Mathematical Induction - Maharashtra Board STD 11 Maths Questions

Q 1MCQ1 Mark

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

A
$1$
B
$2$
C
$3$
✓
$4$

Answer: D.

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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 $\text{x}-\lambda,$ then the least positive integral value of $\lambda$ is:

✓
$1$
B
$2$
C
$3$
D
$4$

Answer: A.

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Q 6Solve the following 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 7Solve the following Question.(1 Marks)1 Mark

State the first principle of mathematical induction.

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Q 8Solve the following 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 9Solve the following Question.(1 Marks)1 Mark

State the second principle of mathematical induction.

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Q 10Solve the following 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 11Solve the Following 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 12Solve the Following 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 13Solve the Following 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 14Solve the Following 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 15Solve the Following 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 16Solve the Following 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 17Solve the Following 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 18Solve the Following 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 19Solve the Following 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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Q 20Solve the Following Question.(5 Marks)5 Marks

Prove the following by the principle of mathematical induction:$\text{a}+(\text{a}+\text{d})+(\text{a}+2\text{d})...+(\text{a}+(\text{n}-1)\text{d})=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$

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Q 21Solve the Following Question.(5 Marks)5 Marks

Prove the following by the principle of mathematical induction:
$1 + 2 + 2^2 + ... + 2^n = 2^{n+1} - 1$ for all $\text{n}\in\text{N}.$

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Q 22Solve the Following Question.(5 Marks)5 Marks

Prove that $\frac{\text{n}^2}{7}+\frac{\text{n}^5}{5}+\frac{\text{n}^3}{3}+\frac{\text{n}^2}{2}-\frac{37}{210}$ n is a positive integer for all $\text{n}\in\text{N}.$

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Q 23Solve the Following Question.(5 Marks)5 Marks

Prove the following by the principle of mathematical induction:
$1.2 + 2.3 + 3.4 + ... +\text{n}(\text{n}+1)=\frac{\text{n}(\text{n}+1)(\text{n}+2)}{3}$

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Q 24Solve the Following Question.(5 Marks)5 Marks

Prove that $\text{x}^{2\text{n}-1}+\text{y}^{2\text{n}-1}$ is divisible by x + y 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