2008 H2 Math Paper 1 Variant

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Questions

1
The diagram shows the curve with equation y=x2.{y = x^2.} The area of the region bounded by the curve, the lines x=1,x=2{x = 1, x = 2} and the x{x}-axis is equal to the area of the region bounded by the curve, the lines y=a,y=4{y = a, y = 4} and the y{y}-axis, where a<4.{a < 4.} Find the value of a.{a.}
[4]
2
The sum of the first n{n} terms of a series is given by
Sn=16n(n+1)(4n+5).S_n = \frac{1}{6} n (n+1) (4 n + 5).
(a)
Find the sum from the 10th term to the 24th term (both inclusive) of the series.
[2]
(b)
Find an expression, in terms of n,{n,} for the n{n}th term of the series.
[3]
3
The complex number z{z} is such that zz+2z=3+4i,{zz^* + 2 z = 3 + 4 \mathrm{i},} where z{z^*} is the complex conjugate of z.{z.} Find z{z} in the form x+yi,{x+y \mathrm{i},} where x{x} and y{y} are real.
[4]
4
The variables x{x} and y{y} satisfy the differential equation
4dydx=23y.4 \frac{\mathrm{d}y}{\mathrm{d}x} = 2 - 3 y.
(a)
Find y{y} in terms of x,{x,} given that y=2{y = 2} when x=0.{x = 0.}
[6]
(b)
State what happens to y{y} for large values of x.{x.}
[1]
5
(a)
Write y=2x+7x+2{\displaystyle y = \frac{ 2 x + 7 }{ x + 2 }} in the form y=A+Bx+2{\displaystyle y = A + \frac{B}{x + 2}} and describe a sequence of transformations which transform the graph of y=1x{\displaystyle y = \frac{ 1 }{ x }} to the graph of y=2x+7x+2.{\displaystyle y = \frac{ 2 x + 7 }{ x + 2 }.}
[4]
(b)
Sketch the graph of y=2x+7x+2,{\displaystyle y = \frac{ 2 x + 7 }{ x + 2 },} giving the equations of any asymptotes and the coordinates of any axial intercepts.
[3]
6
Referred to the origin O,{O,} the position vectors of the points A{A} and B{B} are
ij+2k and 2i+4j+k\mathbf{i} - \mathbf{j} + 2 \mathbf{k} \quad \textrm{ and } \quad 2 \mathbf{i} + 4 \mathbf{j} + \mathbf{k}
respectively.
(a)
Show that OA{OA} and that OB{OB} are perpendicular.
[2]
(b)
Find the position vector of the point V{V} on the line segment AB{AB} such that AV:VB=1:2.{AV:VB = 1:2.}
[3]
(c)
The point C{C} has position vector 4i+2j+2k.{- 4 \mathbf{i} + 2 \mathbf{j} + 2 \mathbf{k}.} Use a vector product to find the exact area of triangle OAC{OAC}
[4]
7
The polynomial P(z){P(z)} has real coefficients. The equation P(z)=0{P(z)=0} has a root reiθ,{r \mathrm{e}^{\mathrm{i} \theta},} where r>0{r > 0} and π<θ<π,{-\pi < \theta < \pi,} θ0.{\theta \neq 0.}
(a)
Write down a second root in terms of r{r} and θ,{\theta,} and hence show that a quadratic factor of P(z){P(z)} is z22rzcosθ+r2.{z^2 - 2 r z \cos \theta + r^2.}
[3]
(b)
Let w=2e16πi.{w = 2 \mathrm{e}^{\frac{1}{6} \pi \mathrm{i}}.} Show that w6=k,{w^6 = -k,} where k{k} is a positive real number to be determined.
[4]
(c)
It is given that 2e56πi{2 \mathrm{e}^{- \frac{5}{6} \pi \mathrm{i}}} is a root of z6=64.{z^6 = -64.} Use this information and parts (i) and (ii) to write z6+64{z^6 + 64} as a product of three quadratic factors with real coefficients, giving each factor in non-trigonometric form.
[5]
8
The line l{l} passes through the points A(1,2,4){A\left( 1 , 2 , 4 \right)} and B(2,3,1).{B\left( - 2 , 3 , 1 \right).} The plane p{p} has equation 3xy+2z=17.{3 x - y + 2 z = 17.} Find
(a)
the coordinates of the point of intersection of l{l} and p,{p,}
[5]
(b)
the acute angle between l{l} and p{p} correct to 1 decimal place,
[3]
(c)
the perpendicular distance from A{A} to p{p} in exact form.
[3]
9
A curve has equation
y=ex3x.y = \mathrm{e}^{x} - 3 x.
(a)
Find the exact volume obtained when the region bounded by the curve, the x{x}-axis and the lines x=4{x=4} and x=5{x=5} is rotated 2π{2 \pi} radians about the x{x}-axis.
[5]
(b)
The two roots of the equation ex3x=0{\mathrm{e}^{x} - 3 x = 0} are denoted by α{\alpha} and β,{\beta,} where α<β.{\alpha < \beta.}
Find the values of α{\alpha} and β,{\beta,} each correct to 3 decimal places.
[2]
(c)
Solve ex3x<0.{\mathrm{e}^{x} - 3 x < 0.}
[2]
(d)
Find the area bounded by the curve and the x{x}-axis, giving your answer correct to 2 decimal places.
[3]
10
A geometric series has common ratio r,{r,} and an arithmetic series has first term a{a} and common difference d,{d,} where a{a} and d{d} are non-zero. The first three terms of the geometric series are equal to the first, fourth and sixth terms respectively of the arithmetic series.
(a)
Show that 3r25r+2=0.{3 r^2 - 5 r + 2 = 0.}
[4]
(b)
Deduce that the geometric series is convergent, and find, in terms of a,{a,} the sum to infinity.
[5]
(c)
The sum of the first n{n} terms of the arithmetic series is denoted by S.{S.} Given that a>0,{a>0,} find the set of possible values of n{n} for which S{S} exceeds 4a.{4 a.}
[5]
11
A curve has parametric equations
x=cos2t,y=sin3t,for 0tπ2.x = \cos^2 t, \quad y = \sin^3 t, \quad \textrm{for } 0 \leq t \leq {\textstyle \frac{\pi}{2}}.
(a)
Sketch the curve
[2]
(b)
The tangent to the curve at the point (cos2θ,sin3θ),{( \cos^2 \theta, \sin^3 \theta ),} where 0<θ<12π,{0 < \theta < \frac{1}{2} \pi,} meets the x{x}- and y-axes {y \textrm{-axes }} at P{P} and Q{Q} respectively. The origin is denoted by O.{O.} Show that the area of triangle OPQ{OPQ} is
112sinθ(3cos2θ+2sin2θ)2.\frac{1}{12} \sin \theta ( 3 \cos^2 \theta + 2 \sin^2 \theta )^2.
[6]
(c)
Show that the area under the curve for 0tπ2{0 \leq t \leq \frac{\pi}{2}} is 20π2costsin4tdt,{\displaystyle 2 \int_0^{\frac{\pi}{2}} \cos t \sin^{4} t \, \mathrm{d} t,} and use the substitution u=sint{u = \sin t} to find this area.
[5]

Answers

1
8143.{\sqrt[3]{\frac{81}{4}}.}
2
(a)
9485.{9485.}
(b)
un=n(2n+1).{u_n = n (2 n + 1).}
3
z=1+2i.{z = - 1 + 2 \mathrm{i}.}
4
(a)
y=23+43e34x.{y = \frac{2}{3} + \frac{4}{3} \mathrm{e}^{- \frac{3}{4} x}.}
(b)
y23.{y \to \frac{2}{3}.}
5
y=2+3x+2{y = 2 + \frac{ 3 }{ x + 2 }}
Transformation 1: Translate by 2 units in the negative x{x}-axis direction.
Transformation 2: Scale by a factor of 3{3} parallel to the y{y}-axis.
Transformation 3: Translate by 2 units in the positive y{y}-axis direction.
6
(b)
13(4i+2j+5k){\frac{1}{3} ( 4 \mathbf{i} + 2 \mathbf{j} + 5 \mathbf{k} )}
(c)
35{\sqrt{35}}
7
(b)
k=64.{k = 64.}
(c)
z6+64=(z223z+4)(z2+23z+4)(z2+4).{z^6 + 64} = {(z^2 - 2 \sqrt{3} z + 4)}\allowbreak{(z^2 + 2 \sqrt{3} z + 4)}{(z^2 + 4)}.
8
(a)
(52,32,112).{\left( \frac{5}{2} , \frac{3}{2} , \frac{11}{2} \right).}
(b)
78.8.{78.8^{\circ}.}
(c)
4714 units.{\frac{4}{7} \sqrt{14}\textrm{ units}.}
9
(a)
π(12e1012e824e5+18e4+183) units3.\pi \left( \frac{1}{2} \mathrm{e}^{10} - \frac{1}{2} \mathrm{e}^{8} - 24 \mathrm{e}^{5} + 18 \mathrm{e}^{4} + 183 \right) \allowbreak \textrm{ units}^3.
(b)
α=0.619,β=1.512.{\alpha = 0.619, \, \beta = 1.512.}
(c)
0.619<x<1.512.{0.619 < x < 1.512.}
(d)
0.18 units2{0.18\textrm{ units}^2}
10
(b)
The geometric series is convergent because 1<r=23<1.{-1 < r = \frac{2}{3} < 1.}
S=3a.{S_\infty = 3 a.}
(c)
{nZ:6n13}.{\{ n \in \mathbb{Z}: 6 \leq n \leq 13 \}.}
11
25 units2.{\frac{2}{5} \textrm{ units}^2.}