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Batory AA HL Paper 3 Homework September 1, 2020

Name:

1. (25 points) This question investigates a special cases of cubic polynomials which can be solved with the aid of trigonometric identities.

(a) Show that 1 +

2 is a solution of the equation:

2x

3

− (2

2 + 6)x

2

+ (4

2 + 5)x −

2 − 1 = 0

and hence find the other two solutions. Your answers should be exact.

(b) Solve the equation

2x

3

− 5x

2

− 6x + 9 = 0 and let x = 3y to obtain the equation:

6y

3

− 5y

2

− 2y + 1 = 0 and write down its solutions.

(c) Show that cos 15

=

3 + 1

2

2 and find a similar expression for sin 15

. (d) Express cos 3α in terms of cos α and hence show that x = cos α is a solution to the equation

4x

3

− 3x − cos 3α = 0

and find the other two solutions in terms of cos α and sin α.

(e) Use parts (c) and (d) and a substitution x = ky for suitable value of k to solve the equation:

y

3

− 3y −

2 = 0

Give your answers in surd form.

(2)

Batory AA HL Paper 3 Homework, page 2 of 3 September 1, 2020

2. (30 points) The question investigates the hyperbolic functions and their graphs.

The hyperbolic functions are defined as follows:

sinh x = e

x

− e

−x

2 cosh x = e

x

+ e

−x

2 tanh x = sinh x cosh x (a) Find cosh x + sinh x and cosh x − sinh x and hence prove that

cosh

2

x − sinh

2

x = 1 (b) Show that:

(i) (sinh x)

0

= cosh x,

(ii) (cosh x)

0

= sinh x.

(c) Decide if sinh x and cosh x are even, odd or neither. Justify your an- swer.

(d) Find the coordinates of any stationary points and inflexion points on the graphs of sinh x and cosh x.

(e) Sketch the graphs of sinh x and cosh x.

(f) Prove the following identities:

sinh(x + y) = sinh x cosh y + cosh x sinh y

cosh(x + y) = cosh x cosh y + sinh x sinh y

(3)

Batory AA HL Paper 3 Homework, page 3 of 3 September 1, 2020

(g) The hyperbolic function sech x is defined by sech x = 1

cosh x . Show that (tanh x)

0

= sech

2

x.

(h) Using the graph of cosh x sketch the graphs of sech x and sech

2

x.

(i) Calculate tanh(0).

(j) State the equation of any asymptotes of the graph of tanh x.

(k) Using parts (h), (i) and (j) sketch the graph of tanh x.

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