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प्रश्न
If the equation `9x^26kx+4=0` has equal roots then k =?
(a)1 or (b)-1 or 4 (c)1 or -4 (d)-1 or -4
उत्तर
(a) `1 or 4`
It is given that the roots of the equation `(x^2+2(k+2)x+9k=0)` are equal.
∴`(b^2-4ac)=0`
⇒`{2(k+2)}^2-4xx1xx9k=0`
⇒`4(k^2+4k+4)-36k=0`
⇒`4k^2+16k+16-36=0`
⇒`4k^2-20+16=0`
⇒`k^2-5k+4=0`
⇒`k^2-4k-k+4=0`
⇒`k(k-4)-(k-4)=0`
⇒`(k-4) (k-1)=0`
⇒` k=4 or k=1`
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\[x^2 - \frac{3x}{10} - \frac{1}{10} = 0\]
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The ratio of fruit trees and vegetable trees in an orchard is 3:4. If 6 more trees of each type are planted, the ratio of trees would be 6:7. Find the number of fruit trees and vegetable trees in the orchard.
The ratio of fruit trees and vegetable trees = 3:4
So, let the number of fruit trees= 3x and the number of vegetable trees = `square`
From the given condition,
`(3x + square)/(square + square) = square/square`
`square (3x + square) = square (square + square)`
`square + square = square + square`
`square - square = square - square`
`- square = - square`
`square = square`
x = `square`
∴ Number of fruit trees in the orchard = 3x = 3 × `square` = `square` and number of vegetable trees in the orchard = 4x = 4 × `square` = `square`
Hence, the number of fruit trees and vegetable trees in the orchard are `square` and `square` respectively.
