Find the value of positive integer- n - for which the quadratic equation- - k -1 n x - k -1 x - k -10 n- has solutions - and -1 for some -A- 12B- 10C- 9D- 11

The correct option is D 11k = 1n∑ (x + k 1)(x + k) = 10n ∑ [x^2 + (2k 1) x + (k 1)k] = 10n n.x^2 + n^2 x+ (n 1)n(n + 1)/3 10n = 0 Let roots be α and ...

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Standard XII

Mathematics

Relations between Roots and Coefficients : Higher Order Equations

Find the valu...

Question

Find the value of positive integer ’ n’ for which the quadratic equation, $n \sum k = 1 (x + k - 1) (x + k) = 10 n,$ has solutions $α$ and $α + 1$ for some $α$

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Solution

The correct option is C 11
$n \sum k = 1 (x + k - 1) (x + k) = 10 n \sum [x^{2} + (2 k - 1) x + (k - 1) k] = 10 n$
$n . x^{2} + n^{2} x + \frac{(n - 1) n (n + 1)}{3} - 10 n = 0$
Let roots be $α$ and $β$
$α + β = n$
$α β = \frac{(n - 1) (n + 1)}{3} - 10$
Since, difference of root is 1,
$S o, (α + β)^{2} - 4 α β = 1$
$n^{2} - 4 (\frac{(n - 1) (n + 1)}{3}) = 1 3 n^{2} - 4 (n^{2} - 1 - 30) = 3 3 n^{2} - 4 n^{2} + 124 = 3 + n^{2} = 121 n = \pm 11 n = 11$

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Find the value of positive integer ’ n’ for which the quadratic equation,n∑k=1(x+k−1)(x+k)=10n, has solutions α and α+1 for some α

Find the value of positive integer ’ n’ for which the quadratic equation, $n \sum k = 1 (x + k - 1) (x + k) = 10 n,$ has solutions $α$ and $α + 1$ for some $α$