Mathematical Induction - Jharkhand Board (JAC) STD 11 Science Maths Questions

Q 1M.C.Q (1 Marks)1 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 2M.C.Q (1 Marks)1 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 3M.C.Q (1 Marks)1 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 4M.C.Q (1 Marks)1 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 5M.C.Q (1 Marks)1 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 61 Marks Question1 Mark

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

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Q 71 Marks Question1 Mark

State the first principle of mathematical induction.

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Q 81 Marks Question1 Mark

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

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Q 91 Marks Question1 Mark

State the second principle of mathematical induction.

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Q 101 Marks Question1 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 112 Marks Questions2 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 123 Marks Question3 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 133 Marks Question3 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 143 Marks Question3 Marks

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

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Q 155 Marks Questions5 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 165 Marks Questions5 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 175 Marks Questions5 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 185 Marks Questions5 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 195 Marks Questions5 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

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