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CBSE Questions for Class 11 Engineering Maths Principle Of Mathematical Induction Quiz 3 - MCQExams.com

Let f(n) equals to the sum of the cubes of three consecutive natural numbers.
f(n) leaves the remainder zero when divided by
  • 11
  • 9
  • 99
  • none of these
For all nI+, the statement P(n)=n77+n55+23n3n105 is a natural number is true, if
  • P(1) is true
  • P(2) is true
  • P(3) is true
  • True nN
The number of values of n, for which p(n)=1!+2!+3!+4!++n! is the square of a natural number, is equal to
  • 0
  • 1
  • 2
  • 3
If nN, then n3+2n is divisible by
  • 2
  • 3
  • 5
  • 6
If P(n)=n55+n33+7n15, then statement "P(n) is a natural number" is true,
  • only for n>1.
  • only for n is an odd positive integer.
  • only for n is an even positive integer.
  • nN
Let P(n)=n3n, the largest number by which P(n) is divisible possible integral values of n is
  • 2
  • 3
  • 5
  • 6
For every natural number n -
  • n>2n
  • n<2n
  • n/2n
  • n/22n
If an=7+7+7+..... having n radical signs, then by method of mathematical induction which of the following is true?
  • an>6,n>1
  • an>3,n>1
  • an>4,n>1
  • an<2,n>1
  • Both (A) & (R) are individually true & (R) is correct explanation of (A).
  • Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
  • (A) is true but (R) is false.
  • (A) is false but (R) is true.
Let P(n):n2+n is an odd integer. It is seen that truth of P(n) the truth of P(n + 1). Therefore, P(n) is true for all -
  • n > 1
  • n
  • n > 2
  • None of these
The inequality n!>2n1 is true - 
  • For all n > 1
  • For all n > 2
  • For all nϵN
  • None of these
Let P(n)=x2n1+y2n1 is divisible by x+y as P(1) is true, then truth of P(k+1) indicates
  • P(n) s divisible by x+y only for odd positive integer
  • P(n) is divisible by x+y only for positive even integer
  • P(n) is true nN
  • P(k) is true
The sequence (xnn1) is defined by x1=0 and xn+1=5xn+24x2n+1 for all n1. Then all xn are 
  • Negative integers
  • Positive integers
  • Rational numbers
  • None of these
Let P(n)=11n+2+122n+1, then the least value of the following which P(n) is divisible by is
  • 19
  • 7
  • 133
  • none of these
For positive integer n, 3n<n! when
  • n6
  • n>7
  • n7
  • n7
Let P(n):n2+n is an odd integer. It is seen that truth of P(n) the truth of P(n + 1). Therefore, P(n) is true for all
  • n > 1
  • n
  • n > 2
  • none of these
For natural number n, 2n(n1)!<nn, if.
  • n<2
  • n>2
  • n2
  • Never
For every positive integer
n, n77+n55+2n33n105 is
  • an integer
  • a rational number
  • a negative real number
  • an odd integer
If P is a prime number then npn is divisible by p when n is a 
  • natural number greater than 1
  • odd number
  • even number
  • None of these
The difference between an +ve integer and its cube, is divisible by
  • 4
  • 6
  • 9
  • None of these
If n is a natural number then (n+12)nn! is true when.
  • n > 1
  • n1
  • n > 2
  • Never
For  every natural number n, n(n+3) is always :
  • multiple of 4
  • multiple of 5
  • even
  • odd
For all n N, n4 is less than
  • 10n
  • 4n
  • 1010
  • None of these
A student was asked to prove a statement by induction. He proved
(i) P(5) is true and
(ii) Trutyh of P(n) truth of p(n + 1), nN
On the basis of this, he could conclude that P(n) is true for
  • no n N
  • all n N
  • all n 5
  • None of these
For every positive integral value of n, 3n>n3 when
  • n > 2
  • n3
  • n4
  • n < 4
For every positive integer n,n77+n55+2n33n105 is
  • an integer
  • a rational number
  • a negative real number
  • an odd integer
P(n):32n+28n9 is divisible by 64, is true for
  • all nϵN{0}
  • n2,nϵN
  • nϵN,n>2
  • none of these
The smallest positive integer for which the statement 3n+1<4n holds is
  • 1
  • 2
  • 3
  • 4
If x > 1, then the statement P(n):(1+x)n>1+nx is true for
  • all nϵN
  • all n > 1
  • all n > 1 and x0
  • None of these
If n is a natural number then (n+12)nn! is true when
  • n > 1
  • n1
  • n > 2
  • never
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