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Question

If A=(cosθisinθisinθcosθ), where i=1, then by principle of Mathematical Induction then An=[cosnθisinnθisinnθcosnθ].


A
True
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B
False
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Solution

The correct option is A True
Given:A=[cosθisinθisinθcosθ]
To Check whether the given statement is true

An=[cosnθisinnθisinnθcosnθ]
Put n=1
A1=[cosθisinθisinθcosθ]
So,An is true for n=1
Let An is true for n=k, so
Ak=[coskθisinkθisinkθcoskθ] ......(1)
Ak+1=[cos(k+1)θisin(k+1)θisin(k+1)θcos(k+1)θ]

Now,Ak+1=Ak×A
=[coskθisinkθisinkθcoskθ][cosθisinθisinθcosθ]
=[coskθcosθ+i2sinkθsinθcoskθsinθ+isinkθcosθisinkθcosθ+icoskθsinθi2sinkθsinθ+cosθcoskθ]
=[coskθcosθsinkθsinθisinkθcosθ+coskθsinθi(sinkθcosθ+coskθsinθ)cosθcoskθsinkθsinθ]
=[cos(k+1)θisin(k+1)θisin(k+1)θcos(k+1)θ]

So,An is true for n=k+1 whenever it is true for n=k
Hence, by principle of mathematical induction An is true for all positive integer.
Hence the given statement is true.

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