Mathematical induction and Divisibility problems

Q 1. If n ∈ N, then 7²ⁿ + 2^{3n – 3}. 3^{n – 1} is always divisible by
A. 25
B. 35
C. 45
D. None of these

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Answer:-A. 25

Q 2. If p is a prime number, then n^p – n is divisible by p when n is a
A. Odd number
B. Irrational number
C. Complex number
D. Natural number greater than 1

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Answer:-D. Natural number greater than 1

Q 3. For every natural number n, n(n + 1) is always
A. Odd
B. Even
C. Multiple of 3
D. Multiple of 4

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Answer:-B. Even

Q 4. If n ∈ N, then 11^{n + 2} + 12^{2n + 1} is divisible by
A. 113
B. 123
C. 133
D. None of these

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Answer:-C. 133

Q 5. For every natural number n, n(n² – 1) is divisible by
A. 4
B. 6
C. 10
D. None of these

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Answer:-B. 6

Q 6. The statement P(n) “1 x 1! + 2 x 2! + 3 x 3! + … + n x n! = (n + 1)! – 1” is
A. True for all n > 1
B. True for all n ∈ N 
C. Not true for any n
D. None of these

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Answer:-B. True for all n ∈ N 

Q 7. For every natural number n
A. n > 2ⁿ
B. n < 2ⁿ
C. n = 2
D. n = 2n²

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Answer:-B. n < 2n

Q 8. The remainder when 5⁹⁹ is divided by 13 is
A. 6
B. 8
C. 9
D. 10

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Answer:-B. 8

Q 9. For each n ∈ N, the correct statement is
A. 2ⁿ < n
B. n² > 2n
C. 2³ⁿ > 7n + 1
D. n⁴ < nⁿ

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Answer:-D. n4 < nn

Q 10. 10ⁿ + 3(4ⁿ⁺²) + 5  is divisible by (n ∈ N)
A. 4
B. 7
C. 9
D. 17

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Answer:-C. 9

Q 11. For natural number n, (n!)² > nⁿ, if
A. n > 3
B. n ³ 3
C. n ³ 4
D.  n > 4

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Answer:-B. n ³ 3

Q 12. For natural number n, 2ⁿ (n – 1)! < nⁿ, if
A. n < 2
B. n > 2
C. n ³ 2
D. Never

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Answer:-B. n > 2

Q 13. For positive integer n, 10^{n – 2} > 81n, if
A. n > 5
B. n ³ 5
C. n < 5
D. n > 6

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Answer:-B. n ³ 5

Q 14. When 2³⁰¹ is divided by 5, the least positive remainder is
A. 2
B. 4
C. 6
D. 8

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Answer:-A. 2

Q 15. For every positive integer n, 2ⁿ < n! when
A. n < 3
B. n ³ 4
C. n < 4
D. None of these

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Answer:-A. n < 3

Q 16. x(x^n–1 – na^n–1) + aⁿ(n–1) is divisible by (x – a)² for
A. n > 1
B. n > 2
C. All n ∈ N
D. None of these

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Answer:-C. All n ∈ N

Q 17. For every positive integral value of n, 3ⁿ > n³ when
A. n > 2
B. n ³ 3
C. n ³ 4
D. n < 4

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Answer:-C. n ³ 4

Q 18. For all positive integral values of n, 3²ⁿ – 2n + 1 is divisible by
A. 2
B. 4
C. 8
D. 12

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Answer:-A. 2

Q 19. If n ∈ N, then x^{2n – 1} + y^{2n – 1} is divisible by
A. x + y
B. x – y
C. x² + y²
D. x² + xy

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Answer:-A. x + y