Mathematics · May / June 2021 · Paper 61 · Question 1

(b) P(Y) = 7

2

3

2

2

42

7

×

+

×

+

=

or

or

12

11

12

11

12

132

22

P(Yellow and two counters) = 7

2

3

2

20

5

or

×

+

×

=

12

11

12

11

132

33

20

(

)

"

"

[

]

[

]

P

Y

∩

R

∪

Y

∩

B

132

=

42

P( )

Y

"

"

132

20

10

=

or

oe

42

21

Notes

(a)

1st B1 for the remaining probs on first set of branches and at least one on 2nd set

2nd B1 for a fully correct tree diagram with all the correct probabilities

1st M1 for a correct ft expression for P(Y) or P(Yellow and two counters)ft their tree

diagram

7

6

+

3

3

7

+

2

1 12

eg

−

×

−

×

11

12

11

NB: The method is implied by the numbers in curly brackets but we do not need to

see them to award the mark.

2nd M1 for a correct ratio formula (symbols or words) and at least one correct ft prob or

fully correct ft ratio. Do not follow through probabilities > 1 or < 0

A1 for 10

&

&)

21 or exact equivalent. (Allow 0.476190

NB if an exact correct fraction is not given and an awrt 0.476…. is given it would

get M1M1A0 if from correct working

Generally if the answer is correct then award full marks (unless from obvious

incorrect working) or notes indicate otherwise

2. (a) B and C

(b) A and C independent gives:

(

)

(

)

P(C) 0.65

×

=

0.13

or

0.65

×

r

+

0.13

=

0.13

or

0.65

×

0.48

−

s

=

0.13

P(C) = 0.2 or

r +

0.13

=

0.2

or 0.48

−

s

=

0.2

r {= 0.2 – 0.13} = 0.07 or s {= 0.48 – 0.2} = 0.28

P(A) + r + s = 1 or 0.65 + "0.07" + s = 1 or 0.65 + "0.28"+ r = 1

s {= 1 – 0.72 =} 0.28 and r {= 1 – 0.93 =} 0.07

(c) P[(

)

B

C

∪

] = "0.2" + q or 0.13 + "0.07" + q

P(

)

{

}

A

∩

C

′

=

p

+

q

=

0.52

{

}

(

)

(

)

(

)

(

)

P

A

∩

C

′

∩

B

∪

C

=

q

⇒

"

p

+

q

" " 0.2

×

+

q

"

=

q

or

(

)

(

)

or (

)

(

)

"

p

q

" " 0.13 "0.07"

q

"

q

"

p

q

" " 1

s

p

"

0.52

p

+

×

+

+

=

+

×

−−

=

−

(

)

(

)

0.52 " 0.2

×

+

q

"

=

q

0.52 0.72

−

p

=

0.52

−

p

[Using p + q = 0.52]

or

q = 13

60

p = 91

300

Notes

(a) B1

B and C seen. If they include A then B0

(b) 1st M1 for a correct equation for P(C) using independence.

1st A1 for P(C) = 0.2 correct linear equation for r or s

2nd A1 for either r = 0.07 or s = 0.28

2nd M1 for using

∑

p

=1 Allow letter r and s or their values for r and s provided they are

probabilities.

3rd A1 for both s = 0.28 and r = 0.07

NB: The quotations around the 0.07 ("0.07") imply that we ft their value

(c) 1st B1ft for an expression (in q) for P(

)

B

∪

C

ft their value of r or their "0.2"

eg 0.13 + "their r" + q Implied by 1st or 2nd M1 below.

2nd B1 for a correct expression for P(

A

∩

C′

)

in terms of p and q or 0.52

Implied by 1st or 2nd M1below

1st M1 for a correct use of independence (ft their probabilities), values or letters.

Implied by 2nd M1

2nd M1 using p + q = 0.52 to gain a linear equation in one variable

1st A1 for a correct fraction for q

2nd A1 for a correct fraction for p

SC: If both p and q are given as equivalent

recurring decimals award A0A1 eg 0.216

and 0.303

3 (a) Width = 2.5 (cm)

1.5 cm2 for freq of 5 so 6 1.5

×

= 9 cm2 for freq of 30 or fd

= w ×h = 9

5

3

So h = 9 ÷2.5 or

6 ÷

5

= 3.6 (cm)

3

16

16.5

Q =

[12]

× allow use of (n + 1) giving

3

[12]

×

3

(b)

+

+

2

25

25

= 13.92 = awrt 13.9

∑

(c)(i)

f

x = ×

5 6.5

+

13 9

×

+

16 11

×

+

25 13.5

×

+

30 17.5

×

+

11 24

×

=

1452

x = 14.52 = awrt 14.5

∑

(ii)

f

2

6.5

2

5

9

2

13

11

2

16

13.5

2

25

17.5

2

30

24

2

11

23 280

M1

x =

× +

×

+

×

+

×

+

×

+

×

=

"23280"

σ =

−

("14.52") or 21.9696

2

x

100

σ = 4.687…= awrt 4.69

x

(d)

1

×

13 16

+

+

25

+

30

+ ×

1

11

2

4

So proportion is 80.25 % or 0.8025 awrt 0.803

0.75 11

×

0.75 11

×

(e) Profit =

2.2 "0.8025"

×

+

0.8

×

−

1.2 " 1

×

−

0.8025

+

"

100

100

= 1.6935 awrt 1.7 (p)

Notes

(a) B1

for width = 2.5 (cm)

for sight of 9 cm2 or w×h = 9 or fd

= (o.e.)

M1

5

3

A1

for height = 3.6 (cm)

(b) M1

for 16

25 × or 9

m

−

12

16

3

25 × or

3

=

15

m

9

−

For any correct equation leading to Q2 or correct fraction as part of Q2

A1

for awrt 13.9 (use of (n + 1) giving 13.98 = awrt 14.0)

(c)(i) M1

for attempt at Σfx with at least 3 correct terms or 900 < Σfx < 1800

for info Σfx = 32.5 + 117 + 176 + 337.5 + 525 + 264

A1

for awrt 14.5 (correct answer only 2/2)

(ii) 1st M1

for attempt at Σfx2 with at least 3 correct terms or 20 000 < Σfx2 < 26 000

for info Σfx2 = 211.25 + 1053 + 1936 + 4556.25 + 9187.5 + 6336

(ft their Σfx2 if clear it is Σfx2 ) Do not allow

2nd M1 for a correct expression including

(Σfx)2 for Σfx2

A1 for awrt 4.69 (allow s = 4.7107… awrt 4.71 ) (correct answer only 3/3)

(d) M1

for attempt at a correct expression (allow 1 error or omission) eg

100 – 5

A1

for awrt 80.3% or 0.803

(e) M1

for a correct expression ft their 0.8025 o.e. eg

(

)

2.2

×

100

−

11.5

−

8.25

+

0.8 8.25 1.2 11.5

×

−

×

÷

100

Condone

2.2 "80" 0.8 (8)

×

+

×

−

1.2

×

(

12

)

÷

100

A1

for awrt 1.7 Allow £0.017 (this must have units)

120 165

−