Question 9 - A-Level Maths

AQA C2 2009 June — Question 9 10 marks

Exam Board	AQA
Module	C2 (Core Mathematics 2)
Year	2009
Session	June
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Laws of Logarithms
Type	Two unrelated log parts: one non-log algebraic part
Difficulty	Moderate -0.8 This is a straightforward C2 logarithms and indices question testing standard rules and techniques. Part (a) involves simple index manipulation (√125 = 125^(1/2) = 5^(3/2)), part (b) uses logarithms in a routine way to solve an exponential equation, and part (c) applies logarithm laws mechanically. All parts are textbook exercises requiring recall and direct application of rules with no problem-solving insight needed, making this easier than average for A-level.
Spec	1.06a Exponential function: a^x and e^x graphs and properties 1.06d Natural logarithm: ln(x) function and properties 1.06f Laws of logarithms: addition, subtraction, power rules 1.06g Equations with exponentials: solve a^x = b

1. Find the value of $p$ for which $\sqrt{125} = 5^p$. [2]
2. Hence solve the equation $5^{2x} = \sqrt{125}$. [1]
Use logarithms to solve the equation $3^{2x-1} = 0.05$, giving your value of $x$ to four decimal places. [3]
It is given that $$\log_a x = 2(\log_a 3 + \log_a 2) - 1$$ Express $x$ in terms of $a$, giving your answer in a form not involving logarithms. [4]

Show mark scheme Show mark scheme source

Question 9:

9

TOTAL

Answer all questions in the spaces provided.

1 The triangle ABC, shown in the diagram, is such that AB ¼ 7cm, AC ¼ 5cm,

BC ¼ 8cm and angle ABC ¼ y.

A

7cm

5cm

y

B 8cm C

(a) Show that y ¼ 38:2(cid:1), correct to the nearest 0.1(cid:1). (3 marks)

(b) Calculate the area of triangle ABC, giving your answer, in cm2, to three significant

figures. (2 marks)

QUESTION

PART

REFERENCE

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Answer	Marks
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1 n

2 (a) Write down the value of n given that ¼ x . (1 mark)

x4

(cid:1) (cid:2)2

3 (b) Expand 1þ . (2 marks)

x2

ð(cid:1) (cid:2)2

3 (c) Hence find 1þ dx. (3 marks)

x2

ð3(cid:1) (cid:2)2

3 (d) Hence find the exact value of 1þ dx. (2 marks)

x2

1 QUESTION

PART

REFERENCE

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Answer	Marks
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3 The nth term of a sequence is u .

n

The sequence is defined by

u ¼ ku þ12

nþ1 n

where k is a constant.

The first two terms of the sequence are given by

u ¼ 16 u ¼ 24

1 2

(a) Show that k ¼ 0:75. (2 marks)

(b) Find the value of u and the value of u . (2 marks)

3 4

(c) The limit of u as n tends to infinity is L.

n

(i) Write down an equation for L. (1 mark)

(ii) Hence find the value of L. (2 marks)

QUESTION

PART

REFERENCE

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4 (a) Use the trapezium rule with four ordinates (three strips) to find an approximate value

ð6

pffiffiffiffiffiffiffiffiffiffiffiffiffi

for x3 þ1dx, giving your answer to four significant figures. (4 marks)

0 pffiffiffiffiffiffiffiffiffiffiffiffiffi

(b) The curve with equation y ¼ x3 þ1 is stretched parallel to the x-axis with scale

1 factor to give the curve with equation y ¼ fðxÞ. Write down an expression for

2 fðxÞ. (2 marks)

QUESTION

PART

REFERENCE

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Answer	Marks
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5 The diagram shows part of a curve with a maximum point M.

y

M

O 15 x

The equation of the curve is

3 5

y ¼ 15x2 (cid:1)x2

dy

(a) Find . (3 marks)

dx

(b) Hence find the coordinates of the maximum point M. (4 marks)

(c) The point Pð1, 14Þ lies on the curve. Show that the equation of the tangent to the

curve at P is y ¼ 20x(cid:1)6. (3 marks)

(d) The tangents to the curve at the points P and M intersect at the point R. Find the

length of RM . (3 marks)

QUESTION

PART

REFERENCE

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6 The diagram shows a sector OAB of a circle with centre O and radius rcm.

A

rcm

O 1.2

rcm

B

The angle AOB is 1.2 radians. The area of the sector is 33.75cm2.

Find the perimeter of the sector. (6 marks)

QUESTION

PART

REFERENCE

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Answer	Marks
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7 A geometric series has second term 375 and fifth term 81.

(a) (i) Show that the common ratio of the series is 0.6. (3 marks)

(ii) Find the first term of the series. (2 marks)

(b) Find the sum to infinity of the series. (2 marks)

1 X

(c) The nth term of the series is u . Find the value of u . (4 marks)

n n

n¼6

QUESTION

PART

REFERENCE

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Answer	Marks
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siny (cid:1)cosy

8 (a) Given that ¼ 4, prove that tany ¼ 5. (2 marks)

cosy

(b)(i) Use an appropriate identity to show that the equation

2 2cos x(cid:1)sinx ¼ 1

can be written as

2sin 2 xþsinx(cid:1)1 ¼ 0 (2 marks)

(ii) Hence solve the equation

2 2cos x(cid:1)sinx ¼ 1

giving all solutions in the interval 0(cid:1)4x4360(cid:1). (5 marks)

QUESTION

PART

REFERENCE

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Answer	Marks
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p

ffiffiffiffiffiffiffiffi p

9 (a) (i) Find the value of p for which 125 ¼ 5 . (2 marks)

p

(ii) Hence solve the equation 52x ¼ ffi 1 ffiffi 2 ffiffi 5 ffiffiffi . (1 mark)

(b) Use logarithms to solve the equation 32x(cid:1)1 ¼ 0:05, giving your value of x to four

decimal places. (3 marks)

(c) It is given that

log x ¼ 2ðlog 3þlog 2Þ(cid:1)1

a a a

Express x in terms of a, giving your answer in a form not involving logarithms.

(4 marks)

QUESTION

PART

REFERENCE

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Question 9:
9
TOTAL
Answer all questions in the spaces provided.
1 The triangle ABC, shown in the diagram, is such that AB ¼ 7cm, AC ¼ 5cm,
BC ¼ 8cm and angle ABC ¼ y.
A
7cm
5cm
y
B 8cm C
(a) Show that y ¼ 38:2(cid:1), correct to the nearest 0.1(cid:1). (3 marks)
(b) Calculate the area of triangle ABC, giving your answer, in cm2, to three significant
figures. (2 marks)
QUESTION
PART
REFERENCE
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1
n
2 (a) Write down the value of n given that ¼ x . (1 mark)
x4
(cid:1) (cid:2)2
3
(b) Expand 1þ . (2 marks)
x2
ð(cid:1) (cid:2)2
3
(c) Hence find 1þ dx. (3 marks)
x2
ð3(cid:1) (cid:2)2
3
(d) Hence find the exact value of 1þ dx. (2 marks)
x2
1
QUESTION
PART
REFERENCE
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3 The nth term of a sequence is u .
n
The sequence is defined by
u ¼ ku þ12
nþ1 n
where k is a constant.
The first two terms of the sequence are given by
u ¼ 16 u ¼ 24
1 2
(a) Show that k ¼ 0:75. (2 marks)
(b) Find the value of u and the value of u . (2 marks)
3 4
(c) The limit of u as n tends to infinity is L.
n
(i) Write down an equation for L. (1 mark)
(ii) Hence find the value of L. (2 marks)
QUESTION
PART
REFERENCE
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4 (a) Use the trapezium rule with four ordinates (three strips) to find an approximate value
ð6
pffiffiffiffiffiffiffiffiffiffiffiffiffi
for x3 þ1dx, giving your answer to four significant figures. (4 marks)
0
pffiffiffiffiffiffiffiffiffiffiffiffiffi
(b) The curve with equation y ¼ x3 þ1 is stretched parallel to the x-axis with scale
1
factor to give the curve with equation y ¼ fðxÞ. Write down an expression for
2
fðxÞ. (2 marks)
QUESTION
PART
REFERENCE
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5 The diagram shows part of a curve with a maximum point M.
y
M
O 15 x
The equation of the curve is
3 5
y ¼ 15x2 (cid:1)x2
dy
(a) Find . (3 marks)
dx
(b) Hence find the coordinates of the maximum point M. (4 marks)
(c) The point Pð1, 14Þ lies on the curve. Show that the equation of the tangent to the
curve at P is y ¼ 20x(cid:1)6. (3 marks)
(d) The tangents to the curve at the points P and M intersect at the point R. Find the
length of RM . (3 marks)
QUESTION
PART
REFERENCE
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6 The diagram shows a sector OAB of a circle with centre O and radius rcm.
A
rcm
O 1.2
rcm
B
The angle AOB is 1.2 radians. The area of the sector is 33.75cm2.
Find the perimeter of the sector. (6 marks)
QUESTION
PART
REFERENCE
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7 A geometric series has second term 375 and fifth term 81.
(a) (i) Show that the common ratio of the series is 0.6. (3 marks)
(ii) Find the first term of the series. (2 marks)
(b) Find the sum to infinity of the series. (2 marks)
1
X
(c) The nth term of the series is u . Find the value of u . (4 marks)
n n
n¼6
QUESTION
PART
REFERENCE
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siny (cid:1)cosy
8 (a) Given that ¼ 4, prove that tany ¼ 5. (2 marks)
cosy
(b)(i) Use an appropriate identity to show that the equation
2
2cos x(cid:1)sinx ¼ 1
can be written as
2sin 2 xþsinx(cid:1)1 ¼ 0 (2 marks)
(ii) Hence solve the equation
2
2cos x(cid:1)sinx ¼ 1
giving all solutions in the interval 0(cid:1)4x4360(cid:1). (5 marks)
QUESTION
PART
REFERENCE
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p
ffiffiffiffiffiffiffiffi p
9 (a) (i) Find the value of p for which 125 ¼ 5 . (2 marks)
p
(ii) Hence solve the equation 52x ¼ ffi 1 ffiffi 2 ffiffi 5 ffiffiffi . (1 mark)
(b) Use logarithms to solve the equation 32x(cid:1)1 ¼ 0:05, giving your value of x to four
decimal places. (3 marks)
(c) It is given that
log x ¼ 2ðlog 3þlog 2Þ(cid:1)1
a a a
Express x in terms of a, giving your answer in a form not involving logarithms.
(4 marks)
QUESTION
PART
REFERENCE
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Show LaTeX source

\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $p$ for which $\sqrt{125} = 5^p$. [2]
\item Hence solve the equation $5^{2x} = \sqrt{125}$. [1]
\end{enumerate}
\item Use logarithms to solve the equation $3^{2x-1} = 0.05$, giving your value of $x$ to four decimal places. [3]
\item It is given that
$$\log_a x = 2(\log_a 3 + \log_a 2) - 1$$
Express $x$ in terms of $a$, giving your answer in a form not involving logarithms. [4]
\end{enumerate}

\hfill \mbox{\textit{AQA C2 2009 Q9 [10]}}

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Q1 5 Q2 8 Q3 7 Q4 6 Q5 13 Q6 6 Q7 11 Q8 9 Q9 10