GATE Maths 2019 PYQ Paper With Answer Key.pdf
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#Q1 Let R=(P(N),∆ ,∩ ) then the zero element of R will be
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N
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𝜙
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A- N, N is the set of Natural numbers
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Ac
#Q2 R=(P(N),∆ ,∩ ) be the ring, then the unity of R is
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𝜙
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N, N = Natural numbers
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A ,A∈P(N)
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Ac ,A∈P(N)
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#Q3 Unity of the zero ring for ( R ,+ ,∙ ) is
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0
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1
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Unity does not exist
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Infinite unity
#Q4 Let R be a commutative ring and a ,b are nilpotent element of R then
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a-b is nilpotent element
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a/b is nilpotent
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a.c is nilpotent ,∀ non-zero c in R
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a+c is nilpotent ,∀ c ∈ R
#Q5 Let a be a nilpotent element of index “r” in R Then 1-a is
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nilpotent
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idempotent
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Unit
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Zero divisor
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#Q9 Let ∪(R) be the set of units in R and Z(R) be the set of units in zero divisor. And R be Ring with unity, then
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∪(R) ⊂Z(R)
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Z(R) ⊂ ∪(R)
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Z(R)∩ ∪(R)≠ϕ
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Z(R)∩ ∪(R)=ϕ
#Q10 Let (Z ,* ,∙ ) be the ring and a *b=a+b and a∙b=a+b-ab. Then unity of (Z ,* ,∙ ) is
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1
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0
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2
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-1
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#Q1 List of zero divisors of Z_(m ) is
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{x∈Z_(m ) |gcd(x,m)≠1}
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{x∈Z_(m ) |gcd(x,m)=1}
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{x∈Z_(m ) |gcd(x,m)=2}
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{x∈Z_(m ) |gcd(x,m)≠2}
#Q5 Let R be an integral domain then
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R[x] is a field
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R[x] is a division ring
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R[x] has inverse of all non-zero elements
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R[x] does not have zero-divisor
#Q6 Let R be any ring, then Unit of R = Units of R [x] if
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R has unity
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R is commutative
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R may have zero divisor
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R is principle ideal domain
#Q7 Let R be a commutative ring with unity then R[x]
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May not have unity
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R[x] has unity but different from R
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R[x] has unity but not commutative.
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Unity of R= unity of R[x]
#Q8 Let R is a CRU and have zero divisors, then
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Units of R = Units of R[x]
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R may have more units than R[x]
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R[x] may have more units than R
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Units of R≠ units of R[x]. always
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