b and c are rational constants to be determined.

[ 2 ]

Question 3(b)

(b)

The curve with equation y=4x2+10x+6y=4 x^{2}+10 x+6 and the line y=k have exactly one point of intersection.

Using your answer to part (a) or otherwise, state the value of the constant k.

[ 1 ]

Question 3

[Maximum number: 2]

The equation of a curve is y=f(x), where f(x)=12x23(x−2)2\mathrm{f}(x)=\frac{1}{2} x^{\frac{2}{3}}(x-2)^{2}. The following points lie on the curve. Non-exact values of the y-coordinates are given correct to 6 decimal places.

A(8,72), B(8.001, k), C(8.01,72.300388), D(8.1,75.038882)

Question 3(b)

(a)

Find the gradient of the chord AD. Give your answer correct to 4 decimal places.

[ 1 ]

Question 3(c)

(b)

State what the values in the table suggest about the value of f'(8).

[ 1 ]

Question 5

[Maximum number: 5]

The equation of a curve is y = 4cos(2x) + 3 for 0 <= x <= 2π.

Question 5(a)

(a)

State the greatest and least possible values of y.

[ 2 ]

Question 5(b)

(b)

Sketch the curve.

[ 2 ]

Question 5(c)

(c)

Hence determine the number of solutions of the equation 4cos(2x) + 3 = 2x - 1 for 0 <= x <= 2π.

[ 1 ]

Question 6

[Maximum number: 7]

Functions f and g are defined by

f(x)=(x+3)2−12 for x⩾0, g(x)=2x−5 for x∈R.\begin{array}{ll} \mathrm{f}(x)=(x+3)^{2}-12 & \text { for } x \geqslant 0, \\ \mathrm{~g}(x)=2 x-5 & \text { for } x \in \mathbb{R} . \end{array}

Question 6(a)

(a)

State the range of f.

[ 1 ]

Question 6(b)

(b)

Find an expression for f−1(x)\mathrm{f}^{-1}(x).

[ 2 ]

Question 6(c)

(c)

Solve the equation gf(x)=69.

[ 4 ]

Question 6

[Maximum number: 7]
Figure for Question 6 — CAIE A-Level Mathematics

The diagram shows a motif formed by the major arc A B of a circle with radius r and centre O, and the minor arc⁡AOB\operatorname{arc} A O B of a circle, also with radius r but with centre C. The point C lies on the circle with centre O.

Question 6(a)

(a)

Given that angle ACB=kπA C B=k \pi radians, state the value of the fraction k.

[ 1 ]

Question 6(b)

(b)

State the perimeter of the shaded motif in terms of π\pi and r.

[ 1 ]

Question 6(c)

(c)

Find the area of the shaded motif, giving your answer in terms of π,r\pi, r and 3\sqrt{3}.

[ 5 ]

Question 8

[Maximum number: 8]

Functions f and g are defined by

f(x)=(x+a)2−a for x⩽−a, g(x)=2x−1 for x∈R,\begin{aligned} & \mathrm{f}(x)=(x+a)^{2}-a \text { for } x \leqslant-a, \\ & \mathrm{~g}(x)=2 x-1 \text { for } x \in \mathbb{R}, \end{aligned}

where a is a positive constant.

Question 8(a)

(a)

Find an expression for f−1(x)\mathrm{f}^{-1}(x).

[ 3 ]

Question 8(b)(i)

(b)

State the domain of the function f−1\mathrm{f}^{-1}.

[ 1 ]

Question 8(b)(ii)

(c)

State the range of the function f−1\mathrm{f}^{-1}.

[ 1 ]

Question 8(c)

(d)

Given that a=72a=\frac{7}{2}, solve the equation gf(x)=0.

[ 3 ]

Question 11

[Maximum number: 11]
Figure for Question 11 — CAIE A-Level Mathematics

The diagram shows the curve with equation x=y2+1x=y^{2}+1. The points A(5,2) and B(2,-1) lie on the curve.

Question 11(a)

(a)

Find an equation of the line A B.

[ 2 ]

Question 11(b)

(b)

Find the volume of revolution when the region between the curve and the line A B is rotated through 360∘360^{\circ} about the y-axis.

[ 9 ]