Advertisements
Advertisements
प्रश्न
If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 – 3abc = – 25.
Advertisements
उत्तर
Given: a + b + c = 5 and ab + bc + ca = 10
We know that: a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
= (a + b + c)[a2 + b2 + c2 – (ab + bc + ca)]
= 5{a2 + b2 + c2 – (ab + bc + ca)}
= 5(a2 + b2 + c2 – 10)
Given: a + b + c = 5
Now, squaring both sides, get: (a + b + c)2 = 52
a2 + b2 + c2 + 2(ab + bc + ca) = 25
a2 + b2 + c2 + 2 × 10 = 25
a2 + b2 + c2 = 25 – 20
= 5
Now, a3 + b3 + c3 – 3abc = 5(a2 + b2 + c2 – 10)
= 5 × (5 – 10)
= 5 × (–5)
= –25
Hence proved.
APPEARS IN
संबंधित प्रश्न
Evaluate the following product without multiplying directly:
103 × 107
Factorise the following using appropriate identity:
`x^2 - y^2/100`
Expand the following, using suitable identity:
`[1/4a-1/2b+1]^2`
Give possible expression for the length and breadth of the following rectangle, in which their area is given:
| Area : 35y2 + 13y – 12 |
Simplify the following expressions:
`(x^2 - x + 1)^2 - (x^2 + x + 1)^2`
If a + b = 10 and ab = 21, find the value of a3 + b3
If \[x^2 + \frac{1}{x^2} = 98\] ,find the value of \[x^3 + \frac{1}{x^3}\]
Evaluate the following:
(98)3
If \[x + \frac{1}{x} = 3\], calculate \[x^2 + \frac{1}{x^2}, x^3 + \frac{1}{x^3}\] and \[x^4 + \frac{1}{x^4}\]
If a + b = 6 and ab = 20, find the value of a3 − b3
Find the following product:
(2ab − 3b − 2c) (4a2 + 9b2 +4c2 + 6 ab − 6 bc + 4ca)
If \[3x + \frac{2}{x} = 7\] , then \[\left( 9 x^2 - \frac{4}{x^2} \right) =\]
Evalute : `((2x)/7 - (7y)/4)^2`
Use the direct method to evaluate the following products :
(y + 5)(y – 3)
Use the direct method to evaluate :
(3b−1) (3b+1)
Expand the following:
(a + 4) (a + 7)
Evaluate the following without multiplying:
(999)2
If a - b = 10 and ab = 11; find a + b.
If x + y = 1 and xy = -12; find:
x2 - y2.
