Edexcel A Level Further Maths
Topic questions Pure mathematics Hyperbolic functions
12 questions · 88 marks · about 2 hours
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Determine the exact values of x for which
The curves with equations
intersect at just one point P
Use algebra to show that the x coordinate of P satisfies the equation
Show that ex 
3 is a solution of this equation.
Hence state the exact coordinates of P.
Show that
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Determine the values of x for which
Give your answers in the form qln 2 where q is rational and in simplest form.
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Determine the values of x for which
Give your answers in the form qln 2 where q is rational and in simplest form.
Explain why 
0
cosh x dx is an improper integral.
Show that 
0
cosh x dx is divergent.
4 sinhx = pcoshxwhere p is a real constant
Given that this equation has real solutions, determine the range of possible values for p
4 sinhx = pcoshxwhere p is a real constant
Given that this equation has real solutions, determine the range of possible values for p
Use a hyperbolic substitution and calculus to show that
where k is an arbitrary constant.
Figure 1 shows a sketch of part of the curve C with equation
The finite region R, shown shaded in Figure 1, is bounded by the curve C, the x‑axis and the line with equation x = 3
Using algebraic integration and the result from part (a), show that the area of R is given by
Use a hyperbolic substitution and calculus to show that
where k is an arbitrary constant.
Figure 1 shows a sketch of part of the curve C with equation
The finite region R, shown shaded in Figure 1, is bounded by the curve C, the x‑axis and the line with equation x = 3
Using algebraic integration and the result from part (a), show that the area of R is given by
Solutions based entirely on graphical or numerical methods are not acceptable.
Figure 1 shows a sketch of part of the curve with equation
y = arsinhx x 
0
and the straight line with equation y = β The line and the curve intersect at the point with coordinates (α, β)
The finite region R, shown shaded in Figure 1, is bounded by the curve with equation y = arsinh x, the y-axis and the line with equation y = β
The region R is rotated through 2π radians about the y-axis.
Use calculus to find the exact value of the volume of the solid generated.
Solutions based entirely on graphical or numerical methods are not acceptable.
Figure 1 shows a sketch of part of the curve with equation
y = arsinhx
x 0
and the straight line with equation y = β The line and the curve intersect at the point with coordinates (α, β)
The finite region R, shown shaded in Figure 1, is bounded by the curve with equation y = arsinh x, the y-axis and the line with equation y = β
The region R is rotated through 2π radians about the y-axis.
Use calculus to find the exact value of the volume of the solid generated.
The curve C has equation
Determine, in terms of natural logarithms, the exact x coordinates of the stationary points of C.
The curve C has equation
Determine, in terms of natural logarithms, the exact x coordinates of the stationary points of C.
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