Mathematics 9709/22 October/November 2020: question paper and mark scheme
Cambridge AS & A Level Mathematics (9709), Paper 2 Pure Mathematics 2, October/November 2020. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
Solve the equation 7 cot 1 = 3 cosec 1 for 0Å < 1 < 90Å. [3]
23x+2 + 8
= 5, find the value of 23x and hence, using logarithms, find the value of x correct
23x − 7
Mark scheme
1 cosθ 1 B1 SOI
Use cot θ = and cosecθ =
sin θ sin θ
Simplify to obtain cos θ = k where 0 < k < 1 M1
3 A1
Obtain cosθ = and hence θ = 64.6 and no other solutions in the
7
range
Alternative method for question 1
Use identity cosec 2θ = 1 + cot 2 θ B1
Simplify to obtain tan θ = k1 or sin θ = k2 where 0 < k2 < 1 M1
1 1 A1
Obtain tan θ = 40 or sin θ = 40 and hence θ = 64.6 and no
3 7
other solutions in the range
3
2 Use 23 x + 2 = 4 × 23 x B1 OE
Solve equation for 23 x M1
Obtain 23 x = 43 A1
Apply logarithms and use power law for 23 x = k where k > 0 M1
Obtain 1.809 A1 AWRT
5
(a) Sketch, on a single diagram, the graphs of y = 12 x − a and y = 32 x − 12 a, where a is a positive constant. [2]
(b) Find the coordinates of the point of intersection of the two graphs. [3]
. .
(c) Deduce the solution of the inequality 12 x − a > 32 x − 12 a. [1]
Mark scheme
3(a) Draw V-shaped graph with vertex on positive x-axis B1
Draw straight line graph correctly positioned with greater gradient B1
2
3(b) 1 3 M1
Solve linear equation with signs ofx and x different
2 2
2 2
1 3 1
or solve non-modulus equation x − a = x − a to obtain
2 2 2
x=
3 A1
Obtain x = a
4
5 A1 And no other point
Obtain y = a
8
3
3(c) 3 B1 FT Following their (single) x-coordinate from part (b)
State x < a
4
1
y
x
O 14
x−2
The diagram shows the curve with equation y = x2 + 8
. The shaded region is bounded by the curve and the lines x = 14 and y = 0.
dy
(a) Find and hence determine the exact x-coordinates of the stationary points. [4]
dx
(b) Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give the answer correct to 2 significant figures. [3]
Mark scheme
4(a) Differentiate using quotient rule (or product rule) *M1
( x 2 + 8) − 2 x( x − 2) A1 OE
Obtain
( x 2 + 8)2
Equate first derivative to zero and attempt solution to get x = … DM1
Obtain 2 ± 12 or exact equivalents A1
4
4(b) 4 8 12 B1 Decimal equivalents need to be to at least 2 decimal places
Use y values (0), , , or decimal equivalents
44 108 204
Use correct formula, or equivalent, with h = 4 M1
4 8 12 A1
Obtain 2 0 + 2 × + 2 × + or equivalent and hence 0.78
44 108 204
3
The equation of a curve is 2e2x y − y3 + 4 = 0.
4e2x y
= 2 dy dx 3y − 2e2x
(a) Show that . [4]
(b) The curve passes through the point 0, 2.
Find the equation of the tangent to the curve at this point, giving your answer in the form ax + by + c = 0. [3]
(c) Show that the curve has no stationary points. [2]
@ A
(a) Find Ô +
8 8
4x + 1 cos2 4x + 1
Mark scheme
5(a) Use product rule to differentiate 2e 2 x y M1 dy
Must be in the form k1 ye 2 x + k 2 e 2 x
dx
dy A1
Obtain 4e 2 x y + 2e 2 x
dx
dy B1
Differentiate − y 3 to obtain −3 y 2
dx
dy 4e2 x y A1 AG
Obtain = 2
dx 3 y − 2e2 x
4
5(b) Substitute 0 and 2 to find gradient of tangent M1
Attempt to find equation of tangent through (0, 2) with numerical M1
gradient
Obtain 4 x − 5 y + 10 = 0 or equivalent of required form A1
3
5(c) Equate numerator of derivative to zero and state that at least one of M1
e2 x and y cannot be zero
Complete argument A1
2
dx. [4]
1π
(b) It is given that Ó 3 + 4 cos2 12 x + k sin 2x dx = 10.
2
0
Find the exact value of the constant k. [6]
Mark scheme
6(a) 8 B1 SOI
Express 2
as 8sec 2 (4 x + 1)
cos (4 x + 1)
Integrate to obtain the form a ln(4 x + 1) M1
Integrate to obtain b tan(4 x + 1) M1
Obtain 2ln ( 4 x + 1) + 2 tan ( 4 x + 1) + c A1 Condone use of brackets rather than modulus signs
4
6(b) 1 M1 For constants where pq ≠ 0
Express 4cos 2 x in the form p + q cos x
2
Obtain correct 2 + 2 cos x A1
Integrate to obtain form px + q sin x + r cos 2 x *M1 For constants where pqr ≠ 0
1 A1 Allow 3x + 2x in place of 5x
Obtain correct 5 x + 2sin x − k cos 2 x
2
Apply limits correctly, equate to 10 and solve for k DM1
5 A1 CWO
Obtain k = 8 − π
2
6
y
x
A O B
A curve has equation y = f x where f x = x4 − 5x3 + 6x2 + 5x − 15. As shown in the diagram, the curve crosses the x-axis at the points A and B with coordinates a, 0 and b, 0 respectively.
(a) Use the factor theorem to show that x − 3 is a factor of f x. [2]
(b) By first finding the quotient when f x is divided by x − 3, show that
_ a=−
5
2−a
. [5]
(c) Use an iterative formula, based on the equation in part (b), to find the value of a correct to
3 significant figures. Give the result of each iteration to 5 significant figures. [3]
Additional Page
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Mark scheme
7(a) Substitute x = 3 and attempt evaluation M1
Obtain 0 and confirm factor x − 3 A1 AG
3
7(b) Divide quartic expression by x − 3 at least as far as x3 + kx2 M1
Obtain x3 − 2 x 2 A1
Obtain x3 − 2 x2 + 5 A1 With no errors seen
Attempt rearrangement of their cubic expression to x = ... M1 Or a = …
5 A1 AG
Confirm a = −
2−a
5
7(c) Use iteration process correctly at least once M1 Need to see 3 values including their starting value.
Obtain final answer −1.24 A1 Answer required to exactly 3 significant figures.
Show sufficient iterations to 5 sf to justify answer or show a sign A1
change in the interval [ −1.245, − 1.235]
3
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Where can I find the 9709/22 October/November 2020 mark scheme?+
On this page, free: the mark scheme for AS & A Level Mathematics 9709/22, October/November 2020 (9709_w20_ms_22), and the question paper (9709_w20_qp_22), to read or download, with each question's marking points written out underneath.
How many marks is 9709/22 out of?+
Mathematics 9709/22 for October/November 2020 is out of 50 marks, with 1 hour 15 minutes to answer it.
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