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Date May 2019 Marks available 4 Reference code 19M.1.AHL.TZ1.H_10
Level Additional Higher Level Paper Paper 1 Time zone Time zone 1
Command term Question number H_10 Adapted from N/A

Question

The function p(x) is defined by p(x)=x33x2+8x24 where xR.

Find the remainder when p(x) is divided by (x2).

[2]
a.i.

Find the remainder when p(x) is divided by (x3).

[1]
a.ii.

Prove that p(x) has only one real zero.

[4]
b.

Write down the transformation that will transform the graph of y=p(x) onto the graph of y=8x312x2+16x24.

[2]
c.

The random variable X follows a Poisson distribution with a mean of λ and 6P(X=3)=3P(X=2)2P(X=1)+3P(X=0).

Find the value of λ.

[6]
d.

Markscheme

p(2)=812+1624       (M1)

Note: Award M1 for a valid attempt at remainder theorem or polynomial division.

= −12     A1

remainder = −12

[2 marks]

a.i.

p(3)=2727+2424 = 0      A1 

remainder = 0

[1 mark]

a.ii.

x=3 (is a zero)     A1

Note: Can be seen anywhere.

EITHER

factorise to get (x3)(x2+8)      (M1)A1

x2+80 (for xR) (or equivalent statement)      R1

Note: Award R1 if correct two complex roots are given.

OR

p(x)=3x26x+8    A1

attempting to show p(x)0       M1

eg discriminant = 36 – 96 < 0, completing the square

no turning points       R1

THEN

only one real zero (as the curve is continuous)      AG

[4 marks]

b.

new graph is y=p(2x)     (M1)

stretch parallel to the x-axis (with x=0 invariant), scale factor 0.5    A1

Note: Accept “horizontal” instead of “parallel to the x-axis”.

[2 marks]

c.

6λ3eλ6=3λ2eλ22λeλ+3eλ     M1A1

Note: Allow factorials in the denominator for A1.

2λ33λ2+4λ6=0    A1

Note: Accept any correct cubic equation without factorials and eλ.

EITHER

4(2λ33λ2+4λ6)=8λ312λ2+16λ24=0       (M1)

2λ=3      (A1)

OR

(2λ3)(λ2+2)=0       (M1)(A1)

THEN

λ = 1.5    A1

[6 marks]

d.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.
[N/A]
d.

Syllabus sections

Topic 4—Statistics and probability » SL 4.8—Binomial distribution
Show 139 related questions
Topic 2—Functions » AHL 2.8—Transformations of graphs, composite transformations
Topic 4—Statistics and probability » AHL 4.17—Poisson distribution
Topic 2—Functions
Topic 4—Statistics and probability

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