Mathematics - Mathematical Induction and Its Application

Question - 1

P(n) is the statement "n2-n+41(n \(\epsilon \) N),is a prime".Then which of the following is not true?

  • A P(1)
  • B P(2)
  • C P(3)
  • D P(4)

Question - 2

For all n \(\epsilon \) ,\(\frac { { n }^{ 7 } }{ 7 } +\frac { { n }^{ 5 } }{ 5 } +\frac { 2{ n }^{ 3 } }{ 3 } -\frac { { n }^{ } }{ 105 } \) ,is

  • A a positive integer
  • B a negative integer
  • C 0
  • D a rational number

Question - 3

\(p^{n+1}+(p+1)^{2n-1}\ \ \ (n\epsilon N)\) ,is divisible by

  • A p
  • B p2
  • C p2+p+1
  • D p2+p

Question - 4

n(n2-1) is divisible by 24, when n is

  • A even
  • B odd
  • C any integer
  • D None of these

Question - 5

The smallest positive integer for which the inequality \(n!<\left( \frac { n+1 }{ 2 } \right) \) is true,is

  • A 1
  • B 2
  • C 3
  • D 4

Question - 6

The greatest positive integer which divides (n+1)(n+2)(n+3)...(n+k) for all n \(\epsilon\) N. is

  • A k
  • B k!
  • C (k+1)!
  • D (k+2)!

Question - 7

The number 2.7n+3.5n-5(n \(\epsilon\) N) is divisible by

  • A 24
  • B 36
  • C 48
  • D None of these

Question - 8

The number np - n ( n?\(\epsilon\)? N, P is a prime number ) is a divisible by

  • A n
  • B n2
  • C p
  • D p2

Question - 9

x(xn-1-nan-1) + an(n-1) is a divisible by (x-a)k for all n>1,then k equals

  • A 4
  • B 3
  • C 2
  • D None of these

Question - 10

Which one of the following is true for all n \(\epsilon \) N?

  • A 1+3+5+....+(2n-1) = n2
  • B 1+3+5+...+(2n-1) = n
  • C 1+3+5+....+(2n-1) = n3
  • D None of the above
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