document stringlengths 121 3.99k | embedding listlengths 384 384 |
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292
Chapter 13
4 Find the following integr
als:
a ∫(4x3 − 3x−4 + r)dx b ∫(x + x − 1 _ 2 + x − 3 _ 2 )dx
c ∫( px4 + 2t + 3x−2)dx
5 Find the following integr
als:
a ∫(3t2 − t−2)dt b ∫(2t2 − 3 t − 3 _ 2 + 1)dt c ∫( pt3 + q2 + px3)dt
6 Find the following integr
als:
a ∫ (2x3 + 3) _______... | [
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-0.06277674436569214,
-0.0893... |
293
Integration
Only one of these curves passes through
this point. Choosing a point on the curve determines the value of c . 13.3 Finding functions
You can find the constant of integration, c, when
you are given (i) any point (x, y) that the curve of the function passes through or (ii) any value that
the function ... | [
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-0.... |
294
Chapter 13
So 5 = 2 __ 5 × 25 − 4 × 2 + c
5 = 64 ___ 5 − 8 + c
5 = 24 ___ 5 + c
So c =
1 __ 5
So y
= 2 __ 5 x 5 __ 2 − 4 x 1 __ 2 + 1 __ 5 Remember 4 5 _ 2 = 25.
Exercise 13C
1 Find the equation of the curv e with the given derivative of y with respec... | [
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0.047931037843227386,
-0.10707344859838486,
0.03536047786474228,
-0.00818... |
295
Integration
7 The displacement of a particle at time
t is given by
the function f(t), where f(0) = 0.
Given that the velocity of the particle is given by f
′(
t) = 10 − 5t,
a find f(t)
b determine the displacement of the particle w
hen t = 3.
8 The height, in metres, of
an arrow fired horizontally from the... | [
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296
Chapter 13
There are three stages when you work out a definite integral:
Write the definite integral
statement with its limits, a and b.Integrate, and write the
integral in square bracketsEvaluate the definite integral
by working out f( b) − f( a).
∫ … dxb
a[ … ]b
a (…) − (…)
■ If f ′(x) is the derivative of f(... | [
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... |
297
Integration
Exercise 13D
1 Evaluate the following definite integrals:
a ∫
2 5 x 3 dx b ∫
1 3 x 4 dx
c ∫
0 4 √ __
x dx d ∫
1 3 3 __ x2 dx
2 Evalua
te the following definite integrals:
a ∫
1 2 ( 2 __ x3 + 3x) dx b ∫
0 2 ( 2x3 − 4x + 5)dx c ∫
4 9 ( √ __
x ... | [
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-0.03814404085278511,
-0.064... |
298
Chapter 13
Example 9
Find the area of the finite region between the curve with equation y = 20 − x − x 2 and the x-axis.
y = 20 − x − x2 = (4 − x )(5 + x )
–5 /four.ss01 O20
xy
Area = ∫ −5 4 ( 20 − x − x2)dx
= [20x − x 2 ___ 2 − x 3 ___ 3 ]
−5 4
= (80
− 8 − 64 ___ 3 ) − (−... | [
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0.005836826749145985,
-0.017... |
299
Integration
Exercise 13E
1 Find the area between the curv e with equation y = f(x), the x-axis and the lines x = a and
x = b in each of the following cases:
a f(x
) = −3x2 + 17x − 10; a = 1, b = 3
b f(x
) = 2x3 + 7x2 − 4x; a = −3, b = −1
c f(x
) = −x4 + 7x3 − 11x2 + 5x; a = 0, b = 4
d f(x
) = 8 ___ x ... | [
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-0.... |
300
Chapter 13
9 The graph sho
ws part of the
curve C with equation y = x2(2 − x).
The region R, shown shaded,
is bounded by C and the x-axis.
Use calculus to find the exact
area of R. (5 marks)E
13.6 Areas under the x-axis
You need to be careful when you are finding areas below the x-axis.
■ When the area boun... | [
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... |
301
Integration
Example 11
Sketch the curve with equation y = x(x − 1)(x + 3) and find the area of the finite region bounded by
the curve and the x -axis.
When x = 0, y = 0
When y = 0, x = 0, 1 or − 3
x → ∞ , y → ∞
x → − ∞, y → − ∞
y
x –3 O 1y = x (x – 1)(x + 3)
The area is given by ∫ −3 0 y dx − ∫ 0 1 ... | [
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-0.014086049981415272,
... |
302
Chapter 13
3 f(x
) = − x 3 + 4 x 2 + 11x − 30
Oy
xy = –x3 + 4x2 – 11x – 30The graph shows a sketch of part of the curve with
equation y = − x 3 + 4 x 2 + 11x − 30.
a Use the factor theorem to show tha
t (x + 3) is a
factor of f(x).
b Write f(x
) in the form (x + 3)(Ax2 + Bx + C).
c Hence, factor... | [
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-0.03... |
303
Integration
Example 13
The diagram shows a sketch of the curve with equation
xy
OC
A
(a, 0)B
(b, 0)y = 2xy = x(x – 3)
y = x(x − 3) and the line with equation y = 2x.
Find the area of the shaded region OAC .x(4 − x ) = x
3x − x2 = 0
x(3 − x ) = 0
x = 0 or 3
Area beneath curve = ∫ 0 3 ( 4x − x2)dx
= [2 x ... | [
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0.0362657755613327,
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-0.03999458625912666,
-0.045... |
304
Chapter 13
= ( 125 _____ 3 − 75 ___ 2 ) − ( 27 ___ 3 − 27 ___ 2 )
= ( 25 ___ 6 ) − (− 27 ___ 6 )
= 26 ___ 3
Sha
ded region is therefore = 25 − 26 ___ 3 = 49 ___ 3
Exercise 13G
1 The diagram shows part of the curve with equation
xy
y = 6y = x2 + 2
AB
O26
y = x2 ... | [
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-0.... |
305
Integration
6 The diagram sho
ws a sketch of part of the curve with
equation y = x2 + 1 and the line with equation y = 7 − x.
The finite region, R1 is bounded by the line and the curve.
The finite region, R2 is below the curve and the line and is
bounded by the positive x - and y -axes as shown in the diag... | [
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-0.0... |
306
Chapter 13
11 The line with equation y
= 10 − x cuts the curve with
A
B
xyy = 2x2 – 5x + 4
y = 10 – xOR
equation y = 2x2 − 5x + 4 at the points A and B,
as shown.
a Find the coordinates of
A and the coordinates
of B. (5 marks)
The shaded region R
is bounded by the line and the
curve as shown.b
Find th... | [
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-0.000... |
307
Integration
10 A cyclist is tra
velling along a straight road. The distance in metres of the cyclist from a fixed
point after t seconds is modelled by the function f(t), where f ′(t
) = 5 + 2t and f(0) = 0.
a Find an expression f
or f(t).
b Calculate the time ta
ken for the cyclist to travel 100 m.
11 The diagr... | [
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308
Chapter 13
16 The diagram sho
ws the curve C with equation
y = x(8 − x) and the line with equation y = 12
which meet at the points L and M.
a Determine the coordina
tes of the point M. (2 marks)
b Given tha
t N is the foot of the perpendicular from
M on to the x-axis, calculate the area of the shaded
region ... | [
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309
Integration
22 The finite region S
, which is shown shaded, is
bounded by the x-axis and the curve with
equation y = 3 − 5x − 2x2.
The curve meets the x-axis at points A and B.
a Find the coordinates of
point A and
point B. (2 marks)
b Find the area of the r
egion S. (4 marks)
23 The graph sho
ws a sketch o... | [
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310
Chapter 13
1 If dy ___ dx = xn, then y = 1 _____ n + 1 x n + 1 + c, n ≠ −1.
Using function notation, if f ′(x
) = xn, then f(x) = 1 _____ n + 1 x n + 1 + c, n ≠ −1.
2 If dy ___ dx = kxn, then y = k _____ n + 1 x n + 1 + c, n ≠ −1.
Using function notation, if f ′(x
) = kxn,... | [
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311
Exponentials and
logarithms
After completing this unit you should be able to:
● Sketch gr
aphs of the form y = ax, y = ex, and transformations of
these graphs → pages 312–317
● Differentiate ekx and understand why this result is important
→ pages 314–317
● Use and interpret m odels that use exponential functio... | [
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-0.... |
312
Chapter 14
14.1 Exponential functions
Functions of the form f( x) = a x, where a is a constant, are called exponential functions . You should
become familiar with these functions and the shapes of their graphs.
For an example, look at a table of values of y = 2x.
x −3 −2 −1 0 1 2 3
y 1 _ 8 1 _ 4 1 _ ... | [
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0.06271... |
313
Exponentials and logarithms
Example 2
Sketch the graph of y = ( 1 _ 2 ) x − 3
. Give the coordinates of the point where the graph crosses the y-axis.
If f(x) = ( 1 _ 2 ) x then y = f( x − 3).
The graph is a translation of the graph
y = ( 1 _ 2 ) x by the vector ( 3 0 ) .
The graph crosses t... | [
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0.0067... |
314
Chapter 14
7 The graph of
y = pqx passes through the points (−3, 150) and (2, 0.048).
a By drawing a sk
etch or otherwise, explain why 0 < q < 1.
b Find the values of
the constants p and q.P
14.2 y = e x
Exponential functions of the form f(x) = ax have a special mathematical property. The graphs of their
grad... | [
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0... |
315
Exponentials and logarithms
Example 3
Differentiate with respect to x.
a e4x b e − 1 _ 2 x c 3e2x
a y = e4x
dy ___ dx = 4e4x
b y = e − 1 __ 2 x
dy ___ dx = − 1 __ 2 e − 1 __ 2 x
c y =
3e2x
dy ___ dx = 2 × 3e2x = 6e2x
Example 4
Sketch the graphs of ... | [
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0.029398... |
316
Chapter 14
c y = 3 + 4 e 1 __ 2 x
Whe
n x = 0, y = 3 + 4 e 1 __ 2 × 0 = 7 so
the graph crosses the y -axis at (0, 7).
The line y = 3 is an asymptote.
y
y = 3 + 4e x
x7
3
O1
2
The graph of y = e 1 _ 2 x has been stretched parallel to
the y-axis with scale factor 4 and then tr... | [
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0.062... |
317
Exponentials and logarithms
7 Find the gradient of the curv
e with equation y = e3x at the point where
a x =
2 b x =
0 c x =
−0.5
8 The function f is defined as f(x
) = e0.2x, x ∈ ℝ. Show that the tangent to the curve at the point
(5, e) goes through the origin.P
14.3 Exponential modelling
You can use ex to m... | [
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0.05... |
318
Chapter 14
Exercise 14C
1 The value of a car is modelled by the formula
V
= 20 000 e − t __ 12
where
V is the value in £s and t is its age in years from new.
a State its va
lue when new.
b Find its value (to the near
est £) after 4 years.
c Sketch the gra
ph of V against t.
2 The population of a country ... | [
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0.01795... |
319
Exponentials and logarithms
5 On Earth, the atmospheric pressur
e, p, in bars can be modelled approximately by the formula
p = e−0.13h where h is the height above sea level in kilometres.
a Use this model to estimate the pr
essure at the top of Mount Rainier, which has an altitude of
4.394 km. (1 mark)
b Demons... | [
-0.008494170382618904,
0.09602805972099304,
0.1061401441693306,
-0.01361599750816822,
-0.03374362364411354,
-0.10648752748966217,
-0.0364370234310627,
0.05207613483071327,
-0.005472090095281601,
0.04854961857199669,
0.014456508681178093,
-0.05177299305796623,
-0.015751276165246964,
0.00000... |
320
Chapter 14
Example 8
Without using a calculator, find the value of:
a log3 81 b log4 0.25 c log0.5 4 d loga (a5)
a log3 81 = 4
b log4 0.25 = −1
c log0.5 4 = −2
d loga (a5) = 5Because 34 = 81.
Because 4−1 = 1 _ 4 = 0 .25.
Because 0.5−2 = ( 1 _ 2 ) −2 = 22 = 4.
Because a5 = a5.
You can use yo... | [
0.02866137959063053,
0.11889641731977463,
-0.1147841066122055,
-0.05856194719672203,
-0.019249327480793,
0.031380701810121536,
-0.0001009116240311414,
-0.002597283571958542,
-0.021287966519594193,
0.09907026588916779,
0.02508963830769062,
-0.09601907432079315,
-0.034280743449926376,
0.0270... |
321
Exponentials and logarithms
3 Without using a calcula
tor, find the value of
a log2 8 b log5 25 c log10 10 000 000 d log12 12
e log3 729 f log10 √ ___ 10 g log4 (0.25) h log0.25 16
i loga (a10) j lo g 2 _ 3 ( 9 _ 4 )
4 Without using a calcula
tor, find the value of x for which
a log5 x = 4 b lo... | [
0.0744384229183197,
0.08955106884241104,
-0.03225674852728844,
0.012092877179384232,
-0.03919794037938118,
-0.00219270889647305,
-0.004185620229691267,
-0.060626205056905746,
0.010773756541311741,
0.051743797957897186,
0.057297442108392715,
-0.04745108261704445,
-0.0671902447938919,
0.0337... |
322
Chapter 14
Example 10
Write as a single logarithm.
a log3 6 + log3 7 b log2 15 − log2 3 c 2log5 3 + 3log5 2 d log10 3 − 4log10 ( 1 _ 2 )
a log3 (6 × 7)
= l
og3 42
b log2 (15 ÷ 3)
=
log2 5
c 2 lo
g5 3 = log5 (32) = log5 9
3 lo
g5 2 = log5 (23) = log5 8
log5 9 + log5 8 = log5 72
d 4 lo
g10 ( 1 __ 2 )... | [
0.021565113216638565,
0.0914715975522995,
-0.10404132306575775,
-0.08084187656641006,
-0.015245390124619007,
0.056219637393951416,
-0.055194079875946045,
-0.0626320019364357,
0.01155114360153675,
0.08137919008731842,
0.01543743722140789,
-0.028753623366355896,
-0.0017046780558302999,
0.007... |
323
Exponentials and logarithms
Example 12
Solve the equation log10 4 + 2 log10 x = 2.
log10 4 + 2 log10 x = 2
log10 4 + log10 x2 = 2
log10 4x2 = 2
4x2 = 102
4x2 = 100
x2 = 25
x = 5Use the power law.
Use the multiplication law.
Rewrite the logarithm using powers.
log10 x is only defined for positive
v
alues of x , s... | [
0.09157591313123703,
0.06256301701068878,
-0.011818905360996723,
0.025537481531500816,
-0.05482561141252518,
-0.013487119227647781,
-0.01238163560628891,
-0.07130257785320282,
-0.010050013661384583,
0.12389561533927917,
-0.05206602066755295,
-0.07711072266101837,
-0.05300439894199371,
0.01... |
324
Chapter 14
4 Solve the follo
wing equations:
a log2 3 + log2 x = 2 b log6 12 − log6 x = 3
c 2 log5 x = 1 + log5 6 d 2 log9 (x + 1) = 2 log9 (2x − 3) + 1
5 a Given tha
t log3 (x + 1) = 1 + 2 log3 (x − 1), show that 3x2 − 7x + 2 = 0. (5 marks)
b Hence, or otherwise, solv
e log3 (x + 1) = 1 + 2 log3 (x − 1). (2 ... | [
0.01271229051053524,
0.13110095262527466,
-0.0567329004406929,
-0.0333261676132679,
0.014966117218136787,
0.03574111685156822,
-0.05198757350444794,
0.07292289286851883,
-0.007001521997153759,
0.059588730335235596,
0.029257820919156075,
-0.05803460627794266,
-0.05011863633990288,
0.0055902... |
325
Exponentials and logarithms
You can solve more complicated equations by ‘taking logs’ of both sides.
■ Whenever f( x) = g( x), loga f(x) = loga g(x)
Example 16
Find the solution to the equation 3x = 2x + 1, giving your answer to four decimal places.
3x = 2x + 1
log 3x = log 2x + 1
x log 3 = (x + 1) log 2
x lo
g 3... | [
0.08601774275302887,
0.13742302358150482,
0.025898804888129234,
0.019976604729890823,
-0.05205421894788742,
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-0.009186477400362492,
-0.053081266582012177,
0.00746889365836978,
0.06137632578611374,
-0.001987326890230179,
-0.10183223336935043,
-0.11204242706298828,
0.033... |
326
Chapter 14
14.7 Working with natural logarithms
■ The graph of y = ln x is a reflection of the graph y = ex in the line y = x.
The graph of y = ln x pa
sses through (1, 0) and does not
cro
ss the y-axis.
The y-axis is an asymptote of the graph y = ln x. This means
that ln x is only defined for po
sitive v... | [
0.07970160990953445,
0.049377817660570145,
0.04991794005036354,
0.06696084141731262,
-0.03220294788479805,
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0.018349705263972282,
0.048857975751161575,
0.11100362986326218,
0.06473139673471451,
0.012540427967905998,
0.0770938... |
327
Exponentials and logarithms
c e2x + 5ex = 14
e2x + 5ex − 14 = 0
(ex + 7)(ex − 2) = 0
ex = −7 o r ex = 2
ex = 2
x = ln 2e2x = (ex)2, so this is a quadratic function of ex.
Start by setting the equation equal to 0 and
factorise. You could also use the substitution u = e
x and write the equation as u2 + 5u − ... | [
0.016947656869888306,
0.0563775897026062,
0.027611611410975456,
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0.014503901824355125,
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0.024254629388451576,
-0.01762135699391365,
-0.09600500762462616,
-0.04952447488903999,
0.0... |
328
Chapter 14
14.8 Logarithms and non-linear data
Logarithms can also be used to manage and explore non-linear trends in data.
■ If y = axn then the graph of log y against log x will be a straight line with gradient n and
vertical intercept log a.
log xOlog y
log a8 The graph of y = 3 + ln (4 − x) is shown to... | [
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0.0651603639125824,
0.05832807719707489,
-0.04007000848650932,
-0.022900497540831566,
0.05024... |
329
Exponentials and logarithms
Example 19
The table below gives the rank (by size) and population of the UK’s largest cities and districts
(London is ranked number 1 but has been excluded as an outlier).
City Birmingham Leeds Glasgow Sheffield Bradford
Rank, R 2 3 4 5 6
Population, P (2 s.f.) 1 000 000 730 000 620 ... | [
0.12025459855794907,
0.04562569409608841,
-0.04132251441478729,
-0.015959633514285088,
-0.0355859249830246,
-0.05353380739688873,
-0.04174921661615372,
0.059487905353307724,
-0.015221820212900639,
0.0893004760146141,
-0.009510914795100689,
-0.005347810219973326,
-0.09322647005319595,
0.006... |
330
Chapter 14
■ If y = ab x then the graph of log y against x will be a straight line with gradient log b and
vertical intercept log a.
xlog y
log a
O For y = ab x you
need to plot log y ag
ainst x
to obtain a linear graph.
If you plot log y ag
ainst
log x yo
u will not get a
linear relationship.Watch o... | [
0.024222899228334427,
0.06282208859920502,
-0.035492926836013794,
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0.0003259084769524634,
-0.028940461575984955,
-0.04683418199419975,
0.030342422425746918,
0.06507482379674911,
0.09648939967155457,
0.06238650530576706,
-0.03303369879722595,
-0.061348799616098404,
0.... |
331
Exponentials and logarithms
Exercise 14H
1 Two variables, S and x satisfy the formula S = 4 × 7x.
a Show that lo
g S =
log 4 +
x log 7.
b The straight line gra
ph of log S against
x is plotted. Write down the gradient and the value
of the intercept on the vertical axis.
2 Two v
ariables A and x satisfy the f... | [
0.04189661517739296,
0.09786305576562881,
-0.039515189826488495,
-0.004359736572951078,
-0.035148512572050095,
-0.006265555042773485,
-0.015506182797253132,
-0.0007047535036690533,
0.02532258629798889,
0.04892211779952049,
0.04438153654336929,
-0.06451766937971115,
-0.1034957692027092,
0.0... |
332
Chapter 14
b Plot a graph of
log R against lo
g m using the v
alues from your table and draw in a
line of best fit. (2 marks)
c Use your gra
ph to estimate the values of a and b to two significant figures. (4 marks)
d Using your va
lues of a and b, estimate the resting metabolic rate of a human male
with a m... | [
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-0.009679335169494152,
0.09397448599338531,
0.009069261141121387,
0.09011393785476685,
0.015747489407658577,
-0.06631612777709961,
0.007764824200421572,
-0.... |
333
Exponentials and logarithms
8 A scientist is modelling the number of
people, N, who have fallen sick with a virus after t days.
tlog N
O1.6(10, 2.55)
From looking at this graph, the scientist suggests that the number of sick people can be
modelled by the equation N = abt, where a and b are constants to be found... | [
0.05219860374927521,
0.05805150419473648,
-0.04002410173416138,
0.02996273711323738,
0.013453750871121883,
-0.12710009515285492,
-0.011712493374943733,
0.023528827354311943,
0.04622252285480499,
0.016523826867341995,
0.01593511551618576,
-0.022541353479027748,
-0.049103908240795135,
0.0553... |
334
Chapter 14
1 Sketch each of the f
ollowing graphs, labelling all intersections and
asymptotes.
a y =
2−x b y = 5ex − 1 c y = ln x
2 a Express log a ( p2q) in terms of log a p and log a q.
b Given tha
t log a ( pq) = 5 and log a ( p2q) = 9, find the values of log a p and log a q.
3 Given tha
t p... | [
0.06501208990812302,
0.08453291654586792,
-0.06963164359331131,
-0.01848771795630455,
0.010336354374885559,
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0.025902610272169113,
0.04157999902963638,
-0.03607666864991188,
-0.013536284677684307,
0.04875... |
335
Exponentials and logarithms
10 The points P and
Q lie on the curve with equation y = e 1 _ 2 x .
The x-coordinates of P and Q are ln 4 and ln 16 respectiv
ely.
a Find an equation for the line
PQ.
b Show that this line passes thr
ough the origin O.
c Calculate the length, to 3 significant figur
es, of ... | [
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0.09681977331638336,
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0.02375524863600731,
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0.0017434972105547786,
0.05487622320652008,
-0.031978789716959,
0.007339731324464083,
0.032171446830034256,
-0.11693038046360016,
-0.030637038871645927,
-0.031178... |
336
Chapter 14
c Find the rate of
increase of the number of views after 100 days, stating the units of your answer.
d Use your answ
er to part c to comment on the validity of the model after 100 days.
15 The moment magnitude scale is used b
y seismologists to express the sizes of earthquakes.
The scale is calculated... | [
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0.059841517359018326,
0.015345769934356213,
-0.044595878571271896,
0.09376969933509827,
... |
337
Exponentials and logarithms
3 loga n = x is equivalent to ax = n (a ≠ 1)
4 The laws o
f logarithms:
• loga x + loga y = loga xy (the multiplication law)
• l
oga x − loga y = loga ( x __ y ) (the division law)
• l
oga (x k) = k loga x (the power la w)
5 You should also l
earn to recognise the following sp... | [
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0.1185493990778923,
0.011390249244868755,
-0.1098771020770073,
-0.010152271017432213,
-0... |
338Review exercise3
1 The vector 9i + qj is parallel to the vector
2i
− j. Find the value of the constant q.
(2 marks)
← Section 11.2
2 Given that |5i − kj| = |2ki + 2j| , find the exact
value of the positive constant k. (4 marks)
← Section 11.3
3 Given the four points X(9, 6), Y(13, −2),
Z(0,
−15), and ... | [
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0.10862459987401962,
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-0.09407343715429306,
-0.03872641175985336,
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0.0064660306088626385,
0.0034515440929681063,
-0.054558947682380676,
0.004658025689423084,
-0... |
339
Review exercise 3
10 Given tha
t y = 4x3 − 1 + 2 x 1 _ 2 , x > 0,
find dy ___ dx . (2 marks)
← Section 12.5
11 The curve C has equation
y = 4x + 3 x 3 _ 2 − 2x2, x > 0.
a Find an expression f
or dy ___ dx (2 marks)
b Show that the point
P(4, 8) lies
on C. (1 mark)
c Show that ... | [
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-0.043566811829805374,
-0.008016... |
340
Review exercise 3
20 Given tha
t ∫
0 6
(x 2 − kx) dx = 0 , find the
value of
the constant k. (3 marks)
← Section 13.4
21 The diagram shows a section of the curve
with equation
y = −x4 + 3x2 + 4. The curve
intersects the x -axis at points A and B . The
finite region R , which is shown shaded, is ... | [
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-0.041313767433166504,
-0.0501190... |
341
Review exercise 3
26 a Find, to 3 significant figures, the v
alue
of x for which 5x = 0.75. (2 marks)
b Solve the equation 2lo
g5x − log53x = 1
(3 marks)
← Sections 14.5, 14.6
27 a Solve 32x − 1 = 10, giving your answer to
3 significant figures. (3 marks)
b Solve log2x + log2(9 − 2x) = 2
(3 marks)
← Section... | [
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3423421 a Given tha
t 4 = 64n, find the value of n. (1)
b Write √ ___
50 in the form k √ __
2 where k is an integer to be determined. (1)
2 Find the equation of the line par
allel to 2x − 3y + 4 = 0 that passes through the point (5, 6).
Give your answer in the form y = ax + b where a and b are rational n... | [
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-0.04556957259774208,
-0.015191040001809597,
-0... |
343
Practice paper
5 A rectangular box has sides measuring
x cm, x
+ 3 cm and 2x cm.
2x cmx cm
x + 3 cm
Figure 1
a Write down an e
xpression for the volume of the box. (1)
Given tha
t the volume of the box is 980 cm3,
b Show that
x3 + 3x2 − 490 = 0. (2)
c Show that
x = 7 is a solution to this equation. (1)
d... | [
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0.03862061724066734,
0.036716800183057785,
-0.1241481751203537,
-0.01648656092584133,
0.023... |
344
Practice paper
10 A triangular lawn ABC
is shown in figure 3:
B
ACDiagram not
to scale
Figure 3
Given that AB = 7.5 m, BC
= 10.6 m and AC =
12.7 m,
a Find angle BAC . (3)
Grass seed costs £1.25 per square metr
e.
b Find the cost of seeding the whole la
wn. (5)
11 g(x
) = (x − 2)2(x + 1)(x − 7)
a Sketch the ... | [
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0.0... |
345
Answers
345Answers
Answers
CHAPTER 1
Prior knowledge check
1 a 2m2n + 3mn2 b 6x2 − 12x − 10
2 a 28 b 24 c 26
3 a 3x + 12 b 10 − 15x c 12x − 30 y
4 a 8 b 2x c xy
5 a 2x b 10x c 5x ___ 3
Exercise 1A
1 a x7 b 6x5 c k d 2p2
e x f y10 g 5x2 h p2
i 2a3 j 2p k 6a9 l 3a2b3
m 27x8 n 24x11 o 63a12 p 32y6
q 4a6 r... | [
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0.023594364523887634,
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0.006021630018949509,
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0.006152608431875706,
-0.05507691577076912,
0.01312678400427103,
... |
346
Answers
346 Full worked solutions are available in SolutionBank.
Online
2 a 5 b 729 c 3 d 1 __ 16
e 1 __ 3 f −1 ____ 125 g 1 h 216
i 125 ___ 64 j 9 __ 4 k 5 __ 6 l 64 __ 49
3 a 8x5 b 5 __ x2 − 2 __ x3 c 5x4
d 1 __ x2 + 4 e 2 __ x3 + 1 __... | [
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-0.031698815524578094,
0.02472320944070816,
0.019751090556383133,
-0.04493129253387451,
-0.10253763943910599,
0.022827155888080597,
0.03088555298745632,
-0.09027034044265747,
-0.01064865943044424,
-0.03... |
347
Answers
3473 c y
x9
18 O d y
x O
4 a x < 3 b x > 9 c x < 2.5 d x > −7
Exercise 2A
1 a x = −1 or x = −2 b x = −1 or x = −4
c x = −5 or x
= −2 d x = 3 or x
= −2
e x = 3 or x
= 5 f x = 4 or x
= 5
g x = 6 or x
= −1 h x = 6 or x
= −2
2 a x = 0 or x
= 4 b x = 0 or x
= 25
c x = 0 or x
= 2 d x = 0 or ... | [
0.017716599628329277,
0.034774065017700195,
0.037739623337984085,
-0.04995901137590408,
0.0031079838518053293,
0.0859120786190033,
0.035284437239170074,
0.015446845442056656,
-0.1327805370092392,
0.053988978266716,
-0.03929651156067848,
-0.0616622194647789,
0.05052229017019272,
-0.04718343... |
348
Answers
348 Full worked solutions are available in SolutionBank.
Online
b y
x O
y = x2 + 2x – 15(3, 0)
(–5, 0)
(0, –15)
Turning point: (−1, −16)
Line of symmetry: x = −1
c y
x Oy = 25 – x2
(–5, 0)(0, 25)
(5, 0)
Turning point: (0, 25)
Line of symmetry: x = 0
d y
x Oy = x2 + 3x + 2
(–1, 0)
(–2, 0)(0, ... | [
-0.08171235769987106,
-0.0643116757273674,
0.034181348979473114,
-0.05814121663570404,
0.008347636088728905,
0.02351064421236515,
-0.005849470384418964,
-0.013341337442398071,
-0.10939627885818481,
-0.00675944285467267,
-0.020795000717043877,
-0.05500122159719467,
0.01088480930775404,
-0.0... |
349
Answers
349Exercise 2H
1 a The height of the bridge above ground level
b x = 1103 and x
= −1103
c 2206 m
2 a 21.8 mph and
75.7 mph
b A = 39.77,
B = 0.01, C = 48.75
c 48.75mph
d −11 mpg; a negative answer is impossible so this
model is not valid for very high speeds
3 a 6 tonnes
b 39.6 kilograms per hec... | [
0.05560750886797905,
0.054463766515254974,
-0.010985961183905602,
-0.023514389991760254,
-0.05735262110829353,
-0.0852833092212677,
-0.0284983292222023,
0.01807441934943199,
-0.13538342714309692,
0.060148999094963074,
0.09056969732046127,
-0.0765831470489502,
0.007468268740922213,
-0.05170... |
350
Answers
350 Full worked solutions are available in SolutionBank.
Online
d a = 1, b = 5 or a = 3, b = −1
e u = 1 1 __ 2 , v = 4 or u = 2, v = 3
f x = −1 1 __ 2 , y = 5 3 __ 4 or x = 3, y = −1
2 a x = 3,
y = 1 __ 2 or x = 6 1 __ 3 , y = −2 5 __ 6
b x = 4 1 __ 2 ... | [
-0.06884238123893738,
-0.04596174135804176,
0.02920406311750412,
-0.09674413502216339,
0.007242766208946705,
0.037808068096637726,
-0.025201871991157532,
-0.03072923980653286,
-0.057093340903520584,
0.09661343693733215,
0.01819494366645813,
-0.03300723060965538,
0.018895741552114487,
0.026... |
351
Answers
351 e {x: −5 < x < −3} ⋃ {x: x > 4}
f {x: −1 <
x < 1} ⋃ {x: 2 < x < 3}
4 a x < 0 or x
> 2 b x < 0 or x
> 0.8
c x < −1 or x
> 0 d x < 0 or x
> 0.5
e x < − 1 __ 5 or x > 1 __ 5 f x < − 2 __ 3 or x > 3
5 a −2 < k < 6 b p < −8 or
p > 0
6 {x:
x < −2} ⋃ {x: x > 7}
7 a {x:
x < ... | [
0.003617358859628439,
0.02979881316423416,
0.02298298478126526,
-0.015808692201972008,
0.0012667098781093955,
0.023505887016654015,
0.0549328587949276,
-0.0162173043936491,
-0.09046171605587006,
0.10245751589536667,
0.02741752192378044,
-0.10725383460521698,
0.04612688347697258,
-0.0096202... |
352
Answers
352 Full worked solutions are available in SolutionBank.
Online
Mixed exercise
1 a 4kx − 2 y = 8
4kx + 3y = −2
−5y = 10
y = −2
b x = 1 __ k
2 x = −4,
y = 3 1 __ 2
3 a Substitute x = 1 + 2
y into 3xy − y2 = 8
b (3, 1) and (− 11 __ 5 , − 8 __ 5 )
4 a Substitute y = 2 −
x into x... | [
-0.07159418612718582,
0.039758432656526566,
0.029994383454322815,
-0.06266020983457565,
-0.06447648257017136,
0.011389545165002346,
-0.02702719159424305,
-0.03692164272069931,
-0.10028614103794098,
0.04789949208498001,
0.007884837687015533,
-0.05697328969836235,
0.030236465856432915,
0.000... |
353
Answers
353 i
xy
22
– 1
212O j
xy
–312O
2 a y
x–1
–11O b y
x–22
1O
c y
x–12
2O d y
x–1–22O
e y
x –2 O f y
x1O
g y
x13
–3O h y
x13
3O
i y
x2O j y
x2O
3 a y = x
(x + 2)(x − 1) b y = x
(x + 4)(x + 1)
y
x1 –2O
–1 –4y
x O
c y = x (x + 1)2 d y = x (x + 1)(3 − x)
–1y
x O
–1 3y
x O e y = x2(x − 1) f y = ... | [
-0.09313216805458069,
-0.009504067711532116,
0.02975899539887905,
-0.021798623725771904,
-0.05843671038746834,
0.06311101466417313,
0.0732831135392189,
-0.03565547615289688,
-0.04057953879237175,
0.03286866098642349,
-0.013768315315246582,
-0.039649106562137604,
0.021541696041822433,
-0.07... |
354
Answers
354 Full worked solutions are available in SolutionBank.
Online
g
3
–27y
x O h
11y
x O
i
28y
x O j
1
218y
x O
5 a b = 4, c = 1, d = −6 b (0, −6)
6 a = 1 __ 3 , b = − 4 __ 3 , c = 1 __ 3 , d = 2
7 a x(x2 − 12x + 32) b x(x − 8)(x − 4)
c y
x O 84
Exercise 4B
1 a y
xO –4 –3 –2 –124... | [
-0.03521917387843132,
-0.032616157084703445,
0.033659614622592926,
-0.05276915803551674,
-0.04875417798757553,
0.04825134575366974,
-0.008693584240972996,
-0.04992702975869179,
-0.10651875287294388,
0.05912495777010918,
0.04343504458665848,
-0.07272208482027054,
-0.05057069659233093,
-0.08... |
355
Answers
355Exercise 4C
1 a y
y =
x4
x
y =2
xO b
y =2
x
y = –2
xy
x O
c
y = –2
x
y = –4
xOy
x d
y =y
x8
x
y =3
xO
e
y = –3
x
y = –8
xOy
x
2 a
y =5
x2
y =2
x2y
x O b
y = –3
x2y
x Oy =3
x2
c y
x O
y = –6
x2y = –2
x2
Exercise 4D
1 a i y
xy = x2
y = x(x2 – 1)–1 1 O
ii 3 iii x2 = x(x2 − 1) b i y
xy = x(x + 2)... | [
-0.014409092254936695,
-0.048172254115343094,
0.007398159708827734,
-0.03783391788601875,
-0.047470901161432266,
0.08058717846870422,
0.027853356674313545,
-0.018648197874426842,
-0.10623576492071152,
0.057712338864803314,
0.0009278937359340489,
-0.05335347354412079,
0.057499587535858154,
... |
356
Answers
356 Full worked solutions are available in SolutionBank.
Online
g i y
x
y = x(x – 4)y = (x – 2)3
42
O
ii 1 iii x(x − 4) = (x − 2)3
h i y
x
y = –x3y = –2
x
O
ii 2 iii −x3 = − 2 __ x
i i y
xy = x2
y = –x3O
ii 2 iii −x3 = x2
j i
y
x
y = –x3y = –x(x + 2)
–2O
ii 3 iii −x3 = −x(x + 2)
k i... | [
-0.05136967822909355,
-0.032262805849313736,
0.010778084397315979,
-0.04483766481280327,
-0.07049275189638138,
0.03231389448046684,
0.06201764568686485,
-0.026966454461216927,
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0.018533414229750633,
0.02798229642212391,
-0.06419938802719116,
-0.031203024089336395,
-0.06... |
357
Answers
3576 a
7y =y = 3x + 7
4
x2y
x O
b 3
c Expand brackets and rearrange.
d (−2, 1), (−1,
4), ( 2 __ 3 , 9)
7 a y
x–1
O4
y = x3 – 3x2 – 4xy = 6x
b (0, 0); (−2, −12); (5, 30)
8 a y
x2
–1 21y = 14x + 2
y = (x2 – 1)(x – 2)
O
b (0, 2); (−3, −40); (5, 72)
9 a y
x–2 2y = (x – 2)(x + 2)2
y = –x2 – 8O
b (0,... | [
-0.07345025986433029,
-0.024132108315825462,
-0.03028907999396324,
-0.03107953816652298,
-0.016820713877677917,
0.12920700013637543,
-0.021195901557803154,
-0.028111690655350685,
-0.022420791909098625,
-0.0027619844768196344,
-0.012350295670330524,
-0.04114476591348648,
-0.016668615862727165... |
358
Answers
358 Full worked solutions are available in SolutionBank.
Online
f y y y
xxx
OOO
(0, 9), (3, 0)
(0, –27), (3, 0)(0, – ), x = 3, y = 01
3ii i iii
2 a
Oy
x–2
–21y = f (x)
b i
Oy
x–44
–1y = f (x + 2) ii
Oy
x –1y = f (x) + 2
c f(x + 2) = (x + 1)(x + 4); (0, 4)
f(x
) + 2 = (x + 1)(x + 2) + 2; (... | [
-0.05754482001066208,
-0.01075490191578865,
0.013117799535393715,
0.02794894389808178,
-0.01729397661983967,
0.08224902302026749,
-0.032595645636320114,
-0.08846425265073776,
-0.020589129999279976,
0.04570527374744415,
0.06389906257390976,
-0.05937379598617554,
-0.05886876583099365,
-0.028... |
359
Answers
359 b
OO Oy
xi y
xii y
xiii
f(–x) f(x) = f(– x)
f(–x)f(x)f(x)
c
OO Oy
xi y
xii y
xiii f(x)
f( x)f(x)
f(x) 1
2f( x)12f( x)12
d
OO Oy
xi y
xii y
xiii f(4x)
f(4x)
f(4x)f(x)
f(x)f(x)
e
OO Oy
xi y
xii y
xiii
f( x)f(x) f(x)
f(x)1
4f( x)14f( x)14
f
OO Oy
xi y
xii y
xiii f(x)
f(x)2f(x)
2f(x)2f(x)
f(... | [
-0.004635103978216648,
0.031784508377313614,
-0.023134341463446617,
-0.037173040211200714,
-0.06392297893762589,
0.09289161115884781,
0.018999839201569557,
-0.05541510134935379,
-0.060722120106220245,
0.010113141499459743,
-0.03683597594499588,
-0.02306658774614334,
0.010177300311625004,
-... |
360
Answers
360 Full worked solutions are available in SolutionBank.
Online
5 y
x O5y = x2 + 3x – 4
y = x2 + 3x – 4
6 y = x2(x – 2)2
3y = –x2(x – 2)2y
x O 2
7 a (1, −3) b (2, −12)
8 (−4, 8)
9 a y
x O 32
y = (x – 2)(x – 3)2
b 2 and 3
Challenge
1 (2, −2)
2 1 __ 4 f ( 1 __ 2 x)
Exercise 4G
1 a y
x (0, 0)... | [
-0.06345223635435104,
-0.06401929259300232,
-0.006433551665395498,
-0.08548048883676529,
0.0010957882041111588,
0.050806328654289246,
-0.022885765880346298,
-0.07518865913152695,
-0.07711315155029297,
0.014278113842010498,
0.011674427427351475,
-0.101340152323246,
0.0008884063572622836,
0.... |
361
Answers
361 f y = 2, x = 2, (0, 0)
y
x O 2y = 2
g y = 1, x = 1, (0, 0)
Oy
x 1y = 1
h y = −2, x = 1, (0, 0)
Oy
x 1
y = –2
3 a A(−2, −6), B(0, 0), C(2, −3), D(6, 0)
Oy
x
(–2, –6)(2, –3)6
b A(−4, 0), B(−2, 6), C(0, 3), D(4, 6)
y
x(–2, 6)
–4(4, 6)
3
O
c A(−2, −6), B(−1, 0), C(0, −3), D(2, 0)
y
x
... | [
-0.0011395610636100173,
-0.04186868667602539,
-0.028336595743894577,
-0.034674689173698425,
-0.04759267717599869,
0.10523037612438202,
0.03353702649474144,
-0.028932224959135056,
-0.0548023022711277,
0.039101533591747284,
0.03794702887535095,
-0.03324824944138527,
-0.011820523999631405,
-0... |
362
Answers
362 Full worked solutions are available in SolutionBank.
Online
j A(4, −6), B(2, 0), C(0, −3), D(−4, 0)
y
x –4
(4, –6)–32O
4 a i x = −2, y = 0, (0, 2)
Oy
–2 x2
ii x = −1, y = 0, (0, 1)
y
–1 x1
O
iii x = 0 y = 0
Oy
x
iv x = −2 y = −1 (0, 0)
Oy
y = –1x–2 v x = 2 y = 0 (0, 1)
... | [
-0.053813476115465164,
-0.08785586804151535,
0.01112822350114584,
-0.03243396058678627,
-0.05499310418963432,
0.007118722423911095,
-0.010372531600296497,
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0.05387353524565697,
0.05218661203980446,
-0.07864665240049362,
-0.02289660833775997,
0.00... |
363
Answers
363 e A(−1, 0.5) B(0, 1.5) C(1, 1.5) D(2, 0.5)
OBC
ADy
x2y – 1 = f(x)
Mixed exercise
1 a y
xy = x2(x – 2)
2x – x22 O
b x = 0, −1, 2; points (0, 0), (2, 0), (−1, −3)
2 a y
x
y = x2 + 2x – 5y = 1 + x
y =6
x1
AB
O
b A(−3, −2), B(2, 3)
c y = x2 + 2x − 5
3 a y
xy = 2
O B(0, 0)A( , 4)3
2
b y
xy = 1
B(0, 0... | [
-0.012487241066992283,
0.016625629737973213,
0.04943002387881279,
-0.0034789175260812044,
-0.041939713060855865,
0.08812687546014786,
0.026446791365742683,
-0.04770047590136528,
-0.08587842434644699,
0.050966501235961914,
0.011647094041109085,
-0.05296263098716736,
-0.022404812276363373,
-... |
364
Answers
364 Full worked solutions are available in SolutionBank.
Online
10 a
y = x2(3x + b)y
x O( , 0) (0, 0)–b
3
y =a
x2
b 1; only one intersection of the two curves
11 a x(x
− 3)2
b
y = x(x – 3)2Oy
x 3
c −4 and −7
12 a
Oy
xy = x(x – 2)2
(2, 2)(0, 0)
b
Oy
x(0, k)y = x(x – 2)2 + x
13 a Asymptotes at ... | [
-0.01777517795562744,
0.028817595914006233,
-0.009379619732499123,
0.01961597427725792,
0.003241601400077343,
0.015858642756938934,
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-0.0024987724609673023,
-0.021341657266020775,
-0.11322513222694397,
0.006697531323879957,
... |
365
Answers
36521 a x = 7 _ 2 , y = −2, x = −3, y = 11
b x < −3 or x
> 3 1 _ 2
22 a Different real roots, discriminant > 0
so k2 – 4k – 12 > 0
b k < −2 or k
> 6
23 −7 < x < 2
24
xy
–3 3914
(5, –16)(1, 8)y = g(x)
y = f(x)7
3
25 a x(x – 2)(x + 2)
b
xy
O –2 2
c
xy
–1 13O
26 a
xy
O 2(3, 2)
4
(2... | [
-0.07792309671640396,
0.030151784420013428,
-0.0040183281525969505,
0.0000266724600805901,
0.043242812156677246,
0.03581976145505905,
-0.0388982780277729,
-0.030336372554302216,
-0.08155082911252975,
0.00399361876770854,
0.00444075558334589,
-0.03304513171315193,
-0.017174962908029556,
-0.... |
366
Answers
366 Full worked solutions are available in SolutionBank.
Online
CHAPTER 5
Prior knowledge check
1 a (−2, −1) b ( 9 __ 19 , 26 __ 19 ) c (7, 3)
2 a 4 √ __
5 b 10 √ __
2 c 5 √ __
5
3 a y = 5 − 2x b y = 2 __ 5 x − 9 __ 5 c y = 3 __ 12 x + 12 __ 7
Exerci... | [
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0.011815020814538002,
0.03904212638735771,
-0.05746825784444809,
-0.014666211791336536,
0.0675702840089798,
-0.033365942537784576,
-0.024509964510798454,
-0.13202022016048431,
0.05757585167884827,
-0.012317740358412266,
-0.046871595084667206,
0.02259398065507412,
-0.0... |
367
Answers
3679 l: y = − 1 __ 3 x − 1, n: y = 3x + 5. Gradients are negative
reciprocals,
therefore lines perpendicular.
10 AB: y
= − 1 __ 2 x + 4 1 __ 2 , CD: y = − 1 __ 2 x − 1 __ 2 , AD: y = 2x + 7,
BC: y
= 2x − 13. Two pairs of parallel sides and lines
with gradients 2 and − ... | [
-0.05025338754057884,
-0.022240646183490753,
0.036309290677309036,
-0.04676760733127594,
-0.061586178839206696,
0.06282756477594376,
0.028959106653928757,
-0.048217251896858215,
-0.08664871752262115,
-0.03229277953505516,
0.0007258870173245668,
-0.0608617439866066,
0.03746010363101959,
-0.... |
368
Answers
368 Full worked solutions are available in SolutionBank.
Online
6 a F = 1.8C + 32 or F = 9 _ 5 C + 32
b a = 1.8 = increase in Fahrenheit temperature when
the Celsius temperature increases by 1°C. b
= 32 temperature in Fahrenheit when
temperature in Celsius is 0°.
c 38.5°C
d −40°C
7 a n = 7... | [
0.06318823993206024,
0.01552538201212883,
0.06028951704502106,
0.03554271161556244,
0.07148338109254837,
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0.017779355868697166,
-0.05162280797958374,
0.05642075464129448,
0.022974004... |
369
Answers
36910 p = 8, q = 7
11 a = −2,
b = 4
Challenge
a p = 9,
q = −1
b y = −x
+ 13
c AC: y = −
x + 8. Lines have the same slope, so they are
parallel.
Exercise 6B
1 a y = 2x + 3 b y = − 1 __ 3 x + 47 __ 3 c y = 5 __ 2 x − 25
d y = 3 e y = − 3 __ 4 x + 37 __ 8 f x = 9
2 y = −... | [
-0.016673151403665543,
0.03510235995054245,
0.025379741564393044,
-0.04865945875644684,
-0.007682542782276869,
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-0.04786821827292442,
-0.03668620064854622,
-0.1098233312368393,
0.027089929208159447,
-0.015026752837002277,
-0.04270963743329048,
0.020857349038124084,
-0.0... |
370
Answers
370 Full worked solutions are available in SolutionBank.
Online
11 a P(−2, 5) and Q(4, 7)
b y = 2x
+ 9 and y = − 1 __ 2 x + 9
c y = −3x
+ 9
d (0, 9)
Challenge
1 y = 1 __ 2 x − 2
2 a ∠ CPR = ∠ CQR = 90° (Angle between tangent and
radius)
CP =
CQ = √ ___
10 (Radii of circle)... | [
-0.02441990189254284,
0.085936039686203,
0.03392624482512474,
0.025710508227348328,
0.0005529384361580014,
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0.017315084114670753,
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0.006619034800678492,
0.03229124844074249,
-0.021908048540353775,
0.052680108696222305,
0.0090... |
371
Answers
371 e 4x3 − 2x2 + 3 f 3x − 4 x2 − 1
g 7x2 ____ 5 − x3 __ 5 − 2 ___ 5x h 2x − 3x3 + 1
i x 7 ___ 2 − 9 x 3 ____ 2 + 2 x 2 − 3 __ x j 3 x 8 + 2 x 5 − 4 x 3 ____ 3 + 2 ___ 3x
2 a x + 3 b x + 4 c x + 3
d x + 7 e x + 5 f x + 4
g x − 4 _____... | [
-0.016584021970629692,
0.026711642742156982,
0.0020620261784642935,
-0.04096461087465286,
-0.0318741649389267,
0.11721539497375488,
0.04342261701822281,
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0.012651643715798855,
-0.07829251140356064,
-0.007970002479851246,
-0.... |
372
Answers
372 Full worked solutions are available in SolutionBank.
Online
Exercise 7D
1 n 2 − n = n(n − 1)
If n is even,
n − 1 is odd and even × odd = even
If n is odd,
n − 1 is even and odd × even = even
2 x ________
(1 + √ __
2 ) × (1 − √ __
2 ) ________
(1 − √ __
2 ) = x ... | [
0.06108298525214195,
0.06655146181583405,
0.03833566606044769,
0.005557755474001169,
-0.05973406136035919,
0.0749589279294014,
0.03334863483905792,
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-0.06544898450374603,
-0.1278267204761505,
-0.04508297145366669,
-0.019872965291142464,
0.013491598889231682,
0.038086693... |
(3n + 1) 3 = 27 n 3 + 27 n 2 + 9n + 1 = 9n(3 n 2 + 3n + 1) + 1
which is one more than a multiple of 9
(3n + 2) 3 = 27 n 3 + 54 n 2 + 36n + 8 = 9n(3 n 2 + 6n + 4) + 8
which is one less than a multiple of 9
5 a For example, when n
= 2, 24 − 2 = 14, 14 is not
divisible by 4.
b Any squ... | [
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0.00882814172655344,
0.131444051861763,
0.094... |
373
Answers
37311 a When x = 2, x3 + x2 − 5x − 2 = 0
b 2, − 3 __ 2 ± √ __
5 ___ 2
12 1 __ 2 , 3
13 a When x = −4,
f(x) = 0
b (x + 4)(x − 5)(x − 1)
14 a f( 2 __ 3 ) = 0, therefore (3x − 2) is a factor of f(x)
a = 2, b = 7 and c = 3
b (3x − 2)(2x + 1)(x + 3)
c x = 2 __ 3 , − ... | [
-0.0577564612030983,
0.11641733348369598,
0.06692392379045486,
-0.028106648474931717,
0.036220453679561615,
0.10602177679538727,
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0.031300514936447144,
-0.02728533372282982,
-0.12264680862426758,
-0.012337922118604183,
-0.04... |
= 1 × 2 × … × (n − 2) × (n − 1) × n _______________________________ 1 × 2 × 1 × 2 × … × (n − 3) × (n − 2) = n(n − 1) ________ 2
10 a = 37
11 p = 17
Challenge
a 10C3 = 10 ! ____ 3 !7 ! = 120 and 10C7 = 10 ! ____ 7 !3 ! = 120
b 14C5 = 14 ! ____ 5 !9 ! = 2002 and 14C9 = 14 ! ____ 9 !5 ... | [
0.012324175797402859,
0.08403260260820389,
0.029683727771043777,
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-0.0851149708032608,
0.07936793565750122,
0.031360380351543427,
-0.08045043796300888,
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-0.09868813306093216,
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0.058278534561395645,
0.013... |
374
Answers
374 Full worked solutions are available in SolutionBank.
Online
d x6 + 12x5 + 60x4 + 160x3 + 240x2 + 192x + 64
e 1 + 8x + 24
x2 + 32x3 + 16x4
f 1 − 2x + 3 _ 2 x2 − 1 _ 2 x3 + 1 __ 16 x4
2 a 1 + 10x + 45 x2 + 120x3
b 1 − 10x + 40 x2 − 80x3
c 1 + 18x + 135 x2 + 540x3
d 256 − 1024x + 1... | [
-0.0494249053299427,
-0.008853430859744549,
0.030064092949032784,
-0.03215424343943596,
-0.006812223698943853,
0.05718117207288742,
-0.05621717497706413,
-0.06187654286623001,
-0.112193264067173,
0.04795798286795616,
-0.010102701373398304,
-0.0988256111741066,
0.01993466541171074,
-0.03071... |
375
Answers
37521 a 128 + 224x + 168 x2
b Substitute x = 0.1 into the expansion.
22 k = 1 __ 2
Challenge
1 540 − 405p = 0,
p = 4 _ 3
2 −4704
CHAPTER 9
Prior knowledge check
1 a 3.10 cm b 9.05 cm
2 a 25.8° b 77.2°
3 a graph of x2 + 3x b graph of (x + 2)2 + 3(x + 2)
c graph of x2 + 3x − 3 b graph of (0... | [
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0.00838424451649189,
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-0.029633214697241783,
0.02965361438691616,
-0.11262048035860062,
0.04361603781580925,
0.040855761617422104,
-0.10074768215417862,
-0.00706632062792778,
-0.0... |
376
Answers
376 Full worked solutions are available in SolutionBank.
Online
g x = 45.3°, y = 94.7°, z = 14.7
or x = 135°,
y = 5.27°, z = 1.36
h x = 7.07,
y = 73.7°, z = 61.3°
or x = 7.07,
y = 106°, z = 28.7°
i x = 49.8°,
y = 9.39, z = 37.0°
2 a ABC = 108°,
ACB = 32.4°, AC = 15.1 cm
Area = 41.2 ... | [
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-0.014788814820349216,
-0.0354... |
377
Answers
377 c The graph of y = sin 1 __ 3 θ is the graph of y = sin θ
stretched by a scale factor 3 in the θ direction.
O90° 180° 270° 360° θy
1
–11
3y = sin θ
Only meets axis at origin
Maximum at (270°, 1)
d The graph of y = tan (
θ − 45°) is the graph of tan θ
translated by 45° in the p... | [
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-0.009611348621547222,
0.... |
378
Answers
378 Full worked solutions are available in SolutionBank.
Online
d Period = 90°
90° 180° 270° 360° –180° –270° –360° –90°Oy
θ2
–2y = tan 2θ
6 a i y = cos (−θ) is a reflection of y = cos θ in the
y-axis, which is the same curve,
so cos θ = cos (−θ).
y = cos θ
Oy
θ–90° –180° –270° 90° 180° 270° ... | [
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-0.011297724209725857,
0.0004015547747258097,
... |
379
Answers
379 c A reflection in the x-axis.
d A translation of −20 in the x direction.
13 a
Oy
x45°1
90° 135° 180°2
–1
–2y = –2cos xy = tan(x – 45°)
b There are no solutions.
14 a 300 b (30, 1) c 60 d √ __
3 ___ 2
15 a p = 5
90° 180° 270° 360°y = f(x)
Oy
x1
–1
b 72°
16 a The four shaded regions ... | [
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0.012247116304934025,
0.02573... |
380
Answers
380 Full worked solutions are available in SolutionBank.
Online
Exercise 10A
1 a
O80°
–80°y
x
P b
OP
80°+100°y
x
c
O
P20°+200°y
x d
15°+165°y
x OP
e
35°
–145°y
x O
P f
45°+225°y
x O
P
g
O
P80°+280°y
x h
O
P30°+330°y
x
i
O 20°
–160° Py
x j
OP
80°–280°y
x
2 a First b Second c Second
d Third e Thir... | [
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0.056726887822151184,
0.07543767988681793,
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-0.006999822333455086,
-0.015700813382864,
-0.06079006567597389,
0.016606662422418594,
-0.02... |
381
Answers
381Answers
Exercise 10C
1 a sin2 θ __ 2 b 5 c −cos2 A
d cos θ e tan x f tan 3A
g 4 h sin2 θ i 1
2 1 1 __ 2
3 tan x − 3 tan y
4 a 1 − sin2 θ b sin2 θ _________ 1 − sin2 θ c sin θ
d 1 − sin2 θ _________ sin θ e 1 − 2 sin2 θ
5 (One outline example of a proof is given)
a LHS = sin... | [
-0.0649929791688919,
0.038551297038793564,
0.019678626209497452,
-0.011075693182647228,
-0.07758218050003052,
0.06506112217903137,
0.044339146465063095,
-0.045378949493169785,
-0.09407655149698257,
-0.02398819662630558,
0.04954768344759941,
-0.009537430480122566,
0.027332130819559097,
0.06... |
382
Answers
382 Full worked solutions are available in SolutionBank.
Online
b (0°, √ __
3 ___ 2 ) , (120°, 0), (300°, 0)
c 86.6°, 333.4°
6 a 0.75
b 18.4°, 108.4°, 198.4°,
288.4°
7 a 2.5
b No: increasing k will bring another
‘branch’ of the
tan graph into place.
Challenge
25°, 65°, 145°
Exercise 10F... | [
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0.039555102586746216,
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0.011888494715094566,
-0.05619869381189346,
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0.025834159925580025,
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0.003383112605661154,
... |
383
Answers
3836 3 √ __
5
7 (−6, 0)
8 (x +
3)2 + (y − 8)2 = 10
9 a (x −
3)2 + (y + 1)2 = 20 (a = 3, b = −1, r = √ ___ 20 )
b Centre (3, −1), radius √ ___ 20
10 a (3, 5) and (4, 2)
b √ ___ 10
11 0 < r < √ __
2 _ 5
12 a (x −
1)2 + (y − 5)2 = 58
b 7y −
3x + 26 = 0
13 a AB = √ ___ 32... | [
0.027320805937051773,
-0.018117578700184822,
0.0018258531345054507,
-0.022952165454626083,
0.010225620120763779,
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0.0002059868857031688,
-0.0317334309220314,
0.11036043614149094,
0... |
384
Answers
384 Full worked solutions are available in SolutionBank.
Online
Exercise 11A
1 a
aa + c
c b
–b
c
–d
cc – d d
b dc
b + c + d
e
3d2c
2c + 3d f
a
–2ba – 2b
g
abdca + b + c + d
2 a 2b b d c b
d 2b e d +
b f d +
b
g −2d h −b i 2d +
b
j −b + 2
d k −b +
d l −d −
b
3 a 2m b 2p c m
d m e p + ... | [
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0.08470707386732101,
0.026508232578635216,
0.004092602990567684,
-0.0545339435338974,
0.033726178109645844,
-0.025870904326438904,
-0.07303007692098618,
-0.16401971876621246,
0.019349470734596252,
-0.007102438248693943,
-0.07686697691679001,
-0.021390538662672043,
-0.0... |
385
Answers
38510 a 36.9° b 33.7° c 70.6°
11 a 67.2° b 19.0
Challenge
Possible solution:
p xy
p + rrqrqr
qr
q
s
sq + spq1
2
pq12rs12
rs12
Area of parallelogram = area of large rectangle − 2(area of
small rectangle) − 2 (area triangle 1) − 2(area triangle 2)
Area of parallelogram = ( p + r
)(q + s) − 2qr − 2( ... | [
0.026528647169470787,
0.011525857262313366,
0.024864403530955315,
-0.024206284433603287,
-0.04078713431954384,
0.05243440717458725,
-0.020369503647089005,
-0.043768201023340225,
-0.06887441873550415,
0.020636245608329773,
-0.0288835596293211,
-0.0351141095161438,
-0.014561627991497517,
-0.... |
386
Answers
386 Full worked solutions are available in SolutionBank.
Online
13 a 18.4° below
b R = (4 + p
)i + (5 − q)j, 4 + p = 3λ and 5 − q = −λ
4 + p = 3(
q − 5) so p + 3q = 11
c 2 √ ___
10 newtons
14 √ ____ 193 _____ 2
Challenge
⟶ OB = 3 __ 2 i + 5 __ 2 j or 99 __ 34 i ... | [
-0.014106039889156818,
0.027097538113594055,
0.06589098274707794,
-0.07422716915607452,
0.0015947961946949363,
0.030816122889518738,
-0.01864241063594818,
0.01514872070401907,
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0.05023103207349777,
0.018388543277978897,
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-0.0036705555394291878,
0.044... |
1 a 4x – 6 b x + 12 c 8x d 16x + 7
e 4 – 10x
2 a 12 b 6 c 7 d 2 1 _ 2
e –2 f 4
3 4, 0
4 (–1, –8)
5 1, –1
6 6, –4 | [
-0.09128184616565704,
0.05619007721543312,
-0.04034635052084923,
-0.054515987634658813,
-0.03408150374889374,
0.050908103585243225,
-0.004828000906854868,
-0.051247864961624146,
-0.05232565477490425,
0.04798399657011032,
-0.057046446949243546,
-0.044985149055719376,
0.031159784644842148,
0... |
387
Answers
3877 a, b
O xy
–2 14
(1, –9)–2y = f (x) y = f9(x)
c At the turning point, the gradient of y = f(x) is zero,
i.e. f9(x) = 0.
Exercise 12E
1 a 4x3 – x−2 b 10x4 – 6x−3 c 9 x 1 _ 2 − x − 3 _ 2
2 a 0 b 11 1 _ 2
3 a (2 1 _ 2 , − 6 1 _ 4 ) b (4, –4) and (2, 0)
c (16, –31) ... | [
-0.03178516775369644,
0.05611780658364296,
-0.05898730829358101,
0.0027553285472095013,
-0.042652811855077744,
0.044635262340307236,
-0.030877454206347466,
-0.01902284286916256,
-0.09275323897600174,
0.0313163660466671,
-0.007577160373330116,
-0.05348066985607147,
-0.03854409605264664,
-0.... |
388
Answers
388 Full worked solutions are available in SolutionBank.
Online
d
Oy
x(– , )1
31427
(3, –18)y = x(x2 – 4x – 3)
5 (1, 1) inflection (gradient is positive either side of point)
Oy
x(1, 1)y = x3 – 3x2 + 3x
6 Maximum value is 27; f(x) < 27
7 a (1, −3): minimum, (−3,
−35): minimum, (− 1 _ 4 ... | [
0.04627145081758499,
0.010959732346236706,
0.03831388056278229,
-0.06612283736467361,
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0.02644680254161358,
0.04862308129668236,
-0.03470100462436676,
-0.023757634684443474,
0.035... |
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