Assertion -A- - n- N- 32n leaves the remainder 1- when divided by 8- Reason- R- 9n-1-8-

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Standard XII

Mathematics

Term Independent of x

Assertion :A:...

Question

Assertion :(A): $\forall n \in N; 3^{2 n}$ leaves the remainder $1$ , when divided by $8$ . Reason: (R): $9^{n} = 1 + 8 λ$

Both (A) & (R) are individually true & (R) is correct explanation of (A).

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Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

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(A) is true but (R) is false.

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(A) is false but (R) is true.

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Solution

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).
$P (n) = 3^{2 n} = {(1 + 8)}^{n}$
$\Rightarrow P (n) = 1 + n_{C_{1}} 8 + n_{C_{2}} 8^{2} + . . . + n_{C_{n}} 8^{n}$
$\Rightarrow P (n) = 1 + 8 (n_{C_{1}} + n_{C_{2}} 8 + . . . + n_{C_{n}} 7^{n - 1}) = 1 + 8 λ$
Therefore, remainder is 1 when divided by 8.

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Assertion :(A): ∀n∈N;32n leaves the remainder 1, when divided by 8. Reason: (R): 9n=1+8λ

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).P(n)=32n=(1+8)n⇒P(n)=1+nC18+nC282+...+nCn8n⇒P(n)=1+8(nC1+nC28+...+nCn7n−1)=1+8λTherefore, remainder is 1 when divided by 8.

Assertion :(A): $\forall n \in N; 3^{2 n}$ leaves the remainder $1$ , when divided by $8$ . Reason: (R): $9^{n} = 1 + 8 λ$