The sum of all positive integral values of -a- a -1-500- for which the equation - x -3- x - a -0 has solution is Where - denote G-I-FA- 812B- 462C- 512D- 784

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Standard XII

Mathematics

How to Find the Inverse of a Function

The sum of al...

Question

The sum of all positive integral values of 'a', $a \in [1, 500]$ for which the equation $[x]^{3} + x - a = 0$ has solution is (Where [.] denote G.I.F)

462

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512

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784

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812

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Solution

The correct option is D 812
a is integer then x must be integer, i.e., [x] = x
$a = x^{3} + x 1 \leq a \leq 500 \Rightarrow 1 \leq x \leq 7, x \in I$
$\sum a_{1} = \sum_{x = 1}^{7} (x^{3} + x) = {(\frac{7 \times 8}{2})}^{2} + (\frac{7 \times 8}{2}) = 812$

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The sum of all positive integral values of 'a', a∈[1,500] for which the equation [x]3+x−a=0 has solution is (Where [.] denote G.I.F)

The correct option is D 812a is integer then x must be integer, i.e., [x] = x a=x3+x1≤a≤500⇒1≤x≤7,x∈I ∑a1=∑7x=1(x3+x)=(7×82)2+(7×82)=812

The sum of all positive integral values of 'a', $a \in [1, 500]$ for which the equation $[x]^{3} + x - a = 0$ has solution is (Where [.] denote G.I.F)

The correct option is D 812
a is integer then x must be integer, i.e., [x] = x
$a = x^{3} + x 1 \leq a \leq 500 \Rightarrow 1 \leq x \leq 7, x \in I$
$\sum a_{1} = \sum_{x = 1}^{7} (x^{3} + x) = {(\frac{7 \times 8}{2})}^{2} + (\frac{7 \times 8}{2}) = 812$