Ruvrcw Exnncrsr 2D'1 Find the gradient of the straight line which passes through the following pairs

of points:

2 Find the equation of the straight line which passes through the following pairsof points:

a (3,4) and (2,6)

c (2,4) and (5, 5)

e (-1,0) and (-3,10)g (8, -3) and (-5, -3)

a (2,0) and (3, 3)

c (3, -1) and (7,0)

e (-1, -6) and (3,2)

a (7,4) and (1, 10)

c (3,0) and (7. -8)

b (3,9) and (5, 10)

d (4,1,) and (1, 16)

f (-7, -2) and (-4, -20)h (2, *4) and (5, -Z)

b (1,,2) and (3, 6)

d (*1,14) and (1.,79)

f (2,5) and (5, 5)

b (1,7) and (5, -3)d (5, -1) and (0, -7)

3 Find the equation of the straight line which:a Passes through (2,4) and has gradient 3 (in the form y : mx * c).

b Passes through (-2,0) and has gradient -6 (in the form y : mx + c).

c Passes through (-1, -2) and has gradient -1(i" the form ax -l by + c : 0).

d Passes through (3, 5) and has gradient ? (t"t the form ax -l by + c : 0).

e Passes through (*2, 3) and has gradient 3 (it't tn" form ax + by -t c : 0).

f Passes through (1, 1) and is perpendicular to 2y : 5x - 1.

(in the form ax + by + c : 0).

4 Find the midpoint of the following pairs of points:

5 a Find the equation of the line which is perpendicular to 5y + 3x : B and whichpasses through (4,6).

b Find the point of intersection of the two lines.

5 a Find (in the form ax + by + c : 0) the equation of the line which passes? through (1,3) and (5,6).

b Find the equation of the line which is perpendicular to the line in part a andwhich passes through (5,6).

c Find the distance between (1,3) and (5,6).

7 a Find the equation of the straight line passing through the origin andperpendicular to the line r -f 5y : 7.

b Find the equation of the straight which passes through the point (2, -5) and isperpendicular to 5x - 4y :11.

8 a Find, by solving simultaneous equations, the point of intersection of the linesx+2y:13andy-2x+1:0.

b Find the gradients of .r * 2y : 13 and y - 2x+ 1 : 0. what is the connectionbetween the two lines?

61

10

a Find the equation of the line which passes through (2,9) and (4,3)'

b Find the coordinates of the points A and B where this line crosses the x andy axes respectively.

c Find the distance between points A and B (in the forrn alb where a and b are

integers).

a Find the point of intersection of 4y + 3x: 34 and 3y - 4x:13.b Find the point of intersection of U + 3x: 59 and 3y - 4x:73-c Find the point of intersection of 4y + 3x: 34 and 3y - 4x - 38.

d Find the point of intersection of 4y * 3x:59 and 3y - 4x :38.e What shape is enclosed between the four lines 4y I 3x:34,3y - 4x:13,

4y+3x:59and 3y-4x:38?

A circle has radius 5 and centre (1, 2).

a Show that (4, 6), (5, -1), (-3,5) and (-2, -2) all lie on the circle.

b Find the equations of the tangents to the circle at (4,6) and (5, -1).c \Atrhat is the connection between these two lines?

d Hence find the area outside the circle but inside the quadrilateral made up ofthe four tangents.

Show that the triangle A(1, 5) B(3, 8) C(4,3) contains a right angle.

Three points R Q and R have coordinates (2,1), (4,7) and (1, a) respectively.Find the value of a rt,

a QR is perpendiculaf to PQ. b QPR is a straight line.

The diagram shows triangle ABC. P is the foot ofthe perpendicular from C to AB and Q is the footof the perpendicular from A to BC.

a Find the equation of the line BC.

b Find the equation of the line AQ.c Write down the equation of the line CP.

d Find the coordinates of R.

e Show that the line BR is perpendicular tothe line AC.

Points A, B and C have coordinates (1, 3), (Z 5) and (9,3) respectively.

a Find the point of intersection of the perpendicular bisector of AB and theperpendicular bisector of BC.

b Explain why the points A, B and C lie on a circle and state the coordinates ofthe centre of the circle.

The points O, P and Q have coordinates (0,0), (5,3) and (a,b) respectively. Theline OQ is perpendicular to QP.

a Show thatla * 3b : a2 + b2.

Given also that a:1,b Find the two possible values of b.

11

13

L4

15

76

62

10

1,1.

t2

The sum of the firstT terms of an arithmetic sequence is five times the seventh

term. If the fourth term is 15, find the first term and the common difference'

Find the sum of the integers between 1 and L00 which are divisible by 3.

How many terms of the arithmetic sequence 2 + 3+ + +l +... are needed tomake a total of 204?

13 The first four terms of an arithmetic progression are 2, a - b,2a * b * 7 anda - 3b respectively, where a and b are constants. Find a andb and hence findthe sum of the first 30 terms of the progression.

nx**$*lHffi *tg*'1 Write down the first four terms of the sequences whose nth terms are given for

n>1.a 2n-t1

1e-n

2 Find an expression for the nth term of each sequence.

ffi'$

b 3n-2

f n(n+2)

cnzn

8 n+I

d 10,

h (-1)'

c 4,7,'10,1.3,...a 4,8,1,2,1,6,...r 123 4

Q l'4r5,6,.,.

b 12,22,32,42,...

e 7,+,+,+,...

3 Write down the first four terms of the following sequences.

4 Write down the terms in each series and then find their sum.

A iln*L: U,l 3,Ur:41

C Un * L: -, L4:3

un

E Un + 1,: Un2, U1 : 1'0

4

a )ar+rr:"1

4

d)r(r-1)r:2

5

s 2 eD'zr

b Ltn * 1: Itn - 5, ur: 59

.un,d. u, * ,: ;,u1: 64

f ,n*7:iln3,Lh:1

5

c \zr-tr:1.

4f >3r:74t Ii)

r:21

4

b 2,,r:O'4

e 22',r:o3-.ht '

,?sr*1,

5 Write each series using the I notation.

a 3*6+9+12 b32+42+52+62

a 3+7+11,+... lOthtermc 3 + 3.1 + ... 101st term

dz+4+6+ 1oo "f,*+.++ #c

f

101+102+103+104

1171.-+-+-+-3456

5 Write down the term stated for each arithmetic series.

b 1+6+11+... 3lsttermd 40 + 39 + 38 + ... 20th term

80

10

11

12

The fifth term of an arithmetic series is 17 and the thirteenthFind the common difference, d ";; th" first term a.

The third term of an arithmetic series is 13 and the tenth termFind the common difference ""d th;;st term.

The tenth term of an arithmetic series is 67 andthe sum of the first twentyterms is 1280' Find the common difference, d andthe firstrerm, a.

If the first term of an arithmetic sequence is 3 and the hundredth term is 807,then find the sum of the first f OO ie'rils.

A girl invests €5 on the first week of the yea4, €5.50 the next, €6.00 the next andso on. How much has she invested by thu end of th" "";i'13 Find how many terms of the arithmetic sequence s,72,19,26 ... shourd beadded together in order to give a totat of more than 3000.

t- f?,ffHffiilffce has thirteen terms whose sum is 143. The third rerm is

tt illlli

the sum of the integers from 1 to 100, inclusive, which are not divisible

16 An arithmetic progression has first term a and.common difference d. Its ninthterm is 22 and the sum of its first ten terms is three times the sum of its firstfive terms. Find the values "fi u-rri a.

17 The third term of an arithmetic series is 11 and the common difference is _3.a Find the first term.b Find the smailest varue of n su'ch that the ruth term is negative.c Find the sum of the first 15 negative terms.

18 The fifth term of an arithmetic series is 3 and the common differenc e is7.a Find the first term.b Find the sum of the first ten positive terms.

Find the sum of the arithmetic series as far as the term given.a 3*7+... 20th term b 300 + 295 + ..,d 2+72+...c 100 + 102 + ... 1001stterm

1 The first term of an arithmetic progression is 14 and the 20thi Find the common difference.

ii Find the sum of the first 500 terms.

41st termnth term

term is 57.

term is 25.4.

tOCR]

81

$'1';,,.i9fi.$[g#:m*#* bciff$

7 y:(x*2)(x+3)

3 y:(2x-3)(x+1)

5 Y: xz(x - 5)

7 y:(x+3)2

9 y:(2x-g)z

7l y:x(x*5)2

13 y: r-z

L5 Y: x-z + x2

717 a:-

a,)19 y:-

3221 y:-+-xx'

123 y:x'*-

1

25 y:x2+7+ "

3l Y:x2+7xatx:3

33 Y : 3x3 - 7x * 10 at (-7, 14)

135 Y:-+ xatx:2

x

findlfor the equations in questions 1 to 30.dxr

27 ,:,(, *1)

/ 7\2,:\**;)

2 y: (x-5)(r+1)

4 Y: x(3x - 16)

6 Y: xz(x * 11)

8 Y:(x-4)2

10 Y: xz(x - 1)2

L2 Y:2(x+3)2

74 y: g+

16 Y: x-1 + x3

118 y:- r

A2O u:1ux3

L?22Y:'-",x

24 a: r'- Lx'

32 y:5x*9atx:10

34 Y:4-3x-3x2at(0,4)4

36 y:--3x +4at(1,5)x.

26

28

30

,: *(, *1)

v:;("*"*r)

v:(z*.+)'

For questions 3L to 37 find the gradient of the curve at the point indicated.

99

37 11 : 1 * ,, at (-2,5)JN x'

38 Find the gradient of the curveA : x2 * 3x - 4 at the two pointswhere y :6.

+,

+S

:r_

39

40

4't

42

Find the r-coordinate of the point where the gradient of the curvea : x2 - 3x -t 5 is equal to 1.

Find the coordinates of the point on the curve U : x2 - 6x * 7 where the

gradient is zero.

Find the r-coordinates of the points on the curve y : x +1at which thetangent is parallel to the x-axis.

A(1,5) and B(4, 11) are two points on the curve ! : x2 - 3x -t 7. Calculate thegradient of the chord AB. The point C is on the curve between A and B wherethe tangent to the curve is parallel to the chord AB. Find the x-coordinate of C.

Find the x-coordinate of the point on the curve V :

+ - 3xz +Z where the

tangent is parallel to the tangent drawn at the point (1, -2).

44 Find the coordinates of the two1

points on the curve A : 4x * : wherex

the gradient is zero.

l,

n

;

:

@ ,,.,lit*i1"1

lJlirmI -

f,ti' ,' ,iliril*,rLrr

rlllj iiilllilrril,.

43

45 Find the coordinates of the point on the curve y : 3x+ { where the gradienris 1.

46 a Calculate$when A:x2-x-11.dxb Find the coordinates of the point on the curve when the gradient is equal :ic Find the equation of the tangent at this point in the forrny : mx * c.

Sr -'

il-

100

47 a Calcuiate$when A : N3 - 9x2 + 17x - 5.

dxb Find the equation of the tangent to the curve when x : 1 in the form

y:mxlc.

49

50

48 A curve is given by y : 3x3 - 75x2 + 66x - 28.

dua Find 3.dx

b Find the coordinates of the points where the gradient of the curve is 42.

A curve is given by y :2x3 * 9xz -f 12x -t 7.

a Find 9.dx

b Find the coordinates of the two points where the gradient of the curve is zero.

A curve is given by y :3x3 + x2 - 2x * 1.

a Find I,dx

b Hence find the equation of the tangent to the curve at (1,3) in the formy:mx*c.

c Find the coordinates of the other point where the tangent hits the curve.

a Look at the graphs above and find the values of the constants a, b, c, p, q, r and. s.

b Find the equation of the normal to each curve at the point where the curvecuts the y-axis.

52 A tangent to the curve A : x2 cuts the y-axis at (0, -9). Find the coordinates of

the point on the curve where the tangent is drawn. state both answers.

i3 A tangent to the curve U : 4 - 12 cuts the y-axis at (0,72). Find the coordinates

of the point on the curve where the tangent is drawn. State both answers.

101

Findy in the following:da dv

a a:6x2 + 7 b :t:6x' -l2x cclx ax

du 2x-t5 dv G+2\k-2)!f - -

--- E - . Idx x' dx x'

Find the following indefinite integrals:

^ ltr, a*

d lli@ + 3) dx

rl+*

rx*4bl - dxI 2xJ

txe l.-dxJVx

^u3-)n l1---:axJ 3xz

n_c l(Vx-lXVx+1)dx

J

rxz-7f | " dx

. lx*1r I -dxJ Zlx

Find an expression f.or y in terms of x in the following:

^ +:2x *3 given thaty: 5 when x:1..dxadu1

b i;: ,li* 4x given thaty: 0 when x: 4.

duc i;: 6xz + 2x + 3 given thaty: 10 when x:2.

r3x -l 4a show that jff dx : lx? + 4) * c.

rx2+7 G2-7\b Show that J "z

d*:

-

L.

The gradient function of a curve it # : Gx - 1,)(x+ 3). The curve passes

through the point (1,,1,2). Find the equation of the curve.

The gradient of a curve it 9d!;: rx + !.The curve passes through the point

(0, n). Find the equation of the curve.

The gradient of a curve t, # : nx -t b.The curve Passes through the points

(1,3) and (0,4). At the point (1,3) the gradient of the curve is 1. Find theequation of the curve.

The gradient of a curve ir # : ax -t b.The curve passes through the points

(2, 8) and (1, -5). At the point (2, 8) the gradient of the curve is 17. Find theequation of the curve.

A curve has an equation which satisfies !: Ar{, - x), where A is a constant.dx

The gradient of the curve at the point (2, -3) is -12.a Find the value of A.

b Find the equation of the curve.

du:" :6x-2 + 2x 3

dx

du /2x -17\zJtl

dx \"')

t2

13

14

II RerhquRemr

1 .i--:

4.f=

- --1

'l-

t; - -

:; ., --

;:lll - --l

1.0

L1

lillll I

110

13

1,4

The gradient of a curve i, # : axz + bx* c. The curve passes through the

points (0,1,), (1,3) and (2,1,5). At the point (0, 1) the gradient of the curve is -1.Find the equation of the curve.

The gradient of a curve ir # : axz + bx* c. The curve passes through the

points (0,3), (-1, -1) and (1,5). At the point (1,5) the gradient of the curve is3. Find the equation of the curve.

The gradient of a curve t, #: # @ + 0).The curve passes through the

points (7 , 20) and (4, 42) . At the point (7 , 20) the gradient of the curve is 7. Findthe equation of the curve.

Rrvrsw Exrncrsn 5CIn questions 1 to 20, findy in terms of r given the following expressions for ff.Remember the integration constant c.

-L x3

4 x5+3

7 2x3 - 3xz

x110 -+-3413 1 -7x16 x2(x - 3)

\9 xz

2x2

5 x'-+

I xa-x

L'L 3xa * 6x

1.4 (x + 7)(x + 2)

17 x(x - 1)(r + 1)

20 x-3

3 xa+x

6 x6+3xx29 --72

72 x*x2+x3

15 (2x - 1Xx + 3)

18 (2x + 1)2

zt lfx,

+ 4) dx

24 I$ - 2x-r x2) dx

r/-x\27 ll3xz -t - I dxJ\ 4)

n lxi

ax

zz lx@

+ 1) dx

Perform the foliowing integrals:

22 l0 *zia, zsj'

r/x 1rtt Jlr * u1o* 26

zg fggr' - 99) dx 2sJ

T [2nx

dx 32

u lzx(x+

4) dx 3s

35 Given that+:2x, findy in terms of x if y : 1 when x :2.cl.-r

:- Given trrat l$ :2x +4, find y in terms of x if A

: _ |when x : 0.

Jl" * 2x) dx

/f"'" * 10) dx

[x-2 dx

Jxloo dx

Jr" - slr' + 3) dx

111

38

39

,tq*:3x2 - 1,, find,yin terms of. xif y:\Iwhen x:'1..

,tH: 3x(x * 2), find,V in terms of. x if U : 72when x : -1..

In questions 40 to 45, find the equation of the curve given the gradient of the curveand the coordinates of a point on the curve.

n, #: 2x * z; (2, -7)

n H:6x -4; (0, -10)

45 +:6x2, ('1,,7)d.x

46 If *: r, * 5 and a :1L when t :2, findu in terms of f.dx

47 If *:Ur'-4 and z:l wheny:0, findzintermsofy.dy

48 If *:Ur'-2t and s:55 when f :3, findswhenf :1.dt

da49 If dr:r, -1and o:8 when /:3, findowhent:2.

Work out the following:

fg: 'jExl

1F

lFu

fi t',filL

. "Tof

ffi TM*q

lj

]il[llilmrlm

no *:2x-3;(0,4)

n #: 4x - 7; (1,, -4)

nu H:3x2 - 2; (2, s)

2F.i

tl

3i

ii

d._

so /aViar

* [xt7ax

sr lxt

ax

s4 lL a,llx

,,!ryo.

s2 I\rx

+ 5

ss /tr,6 - D2 dx

r5x*7ssj " dr'uJ

x*6-\x

t4*3x59 Show that l# dx : lx(x + 4) + cI 2\tx

5o show that lry dx :I x\lx

61 Work outffa*

J&

4x*6

-Vx

+c

62 Acurve has an equation which satisfies f,: o*tr - 2), where a is a constanl

Given that the gradient of the curve at the point (3,5) is 9,

a find the value of a

b find the equation of the curve.

1,1,2