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If y=2[x]+3 = 3[x – 2] +5 then the
sum of digits of the value of
[x+y] is
([.] denotes G.I.F)


By Arya

If y=2[x]+3 = 3[x – 2] +5 then the
sum of digits of the value of
[x+y] is
([.] denotes G.I.F)

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Arya

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1 thought on “If y=2[x]+3 = 3[x – 2] +5 then the<br />sum of digits of the value of<br />[x+y] is<br />([.] denotes G.I.F)<br /><br /><br /><br”

  1. [tex]\large\underline{\sf{Solution-}}[/tex]

    Given that

    [tex]\rm :\longmapsto\:y = 2[x] + 3 – – – (1)[/tex]

    and

    [tex]\rm :\longmapsto\:y = 3[x – 2] + 5 – – – (2)[/tex]

    On equating equation (1) and (2), we get

    [tex]\rm :\longmapsto\:2[x] + 3 = 3[x – 2] + 5 [/tex]

    [tex]\rm :\longmapsto\:2[x] + 3 = 3([x] – 2)+ 5 [/tex]

    [tex]\red{\bigg \{ \sf\because \: [x + n] = [x] + n \: \: where \: n \in \: natural \: number\bigg \}}[/tex]

    [tex]\rm :\longmapsto\:2[x] + 3 = 3[x] -6+ 5 [/tex]

    [tex]\rm :\longmapsto\:2[x] + 3 = 3[x] – 1[/tex]

    [tex]\rm :\implies\:[x] = 4 – – – (3)[/tex]

    [tex]\bf\implies \:4 \leqslant x < 5[/tex]

    Or

    [tex]\bf :\implies\:x = 4 + f \: where \: 0 < f < 1 \: is \: fractional \: part[/tex]

    On substituting equation (3) in equation (1), we get

    [tex]\rm :\longmapsto\:y = 2 \times 4 + 3 [/tex]

    [tex]\rm :\longmapsto\:y = 8 + 3 [/tex]

    [tex]\bf\implies \:y = 11 – – – (4)[/tex]

    Now,

    Consider,

    [tex]\rm :\longmapsto\:[x + y][/tex]

    [tex] \rm \: \: = \: \: [4 + f + 11][/tex]

    [tex] \rm \: \: = \: \: [15 + f][/tex]

    [tex] \rm \: \: = \: \: 15[/tex]

    [tex]\bf\implies \:[x + y] = 15[/tex]

    Additional Information :-

    Definition of Greatest Integer Function

    [tex]\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y = [x] \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 \leqslant x < 1 & \sf 0 \\ \\ \sf 1 \leqslant x < 2 & \sf 1 \\ \\ \sf 2 \leqslant x < 3 & \sf 2 \end{array}} \\ \end{gathered}[/tex]

    [tex]\rm :\longmapsto\:[n] + [ – n] = 0 \: \: if \: n \: \in \: integer[/tex]

    [tex]\rm :\longmapsto\:[n] + [ – n] = – 1 \: \: if \: n \: \cancel\in \: integer[/tex]

    [tex]\rm :\longmapsto\:[x] \geqslant n \implies \: x \geqslant n \: \: if \: n \: \in \: integer[/tex]

    [tex]\rm :\longmapsto\:[x] \leqslant n \implies \: x < n + 1 \: \: if \: n \: \in \: integer[/tex]

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