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PURE MATHEMATICS 1 PRACTICE A SOLUTIONS

1. Given that y = — X4

, express each of the following in the form kxn , where k and 71

are constants.

(a) yi

(b) 2y-1

3

(c) (tly)r (1)

(Total 3 marks)

( 1 De_41:1

14; x (c4.)-;

)

1: (L

xy

6 j

)s x4)1

S5 Har q 0., hokt- = 1

2. Find all the roots of the function f (x) = 4x — 14x1 + 6 (Total 3 marks)

int .j

_1-

% 4cJs - 1 1 1 + 6

cho 0

C Lij -2) 0

-3.0 0 tra,...°

if I I

= 1 2

2- 2-

I

3. Find the set of values of k for which the equation kx2 — (k + 2)x + 3k = 2

has no real roots, except for one value of k which must be stated. Give your answer in set

notation and in exact form (4)

(Total 4 marks)

k xi - —(1c- 2,3c -

k 2- 7c -1- 3 — 0

C_

- 44c < C Ns c P-SAL rzao~s)

(-14.-27 C C 3 2) <o

- I ( IC L 12- lc. -f-

Ilkz — I "2- lc-

C m-t+; - I

la 44cloc-zi) 2 Co )

1 2

22-

e oes.

t.A.r.less kr- 0 as

k e

a~way S 0..3 NO R fLocaT s

,,,,: +.a1 +-k e tin 0- et res-LA c-

1". e ca e cp., c.--L-;

Ic 0 c (C) y_ -I- 3 io)

2.9 6 5 — 1 3 • 9-- •S--

r-

;(tAD..7..

t- S • 2- 3 5- -0.5 1 3

61- Uf4L:1--D

4. A ball is being projected from a point on the top of a building above the horizontal ground.

The height, in metres, of the ball after t seconds can be modelled by the function:

h(t) = 13.7t + 7.8 — 2.9t2

(a) Use the model to find the height of the tower (1)

(b) After how many seconds does the ball hit the ground (2)

(c) Rewrite h(t) in the form A — B (t — C)2where A, B and C are constants (3)

(d) With reference to your answer in part (c) or otherwise, find the maximum height of the ball above

the ground, and the time at which this maximum height is reached (2)

(Total 8 marks)

; k( E- ) (3. 4- E ÷ — z.9P-

k(o) cD 4.E5 C:=0

\NL.12,- 60.11 4-La leemu. - ) IA. (E- ) CD

$ —E.- 4- it'Q ())1

&•sh

b-z ( 8 c

E1)- 6 r

sz3 1.)

-Ls.

as

tc.1Ww —

1, 17- -x-CX- )6c -I•,3 sir 4

r-epeake.01, GA-- 3c

a co, 4-ve r4-1 c

xr.:o 3c:-- 2-

C U r✓e

6:1 ) 0 Lev.. j °Ariz ve

t 110 <O lo•J Lev. O < x< 2-

correct' SlarQ c)(

3 c cwrect-

cc. owci .".c.4.42*

- 2., 0 ) I)

O r; ;,,c r -)L

t- 0.1

0) 00f 12.0.A.10.1 4-fOr'Si C6t:•°1%

2- tA,:A-s 1.242

Co

e??

v e 1, of

C2 .„.„) 4 .i

. Pc, ss; e

L e

5. (a) (i) Sketch the graph of y = x(x — 2)(x + 1)2 stating clearly the points

of intersection with the axes

(ii) State the range of values for which f (x) 0

(iii) The point with coordinates (-2, 0) lies on the curve with equation

y = (x + k)(x + k — 2)(x + k + 1)2 where k is a constant.

Find the possible values of k

(b) On seperate axes, sketch the graph 3y = — x(x — 2)(x + 1)2 stating clearly

the points of intersection with the axes' )cCx - 2-) C +02. 3

cfl -4

(3)

(1)

(1)

(3)

(Total 8 marks)

C A B

O — OA = 2 —

2-

\ 3

)( —

--f 1-

7--

O C_

12- c) C

( 1

J233

ve 0 C

— 1 3; 4

2- 3 3

C 1 3 + •

J23.3

6. A has position vector 5i — 2j and the point /3 has position vector —4i + 3j.

Given that C is the point such that AC = 2A/3

(a) find the unit vector in the direction of OC (3)

D is a point with position vector ai — 3j, where a is a constant.

Given that OD = b0A+ 0.14, where 6 is a constant.

(b) Find the values of a and b (2)

(Total 5 marks)

OA )

z _

7- ) 3

0D = b oB

s r_ 6

(3 ) 2- 3)

O. = cb _

3 l 3

-31 -2.6 4' 3

-3

S, b= 3 b- 3

a S(3)-Lt

I

. ,G 11

b:3/

7. (a) Find the first 3 terms of the expansion (1 + 3px)9 in ascending powers of x leaving

each term in its simplest form (2)

(b) Given that, in the expansion of (1 + 3px)9, the coefficient of x is q and the coefficient

of x2 is 4q, find the value of p and q (3)

(Total 5 marks)

cy C

I -+ 3 1 7-) cl r-- 9C0 19 (3pY)c) t 9C1 1% c 2r-Y5 9C2 14(3g)±1.1

-t (9) CI) ( 3 pn) + 36 c ev,e), ,

2 x 32.4

Caeft at lc

(beRt- ;S

;s

of

Caen. 10- z s

elCpCan o et LS 3a4 13

1-r

-"- 324f.;= Lf

tai r t- ct

gys 1p

SI et-23p

2.1-P C3 —I) to

3r

1"c r 3 NOT JALtD SiNcE

P*°

cL c

77

i,-","" -

C C>

8. The circle C has equation x2 + y2 + 2x — 20y = —51 The line L with equation 3y — 4x = 9 intersects the circle at the points P and Q. Given that the x coordinate of Q is > 0

(a) Find the centre and radius of the circle (2)

(b) Find the equation of the tangent at the point P and the point Q (4)

Points P and Q form the chord PQ of the circle

(c) Find the equation of the perpendicular bisector of the chord PQ giving your answers in the form ax + by + c = 0 where a, b and c are constants

(3)

(d) The perpendicular bisector and the two tangents intersect at a single point. Find the coordinate of the point of intersection.

(3)

(Total 12 marks)

• C- v- -I- C. e_ 3C G., 01 -t- 2,, 4-

t‘o

C pi.e4-;„,T e g

l

are (3c -4-

Oc 5) 1-

Cy t o ) ~~ t 0 -r.5 ---- 0

toy-BSI c)

(CD

-- z 4_ 6 4- 3 154-a

£ £ 4-- ),C

Nrs 9 9-5

i'ailavree) 13 a,avJ

O

x a G SZ, 01 a

.5C.St

— = .69 1:"

C) - =-0,0

().'9)

F

S 4- 3(.., —11;

0 -7: oS - 5, z

4— E--- os t

—ODi .• -4 DaS:9 czr

(-LIE ZD 48r1-4

I 4- V4-7 -I- DC

Nr7

Oix)

ti )-1

C — DC

y C • • -L - • y ,

E x. I r x 4,1$ ca C'Ne. " )•=0-

gz 5 z

1-1

+11,-0 "e14

X

9. (a) Prove that for any positive values of a and b

4a b

b a — >

— 4 (3)

(b) By use of a counter-example, show that this inequality is not true when either a or b is

not positive. (2)

(Total 5 marks)

-1- >

ct

cAl. ) 0,2 + 62 >

G

Li a' -4- 6 2- >. 110,6

— LI c66 4 1:R7:3 n

) (2 G -6) c.:2-

lJ

r -A Cov,sloe , )2- ? 0

40. - 0-6 +62 3

C..1 4.6 461"

$ rep , cx.6

oss:bie sto‘ce

p e + 6 > 1-t

b 0.

4-1 or 10 r) kve

0.

v c,

_4 :s v,ak- > 4

TL: e (Al

is "4=4 e u4. e. whew 42 :41" e r

a 6

pa r; .1614,ca 8

(5)

(Total 8 marks)

; S; + COSL7C = it

c.c.s'', 3 c...s x cp

-1) C CasX—f) =

CaSx= 2

..0 c c>.$ x

DC 7- 60", 3 00' -=

'")(3 Go" -h0'1

o 3 6c

(3 6 o' 0-

PrI

60', 30 0c' 36o-

sin`'+sin2 x cost x 10. (a) Show that 1

cost X - 1

sin4 x x cos2 x (b) Hence solve the equation

cos2 x-1 + 4 = 2sin2 x + 3cosx in the interval

(3)

0 < x < 360°

. 4, . ct..) S S )( C Ss

I. S )4c — (

4 s: 7C -I- S. v1 /4 S;K I 7c.

C 40 1 —

SiN 4 )C. -1' S..% — S;e%4

cz

)c — I

2. C 4- S I s)c

C DS X- •••• S

6/ : V\4

3c 2- I. -

_I- S;" )c CoS 7c•- L.( 2- • DC 3 C S

C •C> szDC -

-y _

— Cf -4- 3 cos ,c

3 = 2 -4- 3 c S.c--

2_ -+ 3 c .s )c - 3 CD

2.0 I - cos Lx)--t- 3 c.s.x. -3 :-_ S ; S.^ 'x + C ,C

-t- 3 c 7c —3 .,- 0

2. c r:".5 7C. 4 3 c )c CD

(3),-s)-t

3C .3)( s) =

3C3,e-s) = e

mx —t5 t

11. Solve the following equations, giving your solutions as exact values

(a) In(3x — 5) = 2 — In 3 (3)

(b) ex + se-x = 6 (3)

(Total 6 ma rl(s)

C3g-s) , a- 1.,.3 \ /1 /4 A- L .12€ m •ele-i-Lee

0 0. -t- I ,b)

C t-,verse o‘i)

9 e f S

e 4-t S

9

7

7

6 e 5 e-n

ar.<4 e e x

en-

cre cal

c e a(' "3 0.

e -

% ilf

-4- -Se-I. ..>"

-I- 5- 6 et

- S

- s 2 C~_-

o

o

CR

12. A container is being filled with water. After t seconds, the height h mm, of the water

present in the container can be modelled by the equation h = abt , where a and b are constants to

be found.

loge h

(10, 11.58)

1.58

The graph passes through the points (0, 1.58) and (10. I 1.58)

(a) Sketch the graph of h against t (2)

(b) Comparing the graph in part (a) to the graph of log2 h against t drawn above, state which graph is

more useful for calculations. Explain your reasoning (2)

(c) Write down an equation of the line (2)

(d) Find the values of a and b, giving your answers to 1 significant figures (4)

(e) Interpret the meaning of a in this model (1)

(I) Suggest one reason why this model in unsuitable for long periods of time and suggest an

improvement to the model (2)

(Total 13 marks)

a k

k

W Ler, Er.0 k CD 'VS"- e stare

( r 0- S .0%6( (

6/

Ike 5(a-,L sf I 05 k f ;s balker foe c ci..--ic.--Kadn-S

4-10-- 4-La VIS

;s f-c-o( 4- L. e Stf ,

I:.e ;s G CLAft/fa

105.„

L to, 0,64- Lka-

log L - 5. CA. -1-

ti .„

L

log L t. ln3

(lc-Ice (05

AA I

( Appal :-.j I cm-ss

A rpb:,.1

( lac Gf or a .

Ow 60-1.1., s;o14.5

a co a mL.0.$)

ckt.-.1s I sayte-Arks.s )

tofu,., ?WY- C ;A.

Co, 1-$a) s C 10, II. cg)

Cro-ohe,--E = ri•s-a - t• s--g I —

-.4ercept- _ I•ss

or-a c a t. ;.^ t&3 t 0 f ,5

_00

ne,

+ I, 5 St (AI)

cli Cnot:e_k ) a

52. %

6 : a

S 14,5 S 21-sa

:-- 2- 9 89 • - •

= 3 Cls•

H =1

d 42•A -

3 ! a eaa -•

act )-e-1 cr r)

3,1 1371

re)

ti

et NO rsel I 0 122 1

zr1 ) Sa:I _A a Nel •0 Cr -) a 1 -a 1 +

1.4 1,1•::;•

pd

• No4 ,s11 I

-I a '‘A o'er a _I-

1"1 Y.° 2f. svN9aNA.,10 a

•

'v-w VI" • X "V W

-4 01-o nn

r Cr

al+ „so a AC)./1 'IrV

I- 1'

2-C — ~1-a

•V C"'" 43,1k r ceol ✓ 01 Dc.

13. Given that f (x) = x 3 — 6x2 + 20x

(a) Prove that the function f (x) is increasing for all real values of x (3)

(b) Sketch the graph of the gradient function y = f'(x) (2)

(Total 5 marks)

(10 = 2t,C3 ,c -f20x 4

(1,) 2 2.x. -t-

C.c,NAID1142t ,0 -ViAe v. r(ic) — )

C.- 3)41. t I

= 2- CX -

Ir.ee

6 r "c 1 2 +

2- I 2- k

sif 7

0. b

c < P-e- P-OraTS

f. or correct' skape

luk.rr6r1 A 1-

COrrecE at-A-A

t( Dr's ail' (0%) 26c - 3)1 +

cc,o, ct, —

C 3, 21)

gm,

I4 —

I Co

14. The diagram below shows a box in the form of a cuboid. The cuboid has a rectangular cross section where the length of the rectangle is equal to three times its width, x cm. The volume of the cuboid is 144 cm3 .

3x

(a) Show that the surface area A of the box is A = 6 — (64 + x3)

(3)

(b) Use calculus to find the minimum value of A (4)

(c) Justify that the value of A yo‘u have found is a minimum (2)

(Total 9 marks)

A let ct.„6„;c1

are V 0( e V 3 pc X

v_ 3"

4 8' _c)

C kAel' cs..0 A ie 3 )4." 3 x.2

2- 2-7tj

()-)c 2" -4

StA e V 0, C e_ A,ta G, t.4- -x,-1 —J

EL)

C) €:) =72 A = x_ 8J___Igiti

384

(X 1 -I- -1 (Z.-) CD( )2_ "4- (. 11 X

_ G 61) ( K3

11-

valta e bf ek „ 6/ Fat

(6),t+321-i sl, 0 (384 3a4)(-')

12x - 3 accx". -

1 2x - 384 r_ 0

t 2.)c 384

= 3A

aJ

35a A = 6x2+ 3R4 tiv1/4 01, v es( v-e 4 -7C

" • mt.th-‘ 1/ CA La-C. J A .= 6 (313-2)14 3 84 M I

8 I. 286.__

= I at C 35. F)

' A

_t_Lk - I - 3 & w-2

cr-A , I I R- 6 R x- 3 Po

41.

6 )c = 3,ra ALA I + 6 a (353:13 ' z.

3(z,

dtA

15. The diagram below shows a sketch of part of the curve C with equation

y = x(x — 2)(x — 4)

x

The point P lies on C and has x coordinate2 — -a. The curve C cuts the x-axis at point B at (2,0).

The normal to the curve at P meets the vertical line at B at the point D.

(a) Show that the equation of the normal to the curve at P is 2y + x = 2 + 312 (4)

(b) Use calculus to find the exact area of the shaded region (4)

(Total 8 marks)

at

)e Cx

-Cx 2- - 2.7, ) C

s X3-4 x2 — 2 x?" +-

)c- -. a -pc.

e P p. x -

t a —12-r2- 1- 12...ra-

— I x t 2 -+ 3 521 2-

41.

(X 2)

a

ro_ote-A v,ef —

J..601 1 ̂ riot t a aF P 1AILer. x.

C At ok•ve. re c•• procoji

J FL) 3 — ( SW Fs" C 2 — FL)

2 J-2-

e C 2_--ri. 2.r23

et. 0.-hav, AOCrv.0 I

QT)

.— 2 J -2_ — I ( 2 et .12: 2.

care.?.-1 cLato

-ix r I - 1

-2 -

— I

3

3 o • S

1) 014,, c. .r 01:".0.1-,2 o.P D \NAiker. c = - 2. _3.172_

_ 3 n

1/4.1

e • 9 C 9 _1(-2-) A' -.L.--

SLo..deJ Preto. = Ar2r.. nT y Curve x3 DP row, K Ec. 2,

2 - (2 -.19

A eta. ...de, Curve

X Es 3

I x 3

a)c)

El̂ es- olecl Greg = Are a. oP Tea e - re CA. 1.4A. Cita/ Latr tet