Let f- - be defined by fn- 2- if n-3 k- k - 10-n- if n-3 k-1- k - 0- if n-3 k-2- k - -If S-n - fn-2- then the sum of the positive elements in S is

F(n) gt;2 is possible only if n=3k+1 Here, n ∈{…, 5, 2,1,4,7,10, …} Let us verify for positive elements f(1)=10 1=9 gt;2 f(4)=10 4=6 gt;2 f(7)=10 7=3 ...

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Standard XII

Mathematics

Area between Two Curves

Let f: ℤ→ℤ be...

Question

Let $f : Z \to Z$ be defined by $f (n) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 2, & if n = 3 k, & k \in Z 10 - n, & if n = 3 k + 1, & k \in Z 0, & if n = 3 k + 2, & k \in Z \end{matrix}$
If $S = {n \in Z : f (n) > 2},$ then the sum of the positive elements in $S$ is

10

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Solution

The correct option is D $12$
$f (n) > 2$ is possible only if $n = 3 k + 1$
Here, $n \in {\dots, - 5, - 2, 1, 4, 7, 10, \dots}$

Let us verify for positive elements
$f (1) = 10 - 1 = 9 > 2 f (4) = 10 - 4 = 6 > 2 f (7) = 10 - 7 = 3 > 2 f (10) = 10 - 10 = 0 ⋮$

So, positive elements in $S$ are $1, 4, 7$
Sum of elements $= 1 + 4 + 7 = 12$

Alternate :
$f (n) > 2$
$\Rightarrow 10 - n > 2$
$\Rightarrow n < 8$
Numbers of the form $n = 3 k + 1$ are $1, 4, 7$
Sum of positive elements $= 1 + 4 + 7 = 12$

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Let f:Z→Z be defined by f(n)=⎧⎪⎨⎪⎩2, if n=3k,k∈Z10−n, if n=3k+1,k∈Z0, if n=3k+2,k∈Z If S={n∈Z:f(n)>2}, then the sum of the positive elements in S is

Let $f : Z \to Z$ be defined by $f (n) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 2, & if n = 3 k, & k \in Z 10 - n, & if n = 3 k + 1, & k \in Z 0, & if n = 3 k + 2, & k \in Z \end{matrix}$
If $S = {n \in Z : f (n) > 2},$ then the sum of the positive elements in $S$ is