Advertisements
Advertisements
प्रश्न
Prove that a2 + b2 + c2 − ab − bc − ca is always non-negative for all values of a, b and c
Advertisements
उत्तर
We have
`a^2 + b^2 + c^2 - ab - bc - ca`
`= 2/2[a^2 + b^2 + c^2 - ab - bc - ca]` [Mulitply and divide by 2]
`= 1/2 [2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca]`
`= 1/2 [a^2 + a^2 + b^2 + b^2 + c^2 - 2ab - 2bc - 2ac]`
`= 1/2[(a^2 + b^2 - 2ab) + (a^2 + c^2 - 2ac) + (b^2 + c^2 - 2bc)]`
`= 1/2 [(a - b)^2 + (b - c)^2 + (c - a)^2]` `[∵ (a - b)^2 = a^2 + b^2 - 2ab]`
`= ((a - b)^2 + (b -c)^2 + (c - a)^2)/2 >= 0`
`∴ a^2 + b^2 + c^2 - ab - bc -ca >= 0`
hence `a^2 + b^2 - ab - bc - ca > 0`
Hence `a^2 + b^2 + c^2 - ab - bc - ca` is always non-negative for all values of a, b and c.
APPEARS IN
संबंधित प्रश्न
Expand the following, using suitable identity:
(3a – 7b – c)2
Factorise the following:
8a3 – b3 – 12a2b + 6ab2
Give possible expression for the length and breadth of the following rectangle, in which their area are given:
| Area : 25a2 – 35a + 12 |
Evaluate following using identities:
991 ☓ 1009
Write in the expanded form:
`(m + 2n - 5p)^2`
Simplify `(x^2 + y^2 - z)^2 - (x^2 - y^2 + z^2)^2`
If a − b = 4 and ab = 21, find the value of a3 −b3
If x = 3 and y = − 1, find the values of the following using in identify:
(9y2 − 4x2) (81y4 +36x2y2 + 16x4)
Find the following product:
(4x − 3y + 2z) (16x2 + 9y2 + 4z2 + 12xy + 6yz − 8zx)
If a + b + c = 9 and a2+ b2 + c2 =35, find the value of a3 + b3 + c3 −3abc
If \[x + \frac{1}{x} = 3\] then find the value of \[x^6 + \frac{1}{x^6}\].
(a − b)3 + (b − c)3 + (c − a)3 =
75 × 75 + 2 × 75 × 25 + 25 × 25 is equal to
If a + b = 7 and ab = 10; find a - b.
If a - `1/a`= 8 and a ≠ 0 find :
(i) `a + 1/a (ii) a^2 - 1/a^2`
Use the direct method to evaluate the following products:
(a – 8) (a + 2)
Use the direct method to evaluate the following products:
(5a + 16) (3a – 7)
Use the direct method to evaluate :
(0.5−2a) (0.5+2a)
Simplify by using formula :
(a + b - c) (a - b + c)
Simplify:
(3x + 5y + 2z)(3x - 5y + 2z)
