BMM10233_Chapter 4_Theory of Equations

8/11/2019 BMM10233_Chapter 4_Theory of Equations

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Chapter 4 | Theory of Equations | BMM10233 | 59 of 204

Wheee lmil eessi is eute t ume the eessi euti is tieF emle + + et e eutis

Degree of an equation:

The highest we f the ile eset i the euti is its egee A fist egee euti is lle lie euti

se egee euti is uti euti thi egee euti is teme s ui euti while futh egee

euti is lle iuti euti

Solution of an equation:

The lue lues f the ile stisfig the euti e lle ts slutis f the euti The ume f ts

f euti is eul t the egee f the eutiHee lie euti hs etl e t uti euti hs etl tw ts; i geel thegee euti

hs ts

Linear Equations:

The geel lie euti i tw iles + + whee e stts lws eesets stight lie

i le If the the euti tkes the fm + Clelc

xa

= stisfies the euti hee is t

Simultaneous linear equations:

If tw me eutis e stisfie the sme lues f ile (tw me iles) the eutis e si t e

simulteus eutisF emle + + e th stisfie f hee e simulteus eutis

T sle simulteus eutis i tw iles we elimite e f the ukw utities fi the lue f the the

ukw sustitute this lue i f the gie eutis whih esults i the sluti

Emle

Sle: + ()

()

Fm () + sustitute i () ( + ) +

+ + sie + we he

Hee

Nte: Gemetill the sluti f simulteus eutis eesets the it f iteseti f the tw lies eesete

the tw gie eutis

Theory of Equations


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Quadratic Equations

Geel uti euti i e ile is + + whee e stts

The ts f this euti e gie 2b b 4ac

x2a

=

The tem etemies the tue f the ts is lle the isimit f the uti euti The Geek

lette (elt) is use t ete the isimit ie 2b 4ac = We kw tht f gie euti etl e f the fllwig is tue

() 20 b 4ac < is egtie Clel 2b 4ac is imgi Hee whe 0 < the ts e imgi

() 20 b 4ac 0. = = Hee 2b 4ac ishes Theefe the ts eb o b

2a 2a

+ =

() Whe 20 b 4ac 0. > > is sitie hee 2b 4ac is el I this se the ts e el istit If i

iti t eig sitie is efet sue the the ts e til; if is t efet sue the ts e

itil

Solving quadratic equations by factorization

If the ts f + + e el til the euti e esle it lie ft hee e esil sle

(The meth hs ee isusse i the eius hte)

Emle: Sle +

We ee t eess s the sum f tw ume whse ut is Clel stisf these itis

Hee +

+

( ) ( ) ( ) ( )

Note:

Quti eutis whse ts e imgi itil t e sle ftizti We use the fmul i suh

se

Sum, difference and product of the roots of a quadratic equation.

If e the ts f + + the2

b c b 4ac, and - =

a a a

+ = =

Forming a quadratic equation.

S f we he le hw t fi the ts f uti euti (gie the euti) Nw we shll wk the ese


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lem ie t fm the uti euti with gie ts If e the ts f uti euti we kw tht( ) ( ) e its fts ( ft theem) Hee the euie euti e witte s( ) ( ) + ( + ) +

ie

(sum f the ts) + (ut f the ts) gies the euie uti euti

SOLVED EXAMPLES

Example: 1.

Sle: +

Sol:

+

+ 20 5

x8 2

= =

Example: 2.

Sle: + [ { ( )}] +

Sol:

+ [ { + }] +

+ [ + ] +

+ [ ] +

+ +

+ +

66

x 233

= =

Example:3.

Sle: ( + )

(

) + 9Sol:

+ +9 + + 9

+ + 9

B tssig like ulike tems 9

Example:4.

Sle:x 2 x 1

x 1 x 5

+=

+

Sol:

( ) ( + ) ( ) ( + )

B ss multilig

+

9


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SOLVING LINEAR SIMULTANEOUS EQUATIONS

Csie the euti + Bth e simle simulteus euti i tw ukws th the

eutis e stisfie the sme lues whih e kw s the slutis ( ts) f the euti

Whe i f simle simulteus eutis i tw ukw s is gie we fist elimite e f the tw

ukw s fi the lue f the ukw The sustitutig the lue f i f the euti the lue

f is fu ut

Example: 5.

Sle: + 9 + 9

Sol:

We he + 9 ()

+ 9 ()

+ 9

Multilig () + 9 ()

Suttig euti () fm euti ()

Sustitutig i () we get + 9 9 9

Hee the euie slutis e

Example: 6.

Sle: + 9

Sol:

We he ()

+ 9 ()

Multilig ()

Fm () + 9Aig

Sustitutig i () we get

( )

+


Example: 7.

Sle:a b a b

6 and 0.2 4 5 2

+ = =

Sol:

We hea b

62 4

+ = ()

a b0

5 2 = ()

Multilig (i) +

Multilig (ii)

B Suttig


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Sustitutig i (i)

a 4 a a6 or 6 1 or 5 or a=10

2 4 2 2+ = = =

Hee

Example: 8.

Sle:7 8 2 12

2, 20p q p q

+ = + =

Sol:

We he7 8

2p q

+ = ()

2 1220

p q+ = ()

Multilig (i) iels + (iii)

Multilig (ii) we ti + (i)Multilig (iii) ; + ()

Fm (i) () we get

+ + + Sustitutig i (i)

7 8 12 or 2p 1 p

p p 2 = = =

sie it fllws tht1 1

q2 2

= =

Hee

1 1p , q

2 2= =

Example: 9.Sle: u + u

Sol:

We he u + (i)

u (ii)

Multilig (i) u + (iii)

Aig (ii) (iii) we get u u

Sustitutig u i (i) () + + 6 3

4 2 =

Theefe u 3

v2

=

Example: 10.

Sle:2 3 5 8 1

2 and 5x y x y 6

+ = + =


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Sol:

We he2 3

2x y

+ = (i)

5 8 15

x y 6+ = (ii)

1 1Put u and v in (i) and (ii)

x y= =

Then 2 u 3v 2+ = (iii)

1and 5u 8 v 5

6+ = (i)

Multil (i) u + Multil (iii) u +

1By Subtracting, 2 u 1 or u in (iii),

2 = =

1 1

2 3v 2 or 1 3v 2 o r v2 3

+ = + = =

Theefe the euie slutis e1 1

u , v2 3

= =

Sie1 1

x we have x 2. Also y y 3.u v

= = = =

Example: 11.

2 2 1 3 2and 0

p q 6 p q+ = + =

Sol:

Solving simultaneous equations by the rule of cross multiplication

The slutis f lie simulteus eutis

+

+

+

+

e gie

1 2 2 1 1 2 2 1 2 21 1

x y 1

b c b c c a c a a b a b= =

Whee

1 2 2 1 1 2 2 1

1 2 2 1 1 2 2 1

b c b c c a c ax , y

a b a b a b a b = =

Emle: Sle: + +

We he + (i)

A + (ii)

Fm (i) (ii) meth f ss multiliti we get


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1 2 2 1 1 2 2 1 2 21 1

x y 1

b c b c c a c a a b a b= =

x y 1

21 10 20 9 3 14= =

+ +

x y 1 11 11or x 1 and y 111 11 11 11 11

= = = = = =


Practical problems leading to simple equations

Example: 12.

A m tels fm A t B t the te f km hu H he tele t the te f2

35

km hu he wul he tke

hus me t e tht iste Fi the iste etwee A B

Sol:

Let the iste etwee A B e km

Time tke t tel kms t km/ hx

= hours4

Time euie t e kms t2 x 5x

3 km/hr 25 17

35

= =

5x x 20 x 17 xGiven that 3 or 3 or x 68

17 4 68

= = =

Hee the iste etwee A B is km

Example: 13.

The mthl imes f tw ess e i the ti :9 thei mthl eeitues e i the ti : If eh ses

Rs e mth fi thei mthl imes

Sol:

Let the mthl imes f the tw ess e Rs & Rs 9 The thei mthl eeitues e Rs ( ) Rs (9

)

Gie tht7x 100 12

9x 100 16

=

Hee Theefe the mthl imes e Rs Rs 9 Rs 9Example: 14.

Puj is fu times s l s Mhu es hee she will e twie s l s Mhu will e Wht e thei eset ges?

Sol:

Let ujs ge e P Mhus ge e M

Gie P M (i)


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(P + ) (M + ) (ii)

Elimitig P etwee (i) (ii) we he

M + (M + ) (sie P M)

M M

P M

Puj is es l Mhu is e l

Example: 15.

hi tles e wth Rs 9 while his tles e wth Rs Wht is the lue f eh hi

tle?

Sol:

Let the st f the hi e Rs the st f tle e Rs

Gie + 9 (i)

+ (ii)

Multil (i) (ii)

We he + 9

+ Suttig

Sustitutig i (i) + 9

Theefe the st f hi is Rs the st f the tle is Rs

Example: 16.

A eti tw igit ume is fu times the sum f its igits if is e t it its igits e eese Fi the ume

Sol:

Let e the igit i the tes le the igit i the uits le

The the ume is +

The ume fme eesig the igits will e +

The sum f the igits is +

Gie tht + ( + ) (i)

A + + + (ii)

Fm (i)

(iii)

fm (ii) 9 + 9 + (i)

B sustitutig + i (iii) ( + )

ie 9

B sustitutig i (iii) Theefe The ume is

Example: 17.

Tw ess A B km t e tgethe i hus if the wlk i the sme ieti e tgethe i hus if the

wlk tws eh the Wht is thei te f wlkig?

Sol:

Let As see e km e hus Bs see e km e hu (let > ) whe wlkig i the sme ieti A gis (

) kmh i hus he gis ( ) kms

( ) ( ) (I)


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whe wlkig tws eh the the lse the iste etwee them eue t the te f ( + ) kmh i hus

the lse iste f ( + ) kms

( + ) + (ii)

(i)

+ (ii)

Aig we get kmh kmh A wlks t kmh B wlks t kmh

Example: 18.

A gu f ele wet t htel If thee h ee me ele i the gu eh wul he i Rs less If thee h

ee less ele eh wul he i Rs me Hw m ele wet t the htel hw muh i eh es

?

Sol:

Let the ume f ele e the mut i t eh es e

The ttl ill mut

If thee wee me ele ie ( + ) ele the eh wul he i Rs less ie eh wul he i Rs ( )

Nte tht the ttl ill mut is ssume t e stt

Hee ( + ) ( ) + ie (i)Simill if thee wee less ele eh wul he i Rs me

Theefe ( ) ( + ) (i)

(i)

(ii)

Multil (i)

Multil (ii) + 9

Aig

B sustitutig i (i) Theefe ess wet t the htel eh i Rs

Example: 19.

Diie it tw ts s tht e futh the gete t m e eul t e thi f the lesse tSol:

Let the gete t e The the lesse t ( )

1 1Given x (greater part) x (lesser part)

4 3=

1 1 x 28 x x x xx (28 x) or

4 3 4 3 3 4 3 28

7 x 28or or x 16 and the part is 28 16 12

12 3

= = + =

= = =

Example: 20.

Seeis Suesh stt t the sme time stes fm Bgle Mse tws eh the t the see

kmh kmh esetiel At the it whee the meet Suesh elizes tht he hs tele kms me th

Seeis Fi the iste etwee Bgle Mse

Sol:

Let the iste tele Seeis e km the iste tele Suesh e ( + ) km

The time tke Seeis t tel km Time tke Suesh t tel ( + ) km


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x x 25or 35 x 25 x 625 or x 62.5

25 35

+= = + =

Diste etwee Bgle Mse + + + + km

Example: 21.

Fi the fti whih is eul t3

5whe th its umet emit e iese whih is eul t

2 2 22a b c whe th e iese 9

Sol:

3

7

Equation reducible to quadratic type

Sme eutis whih e t uti e eue t uti eutis e sustituti e sle

usig e f the eius meths

Example: 22.

21 1

x 8 x 12 0x x

+ + + =

Sol:

2 21Let x y then y 8y 12 0 or y 2y 6y 12 0

x+ = + = + =

( ) ( )

( ) ( ) Theefe

2

21 x 1x 2 or 2 or x 2x 1 0x x++ = = + =

ie + ( ) ( ) ( )

221 x 1x 6 6, i.e., x 6x 1 0

x x

++ = = + =

Hee

2b b 4acx

2a

=

B sustitutig the lues f we get

2

( 6) ( 6) 4.1.1x2

=

6 32 6 4 2x 3 2 2

2 2

= = =

Hee the ts e 1, 1, 3 2 2


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Radical Equations

Eutis ilig sus like ax b+ 2ax bx c+ + 3 ax b+ et e lle il eutis Meths f slig

suh eutis e isusse elw

Example: 23.

Sle: 3x 8 x 2 = Sol:

Suig th sies we get

( ) ( )

+ +

+

+

( ) ( ) ( ) ( ) Hee the ts f the euti e

Example: 24.

Diie it tw ts suh tht the sum f thei eils is1

6

Sol:

Let e e f the tw ts The the the t is

B the gie iti1 1 1

x 25 x 6+ =

225 x x 1or 25 6 25x x

x(25 x) 6

+= =

+

+

( ) ( ) Theefe Hee the euie ts e

Example: 25.

Fi tw seutie ee umes suh tht the sum f thei sues is

Sol:

Let the ume e +

2 2x (x 2) 100+ + =

2 2or, x x 4 4 100+ + + =

2 2or, x x 4x 4 100+ + + =

2or, 2x 4x 4 100 0+ + =

2or, 2x 4x 4 96 0+ + =


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or, 2(x 8) 6(x 8) 0+ + =

or, 2(x 8) 6(x 8) 0+ + =

or, (x 8)(x 8) 0 + =

or, x 8, x 6= =2or, x 2x 48 0+ =

Neglet s ume e

Example: 26.

Fi the sum ut f the ts f the euti +

Sol:

Hee

Let e the ts f the euti The the sum f the ts

b 4 4

a 3 3

+ = = =

ut f the ts

c 5

a 3 = =

Example: 27.

Disuss the tue f the euti 2x 7x 12 0+ + =Sol:

Hee

2 2b 4ac 7 4 1 12 49 48 1 = = =

sie 2b 4ac 0, > the ts e el istitTheefe the ts e el istit

Example: 28.

Disuss the tue f the ts f the euti 2x 4x 4 0+ + =Sol:

Hee

2 2b 4ac 4 4 1 4 16 16 0 = = =Theefe the ts e el eul

Example: 29.

Fi the euti whse ts e { }Sol:

Hee the ts e Theefe the euie uti euti is

(x 3)(x 4) 0 + =

2 2x 4x 3x 12 0 or x x 12 0+ = + =


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Example: 30.

If 2and are the roots of 5x x 2 0, find the value of =

(i)2 2 + () 3 3 +

Sol:

2 2 3 321 31,25 125

+ = + =


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If e the ts f the euti + fi if + 7

4(Aess Ce )

()1

2 ()1

2

()

1

2 () Dt isuffiiet () Ne f these

If e the ts f the euti + + theb

a

+ + =

(Aess Ce )

() () () () ()

If e the ts f the euti + k + suh tht the lue f k is (Aess Ce )

() () 5 () 1 () 6 () 7

Detemie k s tht the euti

+ k hs tw istit ts (Aess Ce )() k> () k < () k < () k > () 2k >

F wht lues f k the euti + (k ) + k hs eul ts? (Aess Ce )

() () () () () Ne f the e

If e el the the ts f the euti ( ) + ( + ) ( ) e (Aess Ce )

() Rel eul () Cmle () Rel ueul () Dt isuffiiet () Ne f these

If is suh tht the2n 2 n (n 4) 16

x satisfiesn 4 n 4

+ + +=

+ +(Aess Ce )

()


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If + the the lue f : is (Aess Ce )() : & : () : & : () : & : () : & : () : & :

If the ts f the euti ( ) () + ( ) f e el eul the the lue f + +

is (Aess Ce )

() () () ze () () 2 2 22a b c

Oe futh f he f ws is i the fest Twie the sue t f the he hs ge t mutis f the emiig

e the ks f ie The ttl ume f ws is (Aess Ce )

() () () () ()

If e the tw ts f uti euti suh tht + the the uti euti hig

s it ts is (Aess Ce )

() + + () + () + () + + 9 () Dt isuffiiet

If e the ts f the uti euti + + the the lue f2 2

+

is (Aess Ce )

()

3

2

2bc a

b c

()

3

2

3abc b

a c

()

2

3

3abc b

a c

()

2

2

ab b c

2b c

() Ne f these

Whih f the fllwig eessis t e eul t ze whe X X ? (Aess Ce )

() X X + () X 9 () X X + () X X + 9 () X X

A B sle uti euti I slig it A me mistke i the stt tem tie the ts s &

while B me mistke i the effiiet f tie the ts s & The et ts f the euti e

(Aess Ce )

() () () () ()

The lue f i the eutix 1

1 x 2 is1 x 2

+ =

(Aess Ce )

() / () / () 9/ ()8

13() Ne f these

9 If + + the whih f the fllwig t e lue f ( + + )? (Aess Ce 9)

() () / () / () ()

If P /+ /the whih f the fllwig is tue? (Aess Ce )

() () + () + () + + () Ne f these


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SCORE SHEET

9

9

Use HB pencil only. Abide by the time-limit


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If th elg t the set { } the the ume f euti f the fm + + hig el ts is

(Aess Ce )

() () () () ()

If + i 3 is t f the euti + + whee e el the ( ) is (Aess Ce )

() ( ) () ( ) () ( ) () ( ) () Ne f these

The ume f uti eutis whih e uhge suig thei t is (Aess Ce )

() () () () ()

Directions for (Que. 4 - 5):I eh f these uestis tw eutis I II e gie Yu he t sle th the eutis

gie swe () if < ; () if ; () if ; () if () if >

I + II + (Aess Ce )

I 9 II 9 + (Aess Ce )

F wht lue f the ts f the euti + + stisf the itis 2 + >

(Aess Ce

)

() < >9

2() > () < < ()

9a

2< () >

If f() + g() + the the ts f the uti euti g[f()] will e (Aess Ce )

() () () 1 2, 1 2 + () ()

If tw uti eutis + + + + he mm t the whih f the fllwig sttemet

hl tue? (Aess Ce )

(A) + (B) (C) / / (D)

() A B () B C D () A C D () A B D () A B C

9 The lue f f whih the sum f the sue f the t f euti ( ) is lest is (Aess Ce

9)

() () / () () () /

If { } the the ume f the eutis f the fm + + hig el ts is (Aess Ce )() () () () ()

Practice Exercise - 2


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()

s ()

()

()

()

()

()

()

9 ()

()

()

()

()

()

()

()

()

()

9 ()

()

()

()

()

()

()

()

()

()

9 ()

()

Answer Key

Practice Exercise -1

Practice Exercise -2

BMM10233_Chapter 4_Theory of Equations

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