Given below are two statements : Statement I : 25 13 + 20 13 + 8 13 + 3 13 25 13 +20 13 +8 13 +3 13 is divisible by 7 7. Statement II : The integral part of ( 7 + 4 3 ) 25 (7+4 3 ? ) 25 is an odd number. In the light

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Q.
Given below are two statements :

Statement I : $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7.

Statement II :The integral part of ${(7 + 4 \sqrt{3})}^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below : [2026]

1 Both Statement I and Statement II are true

2 Statement I is false but Statement II is true

3 Both Statement I and Statement II are false

4 Statement I is true but Statement II is false

Ans.
(1)

Statement I :

Statement II: $R = {(7 + 4 \sqrt{3})}^{25} = I + f$

$R' = {(7 - 4 \sqrt{3})}^{25} = f'$

$∴$ $R + R' = 2 [{}^{25}{C_{0}} 7^{25} + {}^{25}{C_{2}} 7^{23} {(4 \sqrt{3})}^{2} + \dots]$

$I + f + f' = even integer$

$∴$ $I = odd integer$

$∵$ $0 < f + f' < 2 \Rightarrow f + f' = 1$

$\Rightarrow Both the statements are correct$

Solution

Q.
Given below are two statements :

Statement I : $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7.

Statement II :The integral part of ${(7 + 4 \sqrt{3})}^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below : [2026]

1 Both Statement I and Statement II are true

2 Statement I is false but Statement II is true

3 Both Statement I and Statement II are false

4 Statement I is true but Statement II is false

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Solution

Q. Given below are two statements : Statement I : 2513+2013+813+313 is divisible by 7. Statement II :The integral part of (7+43)25 is an odd number. In the light of the above statements, choose the correct answer from the options given below : [2026]

1 Both Statement I and Statement II are true

2 Statement I is false but Statement II is true

3 Both Statement I and Statement II are false

4 Statement I is true but Statement II is false

Ans. (1) Statement I : Statement II: R=(7+43)25=I+f R'=(7-43)25=f' ∴ R+R'=2[C025725+C225723(43)2+⋯] I+f+f'=even integer ∴ I=odd integer ∵ 0<f+f'<2⇒f+f'=1 ⇒Both the statements are correct

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Q.
Given below are two statements :

Statement I : $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7.

Statement II :The integral part of ${(7 + 4 \sqrt{3})}^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below : [2026]