Pearson Edexcel IAL Mathematics Unit P1: Pure Mathematics 1 Question Bank

Practise P1 methods for algebra, coordinate geometry, trigonometry, differentiation and integration using exact working and graphs.

Covered syllabus points

P1.1 - Algebra and functions P1.2 - Coordinate geometry in the (x, y) plane P1.3 - Trigonometry P1.4 - Differentiation P1.5 - Integration

[Maximum number: 3]

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

By substituting $p=2^{x}$ , show that the equation

2 \times 4^{x}-2^{x+3}=17 \times 2^{x-1}-4

can be written in the form

4 p^{2}-33 p+8=0

With $p=2^x$ : $4^x=p^2,\ 2^{x+3}=8p,\ 2^{x-1}=\frac{p}{2}$ .

$2p^2-8p=\frac{17p}{2}-4$ $\Rightarrow4p^2-33p+8=0$

B1 M1 A1*

[Maximum number: 8]

Figure 2

Figure 2 shows a sketch of the straight line l and the curve C.
Given that l cuts the y-axis at -12 and cuts the x-axis at 4, as shown in Figure 2,

Given that C
- has equation y=f(x) where f(x) is a quadratic expression
- has a minimum point at (7,-18)
- cuts the x-axis at 4 and at k, where k is a constant

(a)

find an equation for l, writing your answer in the form y=mx+c, where m and c are constants to be found.

[ 2 ]

(b)

deduce the value of k,

[ 1 ]

(c)

find f(x).

[ 3 ]

y=A(x-4)(x-10) or $y=C(x-7)^2-18$

M1

Using (7,-18): $-18=A(7-4)(7-10)\Rightarrow A=2$

OR using (4,0): $0=C(4-7)^2-18\Rightarrow C=2$

dM1

y=2(x-4)(x-10), or $y=2(x-7)^2-18$

A1

(d)

The region R is shown shaded in Figure 2.
Use inequalities to define R.

[ 2 ]

[Maximum number: 10]

Figure 1

Figure 1 shows the plan view for the design of a stage.
The design consists of a sector OBC of a circle, with centre O, joined to two congruent triangles OAB and ODC.

Given that
- angle B O C=2.4 radians
- area of sector $B O C=40 \mathrm{~m}^{2}$
- AOD is a straight line of length 12.5 m

(a)

find the radius of the sector, giving your answer, in m , to 2 decimal places,

[ 2 ]

$A=\frac{1}{2}r^2\theta$ $40=\frac{1}{2}r^2\times2.4$

M1

$r=\sqrt{\frac{80}{2.4}}=5.77\text{ m}$

A1

(b)

find the size of angle AOB, in radians, to 2 decimal places.

Hence find

[ 1 ]

(c)

the total area of the stage, giving your answer, in $\mathrm{m}^{2}$ , to one decimal place,

[ 3 ]

$\frac{1}{2}\times 5.77\times 6.25\times \sin(0.37)=6.5377\ldots$

M1

Area of stage $=40+2\times\frac{1}{2}\times5.77\times6.25\times\sin(0.37)$

dM1

$=53.1\text{ m}^2$

A1

(d)

the total perimeter of the stage, giving your answer, in m , to one decimal place.

[ 4 ]

Arc length $=5.77\times2.4=13.848$

M1

$AB^2=6.25^2+5.77^2-2\times6.25\times5.77\cos(0.37)$ , so $AB=2.26\ldots$

M1

Perimeter $=12.5+2\times2.26\ldots+13.848$

ddM1

$=30.9\text{ m}$

A1

[Maximum number: 2]

A curve C has equation y=f(x) where

f(x)=(2-k x)^{5}

and k is a constant.
Given that when f(x) is divided by (4 x-5) the remainder is $\frac{243}{32}$

find the gradient of C at the point where x=0

$f'(x)=-32+\frac{128}{5}x$ f'(0)=-32.
M1 A1ft

All question bank results loaded

Covered syllabus points

Unit P1: Pure Mathematics 1 question 1

Unit P1: Pure Mathematics 1 question 2

Question (a)

Question (b)

Question (c)

Question (d)

Unit P1: Pure Mathematics 1 question 3

Question (a)

Question (b)

Question (c)

Question (d)

Unit P1: Pure Mathematics 1 question 4