MATHEMATICSAS MOCK EXAM

December (Edexcel Version) 1 hour and 30 minutes

Instructions to candidates:

• In the boxes above, write your centre number, candidate number, your surname, other names

and signature.

• Answer ALL of the questions.

• You must write your answer for each question in the spaces provided.

• You may use a calculator.

Information to candidates:

• Full marks may only be obtained for answers to ALL of the questions.

• The marks for individual questions and parts of the questions are shown in round brackets.

• There are 11 questions in this question paper. The total mark for this paper is 80.

Advice to candidates:

• You should ensure your answers to parts of the question are clearly labelled.

• You should show sufficient working to make your workings clear to the Examiner.

• Answers without working may not gain full credit.

CM

ASC/3/D17© 2017 crashMATHS Ltd.

1 2 3 3 2 2 1 C 8 D 1 7 4

Surname

Other Names

Candidate Signature

Centre Number Candidate Number

Examiner Comments Total Marks

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1 The equation has two equal roots.

Find the possible values of the constant k. (3)kx2 + 3− k( )x − 4 = 0

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2 Solve the equation

(3)a12 + 4a = 3

4

3

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Figure 1 shows a sketch of the curve with equation y = f(x).

On separate axes, sketch the curves with equation

(i) (3)

(ii) y = f(–x) (3)

On each sketch, you should show clearly the coordinates of any points where the curve crosses or meets the coordinate axes.

y = 12f(x)

y

x

−2

−1−4

Figure 1

5

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6

4

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Find the set of values of x that satisfy

(5)

2 − xx

< 3

7

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8

5

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The function f is defined such that

(a) In ascending powers of x, find the first four terms in the binomial expansion of f(x).

Give each term in its simplest form. (4)

(b) Using your answer to (a), approximate the value of . (3)

(c) Explain how you could make your approximation in part (b) more accurate. (1)

f x( ) = 1− 2x( )8

78

256

9

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TOTAL 8 MARKS

10

6

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(a) Given that , express y in terms of x. (2)(b) Solve the simultaneous equations

(6)

46−3x = 82y

46−3x = 82y

x − 2( )2 + 9y2 = 10

11

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7

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The curve C has the equation y = f(x), where

(a) Show that the curve C crosses the x axis when x = 4. (1)(b) Express f(x) as a product of three linear factors. (4)(c) Sketch the curve with equation y = f(x).

On your sketch, show clearly the coordinates of any points where the curve C crosses or meets

the coordinate axes. (3)(d) Find all the solutions to the equation

(2)

f(x) = −2x3 + 9x2 − x −12

−2 x − 4( )3 + 9 4 − x( )2 − x − 4( )−12 = 0

15

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TOTAL 10 MARKS

18

8

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The lines y = 2 – x and 5x + 2y = 3 are the perpendicular bisectors of the circle C1.

The circle C1 has the same radius as the circle with equation .

(a) Find

(i) the centre of C1 (2) (ii) the radius of C1 (2)(b) Show that the point A(1, –1) lies inside the circle C1. (2)(c) Write down the shortest distance between the point A and the circle C1. (2)

The circle C2 has the same centre as C1 and passes through the point .

(d) Express the equation of the circle C2 in the form

where a, b and r are constants to be found. (2) (e) Hence, or otherwise, determine whether C2 is completely contained within C1. (1)

x2 + y2 = 4x − 2y

2,− 12

⎛⎝⎜

⎞⎠⎟

x − a( )2 + y − b( )2 = r2

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TOTAL 11 MARKS

22

9

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Bernoulli’s inequality states that

for all integers and every real number .

(i) Using the binomial theorem on , prove Bernoulli’s inequality for . (2) (ii) Verify Bernoulli’s inequality for the case x = 0. (1)(iii) Use a counter-example to show that Bernoulli’s inequality is not valid for . (2)

1+ x( )r ≥1+ rxr ≥ 0 x ≥ −1

1+ x( )r x > 0

x < −1

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24

10

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A rectangular barn is to be made out of fence in an open field. The fence must enclose the barn and also split the barn in half, in order to separate the cattle from the chicken. The barn has length l metres and width w metres. An outline of the barn is shown in Figure 2 below.

Given that there is only 270 metres of fence in total,

(a) show that . (2)

(b) Hence, find an expression for the area of the barn, A m2, in terms of l. (1)(c) By completing the square, or otherwise, find

(i) the maximum area of the barn (3) (ii) the length and width of the barn for which the area is maximum. (2)

l

w

Figure 2

w = 90 − 23l

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28

11

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Figure 3 shows two straight lines, l1 and l2.

The line l2 has the equation 2x – 4y = 10. Given that l1 is perpendicular to l2,

(a) find the gradient of l1. (2)The points A and B are where the line l1 crosses the x axis and the y axis respectively. The area of the triangle OAB is 4 square units, where O is the origin.

(b) Find the coordinates of points A and B. (4)(c) Hence, show that the equation of the line l1 is

2x + y + k = 0

where k is a constant to be found. (2)The lines l1 and l2 intersect at the point C. The point D is where l2 intersects the y axis.

(d) Find the exact area of the quadrilateral OACD. (5)

l2l1

y

xA O

C

B

D

Figure 3

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TOTAL FOR PAPER IS 80 MARKS

TOTAL 13 MARKS