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Some CSIR problems

A CSIR question...
Let A be a 2x2 matrix over real satisfies det(A+I)=1+det(A), then
1. det A = 0
2. A= 0
3. tr A = 0
4. A is nonsingular
Basic knowledge on eigen values will give you the answer.
BUT
Is it possible to find the answer, by the method of elimination?
Yes...
This is a part B question, in CSIR examination... (Only one ans is correct).
Suppose option 2 is correct, then 1 and 3 are also correct, this eliminates option 2.
Take A as 0, this eliminates option 4.
So the ans is either 1 or 3...
Now we try to produce example that satisfies any one but not other. If that example satisfies hypothesis of the question, then that should be the answer.
Can you produce such example???

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In CSIR 2018 Dec Mathematics, it was asked to find the number of homomorphisms from A5 to S4.
Immediately our mind thinks about any formula to find the ans, as we have in Zn type groups.
BUT, just think about the fundamental truth!
Why  "simple" group is called simple?
Because its homomorphic images are none other than the whole group and (e)... (As it has no proper normal subgroup, it can not have proper kernels...)
The group A5 is a simple group... Infact a kind of special simple group among the groups of order upto 60...
Now find the number of homomorphisms from A5 to S4?

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