Principle of Mathematical Induction Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. If P(n) : “46
n
 + 16
n
 + k is divisible by 64 for n ? N” is true, then the
least negative integral value of k is.
(a) – 1
(b) 1
(c) 2
(d) – 2
2. A student was asked to prove a statement P(n) by induction. He proved
that P(k + 1) is true whenever P(k) is true for all k > 5  N and also that
P (5) is true. On the basis of this he could conclude that P(n) is true
(a) for all n  N
(b) for all n > 5
(c) for all n  5
(d) for all n < 5
3. Let T(k) be the statement 1 + 3 + 5 + ... + (2k – 1)= k
2
 +10
Which of the following is correct?
(a) T(1) is true
Page 2


PART-I (Single Correct MCQs)
1. If P(n) : “46
n
 + 16
n
 + k is divisible by 64 for n ? N” is true, then the
least negative integral value of k is.
(a) – 1
(b) 1
(c) 2
(d) – 2
2. A student was asked to prove a statement P(n) by induction. He proved
that P(k + 1) is true whenever P(k) is true for all k > 5  N and also that
P (5) is true. On the basis of this he could conclude that P(n) is true
(a) for all n  N
(b) for all n > 5
(c) for all n  5
(d) for all n < 5
3. Let T(k) be the statement 1 + 3 + 5 + ... + (2k – 1)= k
2
 +10
Which of the following is correct?
(a) T(1) is true
(b) T(k) is true T(k + 1) is true
(c) T(n) is true for all
(d) All above are correct
4. Let . Then which of the following is
true?
(a) Principle of mathematical induction can be used to prove the
formula
(b)
(c)
(d) is correct
5. For natural number n, , if
(a)
(b)
(c)
(d) Never
6. For all positive integral values of n,  is divisible by
(a) 2
(b) 4
(c) 8
(d) 12
7. For every natural number  is always
(a) Even
(b) Odd
(c) Multiple of 3
(d) Multiple of 4
8. If a
n
having n radical signs then by methods of
mathematical induction which is true?
Page 3


PART-I (Single Correct MCQs)
1. If P(n) : “46
n
 + 16
n
 + k is divisible by 64 for n ? N” is true, then the
least negative integral value of k is.
(a) – 1
(b) 1
(c) 2
(d) – 2
2. A student was asked to prove a statement P(n) by induction. He proved
that P(k + 1) is true whenever P(k) is true for all k > 5  N and also that
P (5) is true. On the basis of this he could conclude that P(n) is true
(a) for all n  N
(b) for all n > 5
(c) for all n  5
(d) for all n < 5
3. Let T(k) be the statement 1 + 3 + 5 + ... + (2k – 1)= k
2
 +10
Which of the following is correct?
(a) T(1) is true
(b) T(k) is true T(k + 1) is true
(c) T(n) is true for all
(d) All above are correct
4. Let . Then which of the following is
true?
(a) Principle of mathematical induction can be used to prove the
formula
(b)
(c)
(d) is correct
5. For natural number n, , if
(a)
(b)
(c)
(d) Never
6. For all positive integral values of n,  is divisible by
(a) 2
(b) 4
(c) 8
(d) 12
7. For every natural number  is always
(a) Even
(b) Odd
(c) Multiple of 3
(d) Multiple of 4
8. If a
n
having n radical signs then by methods of
mathematical induction which is true?
(a)
(b)
(c)
(d)
9. For every positive integral value of n, 3
n
 > n
3
 when
(a)
(b)
(c)
(d)
10. If  then P(n) is true for
(a) n = 1
(b) n > 0
(c) n < 0
(d) n = 2
11. If , then  is divisible by
(a) x + y
(b) x – y
(c) x
2
 + y
2
(d) x
2
 + xy
12. For a positive integer n,
Let a(n) = 1 +  + … + . Then
(a) a(100) = 100
(b) a(100) > 100
(c) a(200) = 100
(d) a(200) < 100
13. 2
n
 > n
2
 when n ? N such that
Page 4


PART-I (Single Correct MCQs)
1. If P(n) : “46
n
 + 16
n
 + k is divisible by 64 for n ? N” is true, then the
least negative integral value of k is.
(a) – 1
(b) 1
(c) 2
(d) – 2
2. A student was asked to prove a statement P(n) by induction. He proved
that P(k + 1) is true whenever P(k) is true for all k > 5  N and also that
P (5) is true. On the basis of this he could conclude that P(n) is true
(a) for all n  N
(b) for all n > 5
(c) for all n  5
(d) for all n < 5
3. Let T(k) be the statement 1 + 3 + 5 + ... + (2k – 1)= k
2
 +10
Which of the following is correct?
(a) T(1) is true
(b) T(k) is true T(k + 1) is true
(c) T(n) is true for all
(d) All above are correct
4. Let . Then which of the following is
true?
(a) Principle of mathematical induction can be used to prove the
formula
(b)
(c)
(d) is correct
5. For natural number n, , if
(a)
(b)
(c)
(d) Never
6. For all positive integral values of n,  is divisible by
(a) 2
(b) 4
(c) 8
(d) 12
7. For every natural number  is always
(a) Even
(b) Odd
(c) Multiple of 3
(d) Multiple of 4
8. If a
n
having n radical signs then by methods of
mathematical induction which is true?
(a)
(b)
(c)
(d)
9. For every positive integral value of n, 3
n
 > n
3
 when
(a)
(b)
(c)
(d)
10. If  then P(n) is true for
(a) n = 1
(b) n > 0
(c) n < 0
(d) n = 2
11. If , then  is divisible by
(a) x + y
(b) x – y
(c) x
2
 + y
2
(d) x
2
 + xy
12. For a positive integer n,
Let a(n) = 1 +  + … + . Then
(a) a(100) = 100
(b) a(100) > 100
(c) a(200) = 100
(d) a(200) < 100
13. 2
n
 > n
2
 when n ? N such that
(a) n > 2
(b) n > 3
(c) n < 5
(d) n = 5
14. If n ? N and n is odd, then n (n
2
 – 1) is divisible by
(a) 24
(b) 16
(c) 32
(d) 19
15. For each ,  the correct statement is
(a)
(b)
(c)
(d)
16. P(n) : 2.7
n
 + 3.5
n
 – 5 is divisible by
(a) 24,  n ? N
(b) 21,  n ? N
(c) 35,  n ? N
(d) 50,  n ? N
17. By mathematical induction,
 is equal to
(a)
(b)
Page 5


PART-I (Single Correct MCQs)
1. If P(n) : “46
n
 + 16
n
 + k is divisible by 64 for n ? N” is true, then the
least negative integral value of k is.
(a) – 1
(b) 1
(c) 2
(d) – 2
2. A student was asked to prove a statement P(n) by induction. He proved
that P(k + 1) is true whenever P(k) is true for all k > 5  N and also that
P (5) is true. On the basis of this he could conclude that P(n) is true
(a) for all n  N
(b) for all n > 5
(c) for all n  5
(d) for all n < 5
3. Let T(k) be the statement 1 + 3 + 5 + ... + (2k – 1)= k
2
 +10
Which of the following is correct?
(a) T(1) is true
(b) T(k) is true T(k + 1) is true
(c) T(n) is true for all
(d) All above are correct
4. Let . Then which of the following is
true?
(a) Principle of mathematical induction can be used to prove the
formula
(b)
(c)
(d) is correct
5. For natural number n, , if
(a)
(b)
(c)
(d) Never
6. For all positive integral values of n,  is divisible by
(a) 2
(b) 4
(c) 8
(d) 12
7. For every natural number  is always
(a) Even
(b) Odd
(c) Multiple of 3
(d) Multiple of 4
8. If a
n
having n radical signs then by methods of
mathematical induction which is true?
(a)
(b)
(c)
(d)
9. For every positive integral value of n, 3
n
 > n
3
 when
(a)
(b)
(c)
(d)
10. If  then P(n) is true for
(a) n = 1
(b) n > 0
(c) n < 0
(d) n = 2
11. If , then  is divisible by
(a) x + y
(b) x – y
(c) x
2
 + y
2
(d) x
2
 + xy
12. For a positive integer n,
Let a(n) = 1 +  + … + . Then
(a) a(100) = 100
(b) a(100) > 100
(c) a(200) = 100
(d) a(200) < 100
13. 2
n
 > n
2
 when n ? N such that
(a) n > 2
(b) n > 3
(c) n < 5
(d) n = 5
14. If n ? N and n is odd, then n (n
2
 – 1) is divisible by
(a) 24
(b) 16
(c) 32
(d) 19
15. For each ,  the correct statement is
(a)
(b)
(c)
(d)
16. P(n) : 2.7
n
 + 3.5
n
 – 5 is divisible by
(a) 24,  n ? N
(b) 21,  n ? N
(c) 35,  n ? N
(d) 50,  n ? N
17. By mathematical induction,
 is equal to
(a)
(b)
(c)
(d) None of these
18. For every positive integer n, 7
n
 – 3
n
 is divisible by
(a) 7
(b) 3
(c) 4
(d) 5
19. For all n ? N, the sum of  is
(a) a negative integer
(b) a whole number
(c) a real number
(d) a natural number
20. For all n = 1,
 =
(a)
(b)
(c)
(d) None of these
PART-II (Numeric/Integer Type Questions)
21. Let P(n) : “2
n
 < (1 × 2 × 3 × ... × n)”. Then the smallest positive integer
for which P(n) is true is
22. Use principle of mathematical induction to find the value of k, where
(10
2n – 1
 + 1) is divisible by k.
23. If n is a positive integer, then 5
2n + 2
 – 24n – 25 isdivisible by