CAIE P2 2016 November — Question 3 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2016
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
DifficultyStandard +0.3 Part (i) requires knowing the identity tan²θ = sec²θ - 1 and applying reverse chain rule, which is a standard technique. Part (ii) is straightforward integration of trigonometric functions with careful evaluation at limits. Both parts test routine A-level integration skills with no novel problem-solving required, making this slightly easier than average.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
  1. Find \(\int \tan^2 4x \, dx\). [2]
  2. Without using a calculator, find the exact value of \(\int_0^{\frac{\pi}{2}} (4 \cos 2x + 6 \sin 3x) \, dx\). [3]
Question 3:
(ii) ---
3 (i)
AnswerMarks
(ii)Rewrite integrand as sec24x−1
Integrate to obtain 1tan4x−x, condoning absence of +c
4
Integrate to obtain 2sin2x−2cos3x
Apply limits correctly to integral of form k sin2x+k cos3x
1 2
AnswerMarks
Obtain 3− 2 or exact equivalentB1
B1
B1
M1
AnswerMarks
A1[2]
[3]
4 (i)
(ii)
AnswerMarks
(iii)Substitute x= 1 and equate to zero
2
Obtain a=2
Divide by 2x−1 at least as far as x2 +kx
Obtain quotient x2 +2x+5
Calculate discriminant of 3-term quadratic expression or equivalent
Obtain −16 and conclude appropriately
Use logarithms with power law shown in solving 6y = 1
2
AnswerMarks
Obtain −0.387M1
A1
M1
A1
M1
A1
M1
AnswerMarks
A1[2]
[4]
[2]
5 (i)
(ii)
AnswerMarks
(iii)State or imply correct ordinates 2, 1+e, 1+e2 or decimal equivalents
Use correct formula, or equivalent, correctly with h=3 and three ordinates
Obtain answer 12.25 with no errors seen
Refer to top of each trapezium being above curve or equivalent
State or imply volume is ∫π(1+e 1 3 x)dx
Integrate to obtain form k x+k e 1 3 x with or without π
1 2
Obtain correct π(x+3e 1 3 x) or x+3e 1 3 x
AnswerMarks
Obtain π(3+3e2) or exact equivalentB1
M1
A1
B1
B1
M1
A1
AnswerMarks
A1[3]
[1]
[4]
AnswerMarks Guidance
Page 5Mark Scheme Syllabus
Cambridge International AS Level – October/November 20169709 23
6 (i)
(ii)
AnswerMarks
(iii)State dx = 1
dt t+1
Use product rule for derivative of y
Obtain 2tlnt+t or equivalent
Use dy = dy ÷dx
dx dt dt
Obtain (t+1)(2tlnt+t)
Solve 2lnt+1=0
=e−1
Obtain t 2
Identify t =1 only
AnswerMarks
Obtain 2B1
M1
A1
M1
A1
M1
A1
B1
AnswerMarks
B1[5]
[2]
[2]
7 (i)
(ii)
AnswerMarks
(iii)3 4
State +
cosθ sinθ
Use identity for sin2θ and obtain expression of form asinθ+bcosθ
Obtain 6sinθ+8cosθ
State R=10, following their asinθ+bcosθ
Use appropriate trigonometry to find α
Obtain 53.1(3) with no errors seen
Carry out correct process to find one angle between 0 and 360
Obtain 82.4 or 82.5
Carry out correct process to find second angle between 0 and 360
AnswerMarks
Obtain 351.3 and no others between 0 and 360B1
M1
A1
B1
M1
A1
M1
A1
M1
AnswerMarks
A1[3]
[3]
[4]
Question 3:
--- 3 (i)
(ii) ---
3 (i)
(ii) | Rewrite integrand as sec24x−1
Integrate to obtain 1tan4x−x, condoning absence of +c
4
Integrate to obtain 2sin2x−2cos3x
Apply limits correctly to integral of form k sin2x+k cos3x
1 2
Obtain 3− 2 or exact equivalent | B1
B1
B1
M1
A1 | [2]
[3]
4 (i)
(ii)
(iii) | Substitute x= 1 and equate to zero
2
Obtain a=2
Divide by 2x−1 at least as far as x2 +kx
Obtain quotient x2 +2x+5
Calculate discriminant of 3-term quadratic expression or equivalent
Obtain −16 and conclude appropriately
Use logarithms with power law shown in solving 6y = 1
2
Obtain −0.387 | M1
A1
M1
A1
M1
A1
M1
A1 | [2]
[4]
[2]
5 (i)
(ii)
(iii) | State or imply correct ordinates 2, 1+e, 1+e2 or decimal equivalents
Use correct formula, or equivalent, correctly with h=3 and three ordinates
Obtain answer 12.25 with no errors seen
Refer to top of each trapezium being above curve or equivalent
State or imply volume is ∫π(1+e 1 3 x)dx
Integrate to obtain form k x+k e 1 3 x with or without π
1 2
Obtain correct π(x+3e 1 3 x) or x+3e 1 3 x
Obtain π(3+3e2) or exact equivalent | B1
M1
A1
B1
B1
M1
A1
A1 | [3]
[1]
[4]
Page 5 | Mark Scheme | Syllabus | Paper
Cambridge International AS Level – October/November 2016 | 9709 | 23
6 (i)
(ii)
(iii) | State dx = 1
dt t+1
Use product rule for derivative of y
Obtain 2tlnt+t or equivalent
Use dy = dy ÷dx
dx dt dt
Obtain (t+1)(2tlnt+t)
Solve 2lnt+1=0
=e−1
Obtain t 2
Identify t =1 only
Obtain 2 | B1
M1
A1
M1
A1
M1
A1
B1
B1 | [5]
[2]
[2]
7 (i)
(ii)
(iii) | 3 4
State +
cosθ sinθ
Use identity for sin2θ and obtain expression of form asinθ+bcosθ
Obtain 6sinθ+8cosθ
State R=10, following their asinθ+bcosθ
Use appropriate trigonometry to find α
Obtain 53.1(3) with no errors seen
Carry out correct process to find one angle between 0 and 360
Obtain 82.4 or 82.5
Carry out correct process to find second angle between 0 and 360
Obtain 351.3 and no others between 0 and 360 | B1
M1
A1
B1
M1
A1
M1
A1
M1
A1 | [3]
[3]
[4]
\begin{enumerate}[label=(\roman*)]
\item Find $\int \tan^2 4x \, dx$. [2]

\item Without using a calculator, find the exact value of $\int_0^{\frac{\pi}{2}} (4 \cos 2x + 6 \sin 3x) \, dx$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2016 Q3 [5]}}
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