A Level Fur ther Mathematics for AQA

A Level Further Mathematics for AQA Student Book 1
Mixed practice 4
2
1 Find the range of y = 3tanh x + 2.
Choose from these options.
A
2 y < 5
B −1 y 5
C − 1< y < 5
D All real numbers
2 Solve sinh x+ cosh x = k, giving your answer in terms of k.
Choose from these options.
A
arsinh
k
2
3 Simplify tanh (1 + ln p)
.
( ) B k
arcosh (2 ) C e k D ln k
4 Solve cosh ( x + 1) = 3, giving your answer in terms of logarithms.
x −
5 a Express 5sinh x+ cosh x in the form Ae
+ Be
x
, where A and B are integers.
b Solve the equation 5sinh x + cosh x + 5= 0, giving your answer in the form ln a, where a is a
rational number.
[©AQA 2008]
6 a Use the definitions sinh θ =
1
(e θ −e 2 −θ
) and cosh θ =
1
(e θ + e
2 −θ
) to show that 1 + 2sinh θ cosh 2θ
b Solve the equation 3cosh2θ
= 2sinhθ
+ 11, giving each of your answers in the form ln p.
7 Solve sinh 2x = 2cosh x, giving your answer in logarithmic form.
8 Find the exact solutions to cosh 2x+ cosh x =
27
.
8
9 Find the exact solutions to 2cosh x+ sinh x = 2.
10 Prove that 2sinh 2
A≡cosh2A −1.
11 Prove that cosh x > sinh x.
12 Find and simplify an expression for tanh (arsinh x).
13 Use the binomial theorem to show that cosh 4
x ≡
1
+ +
8 cosh 4 x
1
2 cosh 2 x
3
.
8
14 a Sketch the graph of y = tanh x.
[©AQA 2009]
Draft sample
b Given that u = tanh x, use the definitions of sinh x and cosh x in terms of e x −
and e x to show that
x =
1 u
2 ln 1 +
( 1 − u ).
c i Show that the equation
3
+ 7tanh x = 5 can be written as 3tanh 2
x− 7tanh x + 2=
0.
cosh
2
x
ii Show that the equation 3tanh 2
x− 7tanh x + 2=
0has only one solution for x .
Find this solution in the form
1
ln a where a is an integer.
2
90
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