Edexcel IAL Maths P1.1 Algebra and Functions

[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: 5]

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

Solutions relying on calculator technology are not acceptable.

(a)

Show that $\frac{\sqrt{180}-\sqrt{80}}{\sqrt{5}}$ is an integer and find its value.

[ 2 ]

Writes $\sqrt{180}$ as $6 \sqrt{5}$ or $\sqrt{80}$ as $4 \sqrt{5}$
Concludes working $\frac{6 \sqrt{5}-4 \sqrt{5}}{\sqrt{5}}=\frac{2 \sqrt{5}}{\sqrt{5}}=2$
(2)

(b)

Simplify

\frac{4 \sqrt{5}-5}{7-3 \sqrt{5}}

giving your answer in the form $a+b \sqrt{5}$ where a and b are rational numbers.

[ 3 ]

$\frac{4\sqrt5-5}{7-3\sqrt5} =\frac{4\sqrt5-5}{7-3\sqrt5}\times\frac{7+3\sqrt5}{7+3\sqrt5}$ $=\frac{28\sqrt5-35+12(\sqrt5)^2-15\sqrt5}{49-9\times5}$ $=\frac{25}{4}+\frac{13}{4}\sqrt5$
Marking guidance for this part:
M1 for a correct attempt to rationalise by multiplying by $7+3\sqrt5$ .
dM1 for a surd-free denominator and an attempt to multiply out the numerator with four terms.
A1 for $\frac{25}{4}+\frac{13}{4}\sqrt5$ .

[Maximum number: 6]

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)

deduce the value of k,

[ 1 ]

(b)

find f(x).

[ 3 ]

y=A(x-4)(x-10) or $y=C(x-7)^2-18$
Using (7,-18): $-18=A(7-4)(7-10)\Rightarrow A=2$

OR using (4,0): $0=C(4-7)^2-18\Rightarrow C=2$ y=2(x-4)(x-10), or $y=2(x-7)^2-18$

(c)

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

[ 2 ]

[Maximum number: 5]

Find the range of possible values of k for which the curve with equation

y=-\frac{k}{x}+k \quad k>0 \quad x \neq 0

does not touch or intersect the line with equation y=3 x+4

For an intersection of the line and curve:
$3x+4=-\frac{k}{x}+k$ $3x^2+(4-k)x+k=0.$
For two distinct intersections, the discriminant is positive:
$(4-k)^2-12k>0$ $k^2-20k+16>0.$
The critical values are
$k=10\pm2\sqrt{21}.$
Hence the required range is
$10-2\sqrt{21}<k<10+2\sqrt{21}.$
M1 dM1 A1

[Maximum number: 3]

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

Solutions relying on calculator technology are not acceptable.

Figure 2

Figure 2 shows a sketch of the curve C with equation $y=x^{2}-5 x+13$

The point M is the minimum point of C.

The straight line l passes through the origin O and intersects C at the points M and N as shown.

Find, showing your working,

the coordinates of M,

M1 For attempting to complete the square. Look for $(x-2.5)^{2}$
A1 $\quad(x-2.5)^{2}+6.75$ or for correctly extracting x=2.5 as the x coordinate.
A1 $\quad M=(2.5,6.75) \quad$ or $\left(\frac{5}{2}, \frac{27}{4}\right)$

[Maximum number: 2]

Figure 2

Figure 2 shows a sketch of
- the circle C with centre X(4,-3)
- the line l with equation $y=\frac{5}{2} x-\frac{55}{2}$

Given that l is the tangent to C at the point N,

Hence find

the coordinates of N,

M1: Attempts to solve 2 x+5 y+7=0 and $y=\frac{5}{2} x-\frac{55}{2}$ leading to $x=\ldots, y=\ldots$
Allow the values to be just written down, for example via use of a calculator
A1: N=(9,-5) which may be written x=9, y=-5

[Maximum number: 2]

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

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

[Maximum number: 2]

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

Solutions relying on calculator technology are not acceptable.

Figure 2

Figure 2 shows a sketch of the curve C with equation $y=x^{2}-5 x+13$

The point M is the minimum point of C.

The straight line l passes through the origin O and intersects C at the points M and N as shown.

Find, showing your working,

Use inequalities to define the region R.

M1: States any two of $y<x^2-5x+13$ , y>2.7x, and $0\le x<b$ , where $2.5\le b\le5.2$ .
A1ft: States all three inequalities, following through on their line from part (b) and their boundary values.

[Maximum number: 3]

Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, p, and the number of marks achieved on the final exam, f, for 34 students and displayed them on the scatter diagram.

The equation of the least squares regression line for these data is found to be
f=10.8+0.748p

For these students, the mean number of marks on the pre-test is 62.4

Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.

Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam.
The summary statistics used for the model f=10.8+0.748p are corrected to include this information and a new least squares regression line is found.
Given the original summary statistics were,
n=34
sum p=2120
sum pf=133486
Spp=15573.76
Spf=11648.35

Find the range of values of p for which this regression model, f=10.8+0.748p, predicts a greater number of marks on the final exam than on the pre-test.

p< awrt $\underline{42.9}$

A1 (3)

[Maximum number: 2]

Given that

(x-5)(2 x+1)(x+3) \equiv a x^{3}+b x^{2}-32 x-15

where a and b are constants,

find the value of a and the value of b.

[Maximum number: 2]

Sketch the curve with equation

y=-\frac{k}{x} \quad k>0 \quad x \neq 0

[Maximum number: 3]

On a separate diagram, sketch the curve with equation

y=-\frac{k}{x}+k \quad k>0 \quad x \neq 0

stating the coordinates of the point of intersection with the x-axis and, in terms of k, the equation of the horizontal asymptote.

Figure for Answer to Question 12 — Edexcel A-Level Mathematics AS

(b) Note on EPEN it is M1B1A1. We are marking this B1ftB1B1

B1ft For attempting to translate their graph from part (a) up (or a correct graph if (a) is incorrect)

B1 For a correct intercept stated (or allow $1, \frac{k}{k}$ or (1,0) marked on the correct axis with the curve passing through this) OR y=k stated. Do not allow just k to be marked on the line of the asymptote.

B1 For a correct intercept stated (or allow $1, \frac{k}{k}$ or (1,0) marked on the correct axis with the curve passing through this) AND y=k stated. Do not allow just k to be marked on the line of the asymptote.

Note if they have a contradiction between what is stated and what is written on the graph then the graph takes precedence.

Edexcel A-Level Mathematics AS P1.1 Algebra and Functions Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question (c)

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12