2007 H2 Math Paper 1 Variant

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Questions

Solve the inequality

\frac{ - 6 x^2 - 13 x - 8 }{ - 7 x - 6 } > x.

Functions

{f}

and

{g}

are defined by

\begin{align*} &f : x \mapsto \frac{1}{ 7 x - 8 } && \textrm{for } x \in \mathbb{R}, x \neq \frac{8}{7}, \\ &g : x \mapsto x^2 - 2 && \textrm{for } x \in \mathbb{R}. \end{align*}

(a)

Explain why one of the composite functions

{fg}

and

{gf}

does not exist. Give a definition, including the domain, of the composite that exists.

(b)

Find

{f^{-1}(x)}

and state the domain of

{f^{-1}.}

The complex number

{z}

is such that

{zz^* - 4 iz = 33 + 4 \mathrm{i},}

where

{z^*}

is the complex conjugate of

{z.}

Find

{z}

in the form

{x+y \mathrm{i},}

where

{x}

and

{y}

are real.

The variables

{x}

and

{y}

satisfy the differential equation

8 \frac{\mathrm{d}y}{\mathrm{d}x} = 3 - 9 y.

(a)

Find

{y}

in terms of

{x,}

given that

{y = \frac{10}{9}}

when

{x = 0.}

(b)

State what happens to

{y}

for large values of

{x.}

(a)

Write

y = \frac{ - 3 x + 5 }{ x - 1 }

in the form

y = A + \frac{B}{x - 1}

and describe a sequence of transformations which transform the graph of

y = \frac{ 1 }{ x }

to the graph of

y = \frac{ - 3 x + 5 }{ x - 1 }.

(b)

Sketch the graph of

y = \frac{ - 3 x + 5 }{ x - 1 },

giving the equations of any asymptotes and the coordinates of any axial intercepts.

Referred to the origin

{O,}

the position vectors of the points

{A}

and

{B}

are

8 \mathbf{i} - 6 \mathbf{j} + 7 \mathbf{k} \quad \textrm{ and } \quad - 4 \mathbf{i} + 4 \mathbf{j} + 8 \mathbf{k}

respectively.

(a)

Show that

{OA}

and that

{OB}

are perpendicular.

(b)

Find the position vector of the point

{V}

on the line segment

{AB}

such that

{AV:VB = 4:5.}

(c)

The point

{C}

has position vector

{8 \mathbf{i} - 7 \mathbf{j} - \mathbf{k}.}

Use a vector product to find the exact area of triangle

{OBC}

The polynomial

{P(z)}

has real coefficients. The equation

{P(z)=0}

has a root

{r \mathrm{e}^{\mathrm{i} \theta},}

where

{r > 0}

and

{-\pi < \theta < \pi,}

{\theta \neq 0.}

(a)

Write down a second root in terms of

{r}

and

{\theta,}

and hence show that a quadratic factor of

{P(z)}

{z^2 - 2 r z \cos \theta + r^2.}

(b)

Let

{w = 2 \mathrm{e}^{\frac{3}{4} \pi \mathrm{i}}.}

Show that

{w^4 = -k,}

where

{k}

is a positive real number to be determined.

(c)

It is given that

{2 \mathrm{e}^{- \frac{1}{4} \pi \mathrm{i}}}

is a root of

{z^4 = -16.}

Use this information and parts (i) and (ii) to write

{z^4 + 16}

as a product of two quadratic factors with real coefficients, giving each factor in non-trigonometric form.

The line

{l}

passes through the points

{A\left( 2 , 2 , - 3 \right)}

and

{B\left( 2 , - 1 , 2 \right).}

The plane

{p}

has equation

{- 9 x + 18 y + 3 z = - 4.}

Find

(a)

the coordinates of the point of intersection of

{l}

and

{p,}

(b)

the acute angle between

{l}

and

{p}

correct to 1 decimal place,

(c)

the perpendicular distance from

{A}

{p}

in exact form.

A curve has equation

y = \mathrm{e}^{x} - 4 x.

(a)

Find the exact volume obtained when the region bounded by the curve, the

{x}

-axis and the lines

{x=4}

and

{x=5}

is rotated

{2 \pi}

radians about the

{x}

-axis.

(b)

The two roots of the equation

{\mathrm{e}^{x} - 4 x = 0}

are denoted by

{\alpha}

and

{\beta,}

where

{\alpha < \beta.}

Find the values of

{\alpha}

and

{\beta,}

each correct to 3 decimal places.

(c)

Solve

{\mathrm{e}^{x} - 4 x < 0.}

(d)

Find the area bounded by the curve and the

{x}

-axis, giving your answer correct to 2 decimal places.

A geometric series has common ratio

{r,}

and an arithmetic series has first term

{a}

and common difference

{d,}

where

{a}

and

{d}

are non-zero. The first three terms of the geometric series are equal to the first, ninth and eleventh terms respectively of the arithmetic series.

(a)

Show that

{8 r^2 - 10 r + 2 = 0.}

(b)

Deduce that the geometric series is convergent, and find, in terms of

{a,}

the sum to infinity.

(c)

The sum of the first

{n}

terms of the arithmetic series is denoted by

{S.}

Given that

{a>0,}

find the set of possible values of

{n}

for which

{S}

exceeds

{2 a.}

A curve has parametric equations

x = \sin^2 t, \quad y = \cos^9 t, \quad \textrm{for } 0 \leq t \leq {\textstyle \frac{\pi}{2}}.

(a)

Sketch the curve

(b)

The tangent to the curve at the point

{( \sin^2 \theta, \cos^9 \theta ),}

where

{0 < \theta < \frac{1}{2} \pi,}

meets the

{x}

- and

{y \textrm{-axes }}

{P}

and

{Q}

respectively. The origin is denoted by

{O.}

Show that the area of triangle

{OPQ}

\frac{1}{36} \cos^{7} \theta ( 9 \sin^2 \theta + 2 \cos^2 \theta )^2.

(c)

Show that the area under the curve for

{0 \leq t \leq \frac{\pi}{2}}

2 \int_0^{\frac{\pi}{2}} \sin t \cos^{10} t \, \mathrm{d} t,

and use the substitution

{u = \cos t}

to find this area.

Answers

{x < - \frac{6}{7}.}

(a)

{fg}

does not exist because

R_g = {\left[ 0, \infty \right]} \not \subseteq {\left( -\infty , \frac{8}{7} \right) \cup \left( \frac{8}{7} , \infty \right)} = D_f.

\displaystyle gf : x \mapsto \left( \frac{1}{ 7 x - 8 } \right)^2 - 2, \quad \allowbreak {\textrm{for } x \in \mathbb{R}, x \neq \frac{8}{7}.}

(b)

f^{-1}(x) = \frac{1}{7} \frac{1}{ x } + \frac{8}{7}.

D_{ f^{-1} } = R_f = \left( -\infty, 0 \right) \cup \left( 0, \infty \right).

{z = - 1 + 4 \mathrm{i}.}

(a)

{y = \frac{1}{3} + \frac{7}{9} \mathrm{e}^{- \frac{9}{8} x}.}

(b)

{y \to \frac{1}{3}.}

{y = -3 + \frac{ 2 }{ x - 1 }}

Transformation 1: Translate by 1 units in the positive

{x}

-axis direction.
Transformation 2: Scale by a factor of

{2}

parallel to the

{y}

-axis.
Transformation 3: Translate by 3 units in the negative

{y}

-axis direction.

(b)

{\frac{1}{9} ( 24 \mathbf{i} - 14 \mathbf{j} + 67 \mathbf{k} )}

(c)

{2 \sqrt{395}}

(b)

{k = 16.}

(c)

{z^4 + 16} = {(z^2 + 2 \sqrt{2} z + 4)}\allowbreak{(z^2 - 2 \sqrt{2} z + 4)}{}.

(a)

{\left( 2 , 1 , - \frac{4}{3} \right).}

(b)

{19.2^{\circ}.}

(c)

{\frac{13}{138} \sqrt{46}\textrm{ units}.}

(a)

\pi \left( \frac{1}{2} \mathrm{e}^{10} - \frac{1}{2} \mathrm{e}^{8} - 32 \mathrm{e}^{5} + 24 \mathrm{e}^{4} + \frac{976}{3} \right) \allowbreak \textrm{ units}^3.

(b)

{\alpha = 0.357, \, \beta = 2.153.}

(c)

{0.357 < x < 2.153.}

(d)

{1.83\textrm{ units}^2}

(b)

The geometric series is convergent because

{-1 < r = \frac{1}{4} < 1.}

{S_\infty = \frac{4}{3} a.}

(c)

{\{ n \in \mathbb{Z}: 3 \leq n \leq 20 \}.}

{\frac{2}{11} \textrm{ units}^2.}