Mathematical Induction - Gujarat Board (GSEB) STD 11 Science MATHS Questions

Q 1MCQ1 Mark

If p(n): 2n < (1 × 2 × 3 × ... × n). Then the smallest positive integer for which p(n) is true is:

A
1
B
2
C
3
✓
4

Answer: D.

Q 2MCQ1 Mark

If $10^\text{n} + 3 \times 4^{\text{n}+2}+\lambda$ is divisible by 9 for all $\text{n}\in\text{N},$ then the least positive integer value of $\lambda$ is

✓
5
B
3
C
7
D
1

Answer: A.

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Q 3MCQ1 Mark

If $\text{i}^2=-1,$ then the sum $\text{i}+\text{i}^2+\text{i}^3+...$ upto 1000 terms is equal to:

A
1
B
-1
C
i
✓
0

Answer: D.

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Q 4MCQ1 Mark

A student was asked to prove a statement p(n) by induction. He proved p(K + 1) is true whenever p(k) is true for all $\text{k}>5\in\text{N}$ and also p(5) is true. On the basis of this he could conclude that p(n) is true.

A
For all $\text{n}\in\text{N}$
B
For all n > 5
✓
For all $\text{n}\geq5$
D
For all n > 5

Answer: C.

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Q 5MCQ1 Mark

If $x^n-1$ is divisible by $x-\lambda$, then the least positive integral value of $\boldsymbol{\lambda}$ is:

✓
1
B
2
C
3
D
4

Answer: A.

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Q 6(Each question 1 marks)1 Mark

If P(n) is the statement "n(n + 1) is even", then what is P(3)?

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Q 7(Each question 1 marks)1 Mark

State the first principle of mathematical induction.

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Q 8(Each question 1 marks)1 Mark

Write the set of value if n for which the statement p(n): 2n < n! is true.

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Q 9(Each question 1 marks)1 Mark

State the second principle of mathematical induction.

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Q 10(Each question 1 marks)1 Mark

If $p(n): 2 \times 4^{2n-1} + 3^{3n+1}$ is divisible by $\lambda$ for all $\text{n}\in\text{N}$ is true, then find the value of $\lambda$

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Q 11(Each question 2 marks)2 Marks

If $P(n)$ is the statement " $n^3+n$ is divisible by 3 ", prove that $P(3)$ is true but $P(4)$ is not true.

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Q 12(Each question 3 marks)3 Marks

If $P(n)$ is the statement "$n^2 + n$ is even", and if $P(r)$ is true, then $P(r + 1)$ is true.

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Q 13(Each question 3 marks)3 Marks

If $P(n)$ is the statement "$n^2 - n + 41$ is prime", prove that $P(1), P(2)$ and $P(3)$ are true. Prove also that $P(41)$ is not true.

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Q 14(Each question 3 marks)3 Marks

If $P(n)$ is the statement " $2^n \geq 3 n$ " and if $P(r)$ is true, prove that $P(r+1)$ is true.

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Q 15(Each question 4 marks)4 Marks

Prove the following by the principle of mathematical induction: $\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+...+\frac{1}{(\text{4n-1)(4n+3)}}=\frac{\text{n}}{3(\text{4n}+3)}$

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Q 16(Each question 4 marks)4 Marks

Prove the following by the principle of mathematical induction: $1 + 3 + 3^2 + ... + 3^{\text{n}-1}=\frac{3^\text{n}-1}{2}$

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Q 17(Each question 4 marks)4 Marks

Prove that $\cos\alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+...+\cos(\alpha+(\text{n}-1)\beta)\\=\frac{\cos\Big\{\alpha+\big(\frac{\text{n}-1}{2}\big)\beta\Big\}\sin\big(\frac{\text{n}\beta}{2}\big)}{\sin\frac{\beta}{2}}$ For all $\text{n}\in\text{N}.$

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Q 18(Each question 4 marks)4 Marks

Prove the following by the principle of mathematical induction: $1.3 + 3.5 + 5.7 + ... + (2\text{n} - 1)(2\text{n} + 1)= \frac{\text{n}(4\text{n}^2+6\text{n}-1)}{3}$

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Q 19(Each question 4 marks)4 Marks

Prove that $\frac{(2\text{n})!}{2^{2\text{n}}(\text{n}!)^2}\leq\frac{1}{\sqrt{3\text{n}+1}}$ for all $\text{n}\in\text{N}.$

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Mathematical Induction question types

Mathematical Induction questions

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Mathematical Induction MATHS - Mathematical Induction

Mathematical Induction question types

Mathematical Induction questions

Generate a Mathematical Induction paper free