Mark Scheme for PU3H2 Prelim 2_2010
1 / (i) Express in partial fractions.[2](ii)Hence find in terms of n.[3]
(There is no need to express your answer as a single algebraic fraction.)
(iii)Deduce the value of .[2]
[2] / (i)
By cover-up rule,
[3] / (ii)
[2] / (iii)
2 / Functions f and g are defined by
(i)Sketch the graphs of f and g on the same diagram, indicating clearly the equations of any asymptotes and the coordinates of any turning points. Hence, or otherwise, solve . [4]
(ii)Show that the composite function gf exists and find an expression for .[2]
(iii)If the domain of g is restricted to the set , find the least value of a for which exists. Hence, find and state its domain. [4]
[4]
[2]
[4] / (i)
For , by GC,
(ii)Rf = (10, ∞) Dg =
exists. (shown)
(iii)From the graph of g above, exits if .
Hence, the least value of .
Let
, since
,
3 / The fourth, ninth and nineteenth term of an arithmetic progression are consecutive terms of a geometric progression.
(i)Show that the common ratio of the geometric progression is 2.[3]
(ii)The twentieth term of the arithmetic progression is 63. Find its first term and
common difference.[3]
(iii)The sum of the first n terms of the arithmetic progression is denoted by . Using
the results in (ii), find the least value of n for which exceeds 200.[3]
[3] / (i)Let the AP with T1 = a, common difference = d.
T4, T9, T19 are consecutive terms of a GP :
Since
[3] / (ii)Given
Substitute (1) into (2),
[3] / (iii)
(NA) or
Hence, least
4 / The curve C has equation
where a and b are positive constants.
Given that the curve passes through the point and the equations of its asymptotes are and , show that and . [4]
Hence sketch C, stating the equations of any asymptotes and the coordinates of any points of intersection with the axes. [3]
[4] /
Asymptotes :
Comparing with and ,
------(1)
And the curve passes through the point ,
(shown), since a is positive.
Substitute into (1), (shown)
[3] /
5 / The equation of a curve Cis , where p is a constant. Find .[2]
It is given that C has a tangent which is parallel to the y-axis. Show that the y-coordinate of the point of contact of the tangent with C must satisfy
.
Hence show that .[3]
Find the values of p in the case where the line is a tangent to C.[3]
It is given instead that C has a tangent which is parallel to the x-axis. Show that in this case also. [2]
[10] /
If tangent is parallel to the y-axis, .
Substitute into (1),
(shown)
For the point of contact of the tangent with C,
(shown)
When is a tangent to C,
When y = 2, .
When y = −2,
If the tangent is parallel to the x-axis, .
Substitute into (1),
For the point of contact of the tangent with C,
(shown)
6 / A disease is spreading through a population of N individuals. It is given that the rate of increase of the number of infected individuals at any time is proportional to the product of the number of infected individuals and the number of uninfected individuals at that time. At any time t, x is the number of infected individuals.
Given that initially only one person is infected, show that , where k is a positive constant.[7]
[7] / , k is a positive constant.
When t = 0, x = 1,
(shown)
7 / (i)Given that , find . Hence obtain the first three non-zero terms in the Maclaurin’s series for . [5]
(ii)Hence, or otherwise, show that the first three non-zero terms in the expansion of are , where a and b are constants to be found. [3]
[5] / (i)
[3] / (ii)
8 / The diagram below shows the graph of with a vertical asymptote . The points and are the point of inflexion and the minimum point respectively.
Sketch, on separate diagrams, the graphs of
(i),[3]
(ii).[3]
[3]
[3] / 8(i)
(ii)
9 / Given that the plane π : and the line l : intersect at a point, show that .[2]
Find
(i)the value of when l lies in π,[2]
(ii)the position vector of the point of intersection of l and π.[2]
[2] /
Substitute (2) into (1),
(Shown)
[2] / (i)If l is in , is on , .
Hence, has infinitely many solutions.
[2] / (ii)When l and π intersect at one point, , from above. Substitute this into the equation of the line l, the position vector required is
10 / (a)Given that the equation has aroot of the form , wherek is a non-zero real number, find the possible values of k.
Hence solve the equation .[5]
(b)In an Argand diagram, the point P represents the complex number z such that
and .
(i)Sketch the locus of P.[3]
(ii)Hence, or otherwise, show that .[3]
[5] / (a) Given that is a root, so substitute into the given equation
Comparing the real or imaginary parts on both sides,
OR,
Hence, is a factor of the given equation
[3] / (b) (i)
[3] / (ii) min. (arg z) = argument of any complex numbers along CD
At A, . -----(1)
max. (arg z) = argument of a,where A ≡ a
Hence, (Shown)
11 / (a)Use the substitution to find the exact value of
.[5]
(b)(i)Find .[4]
(ii)Hence find the exact value of .[2]
[5] / (a)Let
When When
=
=
[4] / (b)(i)
[2] / (b)(ii)
12 /
The diagram shows the curve C with parametric equations
.
(i)Find gradient of the curve at the point where t = 1.[3]
(ii)Show that the cartesian equation of C is .[2]
Two points, P and Q, lie on the curve C with coordinates and respectively. Point R lie on the x-axis. The region S is bounded by the lines QR and PR and the arc PQ of the curve C.
(iii)Find the exact value of the volume of revolution when S is rotated completely about thex– axis. [3]
[3] / (i)
When t = 1,
[2] / (ii)
(1) + (2),
(1) – (2),
(3) (4),
(Shown)
[3] / 12 (iii) Volume of the solid = + volume of cone
=
= or (exact value)
1