Further Revision Questions Mark scheme
!
1. A function 𝑓 is defined by 𝑓(𝑥) = " ! #$"#%, where 𝑥 ∈ ℝ, 𝑥 ≠ −1, 𝑥 ≠ 3.
(a)
Sketch the curve 𝑦 = 𝑓(𝑥), clearly indicating any asymptotes with their equations.
State the coordinates of any local maximum or minimum points and any points of
intersection with the coordinate axes.
[6]
Markscheme
!
𝑦-intercept "0, − "& A1
!
Note: Accept an indication of − on the 𝑦-axis.
"
vertical asymptotes 𝑥 = −1 and 𝑥 = 3 A1
horizontal asymptote 𝑦 = 0 A1
uses a valid method to find the 𝑥-coordinate of the local maximum point
(M1)
Note: For example, uses the axis of symmetry or attempts to solve 𝑓′(𝑥) =
0.
!
local maximum point "1, − #& A1
Note: Award (M1)A0 for a local maximum point at 𝑥 = 1 and coordinates
not given.
three correct branches with correct asymptotic behaviour and the key
features in approximately correct relative positions to each other A1
[6 marks]
!
2. A function 𝑔 is defined by 𝑔(𝑥) = " ! #$"#%, where 𝑥 ∈ ℝ, 𝑥 > 3.
The inverse of 𝑔 is 𝑔#! .
(b.i)
√'" ! ("
Show that 𝑔#! (𝑥) = 1 + "
.
[6]
Markscheme
!
𝑥= M1
$ ! %&$%"
Note: Award M1 for interchanging 𝑥 and 𝑦 (this can be done at a later
stage).
EITHER
attempts to complete the square M1
𝑦 & − 2𝑦 − 3 = (𝑦 − 1)& − 4 A1
1
𝑥=
(𝑦 − 1)& − 4
! !
(𝑦 − 1)& − 4 = "(𝑦 − 1)& = 4 + & A1
' '
1 4𝑥 + 1
𝑦 − 1 = ±34 + 5= ±3 6
𝑥 𝑥
OR
attempts to solve 𝑥𝑦 & − 2𝑥𝑦 − 3𝑥 − 1 = 0 for 𝑦 M1
%(%&')±+(%&')! ,#'("',!)
𝑦= &'
A1
Note: Award A1 even if − (in ±) is missing
&'±√!.' ! ,#'
= &'
A1
THEN
√#' ! ,'
=1± '
A1
√#' ! ,'
𝑦 > 3 and hence 𝑦 = 1 − is rejected R1
'
Note: Award R1 for concluding that the expression for 𝑦 must have the ‘+’
sign.
The R1 may be awarded earlier for using the condition 𝑥 > 3.
√4𝑥 & + 𝑥
𝑦 =1+
𝑥
√#' ! ,'
𝑔%! (𝑥) = 1 + '
AG
[6 marks]
(b.ii)
State the domain of 𝑔#! .
[1]
Markscheme
domain of 𝑔%! is 𝑥 > 0 A1
[1 mark]
"
A function ℎ is defined by ℎ(𝑥) = arctan $, where 𝑥 ∈ ℝ.
(c)
)
Given that (ℎ ∘ 𝑔)(𝑎) = ' , find the value of 𝑎.
*
Give your answer in the form 𝑝 + $ √𝑟, where 𝑝, 𝑞, 𝑟 ∈ ℤ( .
[7]
Markscheme
attempts to find (ℎ ∘ 𝑔)(𝑎) (M1)
/(0) !
(ℎ ∘ 𝑔)(𝑎) = arctan " & =(ℎ ∘ 𝑔)(𝑎) = arctan "&(0! %&0%")&> (A1)
&
𝑔(𝑎) 𝜋 1 𝜋
arctan = >= @arctan @ A= A
2 4 2(𝑎 − 2𝑎 − 3)
& 4
attempts to solve for 𝑔(𝑎) M1
1
⇒ 𝑔(𝑎) = 2 @ = 2A
(𝑎& − 2𝑎 − 3)
EITHER
⇒ 𝑎 = 𝑔%! (2) A1
attempts to find their 𝑔%! (2) M1
+#(&)! ,&
𝑎 =1+ A1
&
Note: Award all available marks to this stage if 𝑥 is used instead of 𝑎.
OR
⇒ 2𝑎& − 4𝑎 − 7 = 0 A1
attempts to solve their quadratic equation M1
%(%#)±+(%#)! ,#(&)(1) #±√1&
𝑎= #
"= #
& A1
Note: Award all available marks to this stage if 𝑥 is used instead of 𝑎.
THEN
"
𝑎 = 1 + & √2 (as 𝑎 > 3) A1
(𝑝 = 1, 𝑞 = 3, 𝑟 = 2)
!
Note: Award A1 for 𝑎 = 1 + & √18 (𝑝 = 1, 𝑞 = 1, 𝑟 = 18)
[7 marks]
" ! #!'"($'
Consider the function defined by 𝑓(𝑥) $"(+
, where 𝑥 ∈ ℝ, 𝑥 ≠ −3.
(a)
State the equation of the vertical asymptote on the graph of 𝑓.
[1]
Markscheme
(vertical asymptote equation) 𝑥 = −3 A1 Note: Accept 2𝑥 +
6 = 0 or equivalent.
[1 mark]
(b)
Find the coordinates of the points where the graph of 𝑓 crosses the 𝑥-axis.
[2]
Markscheme
(2,0) and (12,0) A1A1 Note: Award A1 for (2,0) and A1
for (12,0). Award A1A0 if only 𝑥 values are given.
[2 marks]
The graph of 𝑓 also has an oblique asymptote of the form 𝑦 = 𝑎𝑥 + 𝑏, where 𝑎, 𝑏 ∈ ℚ.
(c)
Find the value of 𝑎 and the value of 𝑏.
[4]
Markscheme
! ' ! % !#' , &#
METHOD 1 𝑎 = & A1 attempt at ‘long division’ on &' , .
' ! % !#' , &# ! !1 … !1
(M1) = 𝑥 − "+ & (A1) 𝑏 = −
&' , . & & &' , . &
! !1 !
A1 Note: Accept 𝑦 = &
𝑥 − &
. METHOD 2 𝑎 = & A1
'! % !#' , &# ! 4
&' , .
≡ & 𝑥 + 𝑏 + &' , . (A1) 𝑥 & − 14𝑥 + 24 ≡
!
&
𝑥(2𝑥 + 6) + 𝑏(2𝑥 + 6) + 𝑐 attempt to equate coefficients of 𝑥:
!1 ! !1
(M1) −14 = 3 + 2𝑏 𝑏 = − &
Note: Accept 𝑦 = &
𝑥 − &
. METHOD
! ' ! % !#' , &# ! %!1 , &#
3𝑎=& A1 &' , .
− & 𝑥 ≡ &' , . (A1) attempt
to find the limit of 𝑓(𝑥) − 𝑎𝑥 as 𝑥 → ∞ (M1) 𝑏 =
%!1' , &# !1 ! !1
lim &' , . = −& A1 Note: Accept 𝑦 = & 𝑥 − & .
'→6
[4 marks]
(d)
Sketch the graph of 𝑓 for −50 ≤ 𝑥 ≤ 50, showing clearly the asymptotes and any
intersections with the axes.
[4]
Markscheme
two branches with approximately correct shape (for −50 ≤ 𝑥 ≤ 50)
A1 Note: For this A1 the graph must be a function. their vertical and
oblique asymptotes in approximately correct positions with both branches
showing correct asymptotic behaviour to these asymptotes
A1A1 Note: Award A1 for vertical asymptote and behaviour and A1 for
oblique asymptote and behaviour. If only top half of the graph seen only
award A1A0 if both asymptotes and behaviour are seen. their axes
intercepts in approximately the correct positions A1 Note:
Points of intersection with the axes and the equations of asymptotes do not
need to be labelled. Ignore incorrect labels
[4 marks]
(e)
Find the range of 𝑓.
[4]
Markscheme
S−10 − 5√3 =T − 18.6602 … OR S− 10 + 5√3 =T − 1.33974 … seen
anywhere (A1) attempt to write the range using at least one
value in an interval or an inequality in 𝑦 or 𝑓(𝑥) (M1) 𝑦 ≤
−18.7, 𝑦 ≥ −1.34 A1A1 Note: Award A1 for each inequality.
Award A1A0 for strict inequalities in both. Do not award FT from (d). Accept
equivalent set notation.
[4 marks]
(f)
Solve the inequality 𝑓(𝑥) > 𝑥.
[4]
Markscheme
S−10 − 2√31 =T − 21.1355 … OR S−10 + 2√31 =T1.13522 … seen
anywhere (A1) 𝑥 < −21.1, −3 < 𝑥 < 1.14
A1A1A1 Note: Award A1 for 𝑥 < −21.1, A1 for correct endpoints of a
single interval −3 and 1.14 and for A1 for −3 < 𝑥 < 1.14. Do not award
FT from (d). Accept equivalent set notation.
[4 marks]
3.
4.
5.
