Edexcel A-Level Mathematics AS Unit P1 Pure Mathematics 1 Questions

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

Covered syllabus points

[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: 6]

Figure 2

Figure 2 shows the points P, Q and R.
Points P and Q have coordinates (-1,4) and (4,7) respectively.

(a)

Find an equation for the straight line passing through points P and Q.

Give your answer in the form ax+by+c=0 where a,b and c are integers.

[ 3 ]

\text{Gradient }PQ=\frac{7-4}{4-(-1)}=\frac35

y-4=\frac35(x+1)

3x-5y+23=0

(b)

The point R has coordinates (p,-3), where p is a positive constant.

Given that angle $QPR=90^\circ$ ,

find the value of p.

(Solutions relying on calculator technology are not acceptable.)

[ 3 ]

\text{gradient }PR=-\frac53

\frac{-3-4}{p-(-1)}=-\frac53

5p+5=21

p=\frac{16}{5}

[Maximum number: 5]

The sides of a triangle are in the ratio 2: 4: 5

(a)

Find the size of the largest angle of the triangle, in degrees, to one decimal place.

Given that the area of the triangle is $140 \mathrm{~cm}^{2}$

[ 2 ]

$5^2=2^2+4^2-2(2)(4)\cos\theta$ $\theta=108.2^\circ$

(b)

find the length of the shortest side, in cm , to one decimal place.

[ 3 ]

$140=\frac12(2x)(4x)\sin108.2^\circ$ $x^2=\frac{280}{8\sin108.2^\circ}=36.8\ldots$
Shortest side $=2\sqrt{36.8\ldots}=12.1\text{ cm}$
B1ft, M1, 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

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