OCR Further Pure Core 2 2024 June — Question 7 10 marks

Exam BoardOCR
ModuleFurther Pure Core 2 (Further Pure Core 2)
Year2024
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicHyperbolic functions
TypeVolume of revolution with hyperbolics
DifficultyChallenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated techniques: simplifying hyperbolic expressions using exponential definitions, setting up a volume of revolution integral, choosing and executing a non-obvious substitution (likely u = e^x), and manipulating the result to match an inverse tan form. The 7-mark part (b) demands extended reasoning and integration of several concepts, placing it well above average difficulty but not at the extreme end since the techniques are standard for FM students who have studied the relevant material.
Spec4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08d Volumes of revolution: about x and y axes4.08h Integration: inverse trig/hyperbolic substitutions
  1. Express \(17\cosh x - 15\sinh x\) in the form \(e^{-x}(ae^{bx} + c)\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]
A function is defined by \(f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}\). The region bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. In this question you must show detailed reasoning. Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k\pi\tan^{-1} q\) where \(k\) and \(q\) are rational numbers to be determined. [7]
Question 7:
AnswerMarks Guidance
7(a) e x + e − x e x − e − x
1 7 c o s h x − 1 5 s i n h x = 1 7 − 1 5
2 2
= 17−15 e x+ 17+15 e −x( =e x+16e −x)
2 2
= e − x ( e 2 x + 1 6 ) (so a = 1, b = 2 and c = 16)
AnswerMarks
wwwM1
M1
A1
AnswerMarks
[3]3.1a
1.1
AnswerMarks
2.2aSubstituting correct exponential
definitions of hyperbolic
functions into expression
Collecting terms
AnswerMarks
Answer can be embedded. ISW.egsex+te−x where s, t are non-zero
constants, possibly unsimplified.
Y541/01 Mark Scheme June 2024
(b)
18
V
ln
0
3
1 7 c o s h
1
x 1 5 s i n h x
2
d x  = 

−
 M1 1.1 DR. Formula for VoR correctly dx must be seen here but can be
used in solution (limits and/or  implied later.
may come later) soi
ln
0
3
e x ( e
12
x 1 6 )
d x  = 
− +
or
ln
0
3
e 2
ex
x
1 6
d x  
+
M1 1.1 Squaring out and using their part
(a) formula
u = ex M1 3.1a Useful substitution stated or eg u = e–x or ex = 4tanu etc
used
du = exdx => V
...
...
e 2
ex
x
1 6
d x
...
...
u 2
1
1 6
d u    = 
+
= 
+
M1* 1.1 Making the substitution
(including dx) to reduce to exdx=4sec2uduand
integrable form. Ignore limits
here.
V
4
d u
=  if
using ex = 4tanu
...
...
u 2
1
+ 1 6
d u =
 1
4
t a n − 1
u
4
 ...
...
B1FT 1.1 Correct arctan integration Could be implicit in the
(ignore limits etc). FT their 16 substitution.
and 16 (ie  d u = [ u ] if using ex = 4tanu)
x
V
0
4
u
t a n
e 0
1 u
4
1 ,
3
1
x ln
4
3
t a n
u
1 3
4
e ln 3
t a n
3
1 1
4
  
=
=

=
−
=
 =
=

−
=
−
=
− 
PMT
depM1* 1.1 Dealing with limits correctly ln3  e x
(either converting to u-space or Could see tan −1 
substituting back to x-space – if 4 4  0
the latter ignore missing “x =” in
the limits if correct at
resubstitution)
3 1 A1 3.1a Formula for tan(A – B) must be 0.3129987969... alone or used to
−
tan  tan −1 3 −tan −1 1 = 4 4 = 12−4 = 8 used. derive k and/or q gets A0
 4 4 3 1 16+3 19 (NB It is possible to find 8/19
1+ 
4 4 using tan and tan–1 on a
tan −1 3 −tan −1 1 =tan −1 8 calculator)
4 4 19
1 8  1 8 
V = tan −1 so k = and q= 
4 19  4 19
Question 7:
7 | (a) | e x + e − x e x − e − x
1 7 c o s h x − 1 5 s i n h x = 1 7 − 1 5
2 2
= 17−15 e x+ 17+15 e −x( =e x+16e −x)
2 2
= e − x ( e 2 x + 1 6 ) (so a = 1, b = 2 and c = 16)
www | M1
M1
A1
[3] | 3.1a
1.1
2.2a | Substituting correct exponential
definitions of hyperbolic
functions into expression
Collecting terms
Answer can be embedded. ISW. | egsex+te−x where s, t are non-zero
constants, possibly unsimplified.
Y541/01 Mark Scheme June 2024
(b)
18
V
ln
0
3
1 7 c o s h
1
x 1 5 s i n h x
2
d x  = 

−
 M1 1.1 DR. Formula for VoR correctly dx must be seen here but can be
used in solution (limits and/or  implied later.
may come later) soi
ln
0
3
e x ( e
12
x 1 6 )
d x  = 
− +
or
ln
0
3
e 2
ex
x
1 6
d x  
+
M1 1.1 Squaring out and using their part
(a) formula
u = ex M1 3.1a Useful substitution stated or eg u = e–x or ex = 4tanu etc
used
du = exdx => V
...
...
e 2
ex
x
1 6
d x
...
...
u 2
1
1 6
d u    = 
+
= 
+
M1* 1.1 Making the substitution
(including dx) to reduce to exdx=4sec2uduand
integrable form. Ignore limits
here.
V
4
d u

=  if
using ex = 4tanu
...
...
u 2
1
+ 1 6
d u =
 1
4
t a n − 1
u
4
 ...
...
B1FT 1.1 Correct arctan integration Could be implicit in the
(ignore limits etc). FT their 16 substitution.
and 16 (ie  d u = [ u ] if using ex = 4tanu)
x
V
0
4
u
t a n
e 0
1 u
4
1 ,
3
1
x ln
4
3
t a n
u
1 3
4
e ln 3
t a n
3
1 1
4
  
=
=


=
−
=
 =
=


−
=
−
=
− 
PMT
depM1* 1.1 Dealing with limits correctly ln3  e x
(either converting to u-space or Could see tan −1 
substituting back to x-space – if 4 4  0
the latter ignore missing “x =” in
the limits if correct at
resubstitution)
3 1 A1 3.1a Formula for tan(A – B) must be 0.3129987969... alone or used to
−
tan  tan −1 3 −tan −1 1 = 4 4 = 12−4 = 8 used. derive k and/or q gets A0
 4 4 3 1 16+3 19 (NB It is possible to find 8/19
1+ 
4 4 using tan and tan–1 on a
tan −1 3 −tan −1 1 =tan −1 8 calculator)
4 4 19
1 8  1 8 
V = tan −1 so k = and q= 
4 19  4 19
\begin{enumerate}[label=(\alph*)]
\item Express $17\cosh x - 15\sinh x$ in the form $e^{-x}(ae^{bx} + c)$ where $a$, $b$ and $c$ are integers to be determined. [3]
\end{enumerate}

A function is defined by $f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}$. The region bounded by the curve $y = f(x)$, the $x$-axis, the $y$-axis and the line $x = \ln 3$ is rotated by $2\pi$ radians about the $x$-axis to form a solid of revolution $S$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item \textbf{In this question you must show detailed reasoning.}

Use a suitable substitution, together with known results from the formula book, to show that the volume of $S$ is given by $k\pi\tan^{-1} q$ where $k$ and $q$ are rational numbers to be determined. [7]
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 2 2024 Q7 [10]}}
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