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AP EAMCET (EAPCET) Mathematic Multiple Choice Questions (MCQs)

Mathematical Induction - Multiple Choice Questions

11. What is the principle of mathematical induction used for?

a) Proving inequalities
b) Proving divisibility properties
c) Proving statements for all positive integers
d) All of the above
Answer: d) All of the above

12. Which step is not part of mathematical induction?

a) Verifying the base case
b) Assuming the statement for \(n = k\)
c) Verifying the statement for \(n = k+1\)
d) Checking for all negative integers
Answer: d) Checking for all negative integers

13. Using mathematical induction, prove \(1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}\). What is the base case?

a) \(n = 1\)
b) \(n = 0\)
c) \(n = 2\)
d) None of the above
Answer: a) \(n = 1\)

14. Which of the following is true for the principle of mathematical induction?

a) The base case must always hold.
b) The inductive hypothesis assumes \(n = k+2\).
c) Induction works only for prime numbers.
d) Induction does not require proof.
Answer: a) The base case must always hold.

15. For \(n \geq 1\), \(2^n - 1\) is divisible by:

a) 3
b) \(2^n\)
c) \(2^{n-1}\)
d) None of the above
Answer: c) \(2^{n-1}\)

16. Prove using induction: \(n^3 + 2n\) is divisible by 3. What is the inductive step?

a) Prove \(n = k+1\) assuming \(n = k\) holds true.
b) Check divisibility for \(n = 2\).
c) Add 3 to the expression for \(n\).
d) Replace \(k\) with \(k-1\) in the expression.
Answer: a) Prove \(n = k+1\) assuming \(n = k\) holds true.

17. Which mathematical property does induction rely on?

a) Commutativity
b) Associativity
c) Well-ordering principle
d) Distributivity
Answer: c) Well-ordering principle

18. The formula \(1^3 + 2^3 + 3^3 + \ldots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\) can be proved using:

a) Mathematical induction
b) Integration
c) Substitution method
d) Differentiation
Answer: a) Mathematical induction

19. To prove \(n^2 - n\) is divisible by 2 for all \(n \geq 1\), what assumption is made in the inductive step?

a) \(k^2 - k\) is divisible by 2
b) \(k^2 - 1\) is divisible by 2
c) \(k(k-1)\) is divisible by 3
d) None of the above
Answer: a) \(k^2 - k\) is divisible by 2

20. For \(n \geq 1\), prove \(5^n - 1\) is divisible by 4. What is the base case?

a) \(n = 0\)
b) \(n = 1\)
c) \(n = 2\)
d) \(n = 4\)
Answer: b) \(n = 1\)

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AP EAMCET (EAPCET) Mathematic Multiple Choice Questions (MCQs)

Mathematical Induction - Multiple Choice Questions

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