· PDF file... If a = 4.965, b = 2.343 & c = 2.622 then find the a3 3- b3 - c ... a3 –...

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Algebra - 1

Algebra: A branch of mathematics in which symbols, usually

letters of the alphabet, represent numbers or quantities and

express general relationships that hold for all members of a

specified set.

Important formulas:

( a + b)2 = a2 + 2ab + b2

( a − b)2 = a2 − 2ab + b2

( a + b)2 + ( a − b)2 = 2(a2 + b2)

( a + b)2 − ( a − b)2 = 4ab

( a + b)3 = a3 + b3 +3ab(a+ b)

( a − b)3 = a3 − b3 − 3ab(a − b)

( a2− b2) = (a + b)(a − b)

( a3 + b3) = (a + b) (a2 − ab + b2)

( a3 − b3) = (a − b)(a2+b2 +ab)

( a + b+ c)2 = a2 + b2 + c2+2(ab + bc + ca)

( a – b − c)2 = a2 + b2 +c2 − 2(ab + ac − bc)

( a + b + c)3 = a3 + b3 + c3 +3(a+b)(b+c)(c+a)

( a3 + b3 + c3 – 3abc) = (a+b+c)(a2+b2+c2 − ab−bc −ca)

If a + b + c = 0 then, a3 + b3 + c3 = 3abc



Previous year Question EX-1: If a3+ b3 = 9 and a + b = 3 , then find

𝟏

𝒂 +

𝟏

𝒃 ?

Sol:

(a + b)3 = a3+ b3+3ab(a+b)

33= 9 + 3ab × 3

ab = 2

1

𝑎 +

1

𝑏

= a+𝑏

ab

= 3

2

EX-2: If a = 4.965, b = 2.343 & c = 2.622 then find the

a3 - b3 - c3- 3abc

Sol:

Sol: a – b – c = 4.965 – 2.343 – 2.622 = 0

a3 – b3 – c3 – 3abc = (a - b – c )(a2+b2+c2- ab +bc-ca)

a3 – b3 – c3 – 3abc = 0

EX-3: If x2 + 4y2 + z2 + 3 = 2x + 4y + 2z, then find the value of x +

y + z .

Sol:

x2 + 4y2 + z2 + 3 = 2x + 4y + 2z

x2 – 2X+1 + 4y2 – 4y +1+ z2 − 2z +1 = 0



(x−1)2 + (2y − 1)2 + (z −1)2 = 0

( x − 1) = 0 ,

( 2y − 1) = 0 ,

x = 1

y =1

2

( z –1) = 0 ,z = 1

x + y + z = 1+ 1

2+ 1 = 2

1

2

EX-4: If a3 – b3 – c3 = 0 then find the value of a9 – b9 – c9 -3a3b3c3

Sol:

a3 – b3 – c3 = 0

a3 – b3 = c3

Cubing both side

a9 – b9 -3a3b3(a3 – b3 ) = c9

a9 – b9 - c9 – 3a3b3c3= 0

EX- 5 : If x2 = y + z, y2 = z + x and z2 = x + y, then the value of

𝟏

𝟏+𝒙+

𝟏

𝟏+𝒚+

𝟏

𝟏+𝒛 is

Sol:

𝑥2 = 𝑦 + 𝑥

𝑥2 + 𝑥 = 𝑥 + 𝑦 + 𝑧

𝑥(𝑥 + 1) = 𝑥 + 𝑦 + 𝑧



𝑥 + 1 =𝑥 + 𝑦 + 𝑧

𝑥

1

𝑥 + 1=

𝑥

𝑥 + 𝑦 + 𝑧

𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑦,1

𝑦 + 1=

𝑦

𝑥 + 𝑦 + 𝑧

1

𝑧 + 1=

𝑧

𝑥 + 𝑦 + 𝑧

1

1 + 𝑥+

1

1 + 𝑦+

1

1 + 𝑧

=𝑥

𝑥 + 𝑦 + 𝑧+

𝑦

𝑥 + 𝑦 + 𝑧+

𝑧

𝑥 + 𝑦 + 𝑧

=𝑥 + 𝑦 + 𝑧

𝑥 + 𝑦 + 𝑧

= 1

EX-6: If x2 + 1 = 2x, then the value of𝒙𝟒+

𝟏

𝒙𝟐

𝒙𝟐 −𝟑𝒙+𝟏

Sol:

𝒙𝟐 + 𝟏 = 𝟐𝒙 (𝑮𝒊𝒗𝒆𝒏)

𝒙 +𝟏

𝒙= 𝟐. . . . . . (i)

𝒙𝟒 +𝟏𝒙𝟐

𝒙𝟐 − 𝟑𝒙 + 𝟏=

𝒙𝟔 + 𝟏𝒙𝟐

(𝒙𝟐 − 𝟑𝒙 + 𝟏)

=𝒙𝟔 + 𝟏

(𝒙𝟐 + 𝟏 − 𝟑𝒙). 𝒙𝟐

=𝒙𝟔 + 𝟏

(𝟐𝒙 − 𝟑𝒙)𝒙𝟐 =𝒙𝟔 + 𝟏

−𝒙𝟑



= − (𝒙𝟔 + 𝟏

𝒙𝟑 ) = − (𝒙𝟔

𝒙𝟑 +𝟏

𝒙𝟑)

= − (𝒙𝟑 +𝟏

𝒙𝟑)

= − [(𝒙 +𝟏

𝒙)

𝟑

− 𝟑 (𝒙 +𝟏

𝒙)]

= −[𝟐𝟑 − 𝟑 × 𝟐]

= −𝟐

EX-7: Ex: If a2 + b2 +c2 = ab + bc + ca then find the value of 𝒂+𝒄

𝒃

Sol:

Sol: a2 + b2 + c2 = ab + bc + ca

2(a2 + b2 + c2 = ab + bc + ca)

a2 + b2 – 2ab + b2 + c2 – 2bc + c2 +a2 – 2ac = 0

( a – b ) 2 + ( b – c ) 2 + (c – a) 2 = 0

a = b , b = c , c = a

= 𝒂 + 𝒄

𝒃= 𝟐

EX-8 : Ex: If ab + bc +ca = 0 then find the value of 2 2 2

1 1 1

a bc b ca c ab

Sol:

Sol: ab + bc +ca = 0 so ab + ca = -bc , bc + ca = - ab , ab + bc = - ac

2 2 2

1 1 1

a bc b ca c ab

2 2 2

1 1 1

a ab ac b bc ab c bc ca

= 𝟏

𝒂+𝒃+𝒄(

𝟏

𝒂+

𝟏

𝒃 +

𝟏

𝒄 )



= 0

EX-9: a + 𝟏

𝒂= √𝟑, then the value of a6 -

𝟏

𝒂𝟔 + 𝟐 will be

Sol:

𝑎 +1

𝑎= √3

𝑜𝑛 𝑐𝑢𝑏𝑖𝑛𝑔 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠,

𝑎3 +1

𝑎3+ 3𝑎

1

𝑎(𝑎 +

1

𝑎) = 3√3

𝑎3 +1

𝑎3+ 3√3 = 3√3

𝑎3 +1

𝑎3= 0. . . . . (𝑖)

𝑎6 −1

𝑎6+ 2

= (𝑎3 +1

𝑎3) (𝑎3 −

1

𝑎3) + 2 = 2

EX-10 : Ex: If a +b + c = 0, then find s

Sol:

( a + b+ c)2 = a2 + b2 + c2+2(ab + bc + ca)

∵ a +b + c = 0 ,

a2 + b2 + c2 = − (2ab + 2bc +2ca)

2 2 2

2 2 2

a b c

a b b c c a

2 2 2

2 2 22 2 2 2

a b c

a b c ab bc ca

2 2 2

2 2 2

1

33

a b c

a b c

· PDF file... If a = 4.965, b = 2.343 & c = 2.622 then find the a3 3- b3 - c ... a3 –...

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