Rings and Polynomial Domains - Algebra - Mathematics Previous Year Questions

1

2008 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2008

The number of maximal ideals in \(\mathbb{Z}_{27}\) is

A

0

B

1

C

2

D

3

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2

2008 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2008

The ring \(\mathbb{Z}[\sqrt{-11}]\) is

A

a Euclidean Domain

B

a Principal Ideal Domain, but not a Euclidean Domain

C

a Unique Factorization Domain, but not a Principal Ideal Domain

D

not a Unique Factorization Domain

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3

2008 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2008

Let \(R\) be a Principal Ideal Domain and \(a, b\) any two non-unit elements of \(R\). Then the ideal generated by \(a\) and \(b\) is also generated by

A

a + b

B

ab

C

gcd(a, b)

D

lcm(a, b)

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4

2008 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2008

Let \(I = \{x \in X : x(0) = 0\}\). Then

A

\(I\) is not an ideal of \(X\)

B

\(I\) is an ideal, but not a prime ideal of \(X\)

C

\(I\) is a prime ideal, but not a maximal ideal of \(X\)

D

\(I\) is a maximal ideal of \(X\)

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5

2009 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2009

The number of elements of a principal ideal domain can be

A

15

B

25

C

35

D

36

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6

2009 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2009

Which one of the following ideals of the ring \( \mathbb{Z}[i] \) of Gaussian integers is NOT maximal ?

A

\(\langle 1+i \rangle \)

B

\(\langle 1-i \rangle \)

C

\(\langle 2+i \rangle \)

D

\(\langle 3+i \rangle \)

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7

2009 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2009

\( P(x) = \)

A

\(1 + x \)

B

\(-1 - x \)

C

\(\frac{1+x}{x} \)

D

\(\frac{-1-x}{x} \)

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8

2010 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2010

Consider the polynomial ring \( \mathbb{Q}[x] \). The ideal of \( \mathbb{Q}[x] \) generated by \( x^2 - 3 \) is

A

maximal but not prime

B

prime but not maximal

C

both maximal and prime

D

neither maximal nor prime

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9

2010 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2010

If 137 + 276 = 435 how much is 731 + 672?

A

534

B

1403

C

1623

D

1513

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10

2010 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2010

5 skilled workers can build a wall in 20 days; 8 semi-skilled workers can build a wall in 25 days; 10 unskilled workers can build a wall in 30 days. If a team has 2 skilled, 6 semi-skilled and 5 unskilled workers, how long will it take to build the wall?

A

20 days

B

18 days

C

16 days

D

15 days

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11

2011 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2011

If \( \mathbb{Z}[i] \) is the ring of Gaussian integers, the quotient \( \mathbb{Z}[i]/(3-i) \) is isomorphic to

A

\(\mathbb{Z} \)

B

\(\mathbb{Z}/3\mathbb{Z} \)

C

\(\mathbb{Z}/4\mathbb{Z} \)

D

\(\mathbb{Z}/10\mathbb{Z} \)

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12

2011 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2011

For the rings \( L = \frac{\mathbb{R}[x]}{\langle x^2 - x + 1 \rangle} \); \( M = \frac{\mathbb{R}[x]}{\langle x^2 + x + 1 \rangle} \); \( N = \frac{\mathbb{R}[x]}{\langle x^2 + 2x + 1 \rangle} \), which one of the following is TRUE?

A

\(L \) is isomorphic to \(M \); \(L \) is not isomorphic to \(N \); \(M \) is not isomorphic to \(N \)

B

\(M \) is isomorphic to \(N \); \(M \) is not isomorphic to \(L \); \(N \) is not isomorphic to \(L \)

C

\(L \) is isomorphic to \(M \); \(M \) is isomorphic to \(N \)

D

\(L \) is not isomorphic to \(M \); \(L \) is not isomorphic to \(N \); \(M \) is not isomorphic to \(N \)

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13

2011 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2011

There are two candidates P and Q in an election. During the campaign, 40% of the voters promised to vote for P, and rest for Q. However, on the day of election 15% of the voters went back on their promise to vote for P and instead voted for Q. 25% of the voters went back on their promise to vote for Q and instead voted for P. Suppose, P lost by 2 votes, then what was the total number of voters?

A

100

B

110

C

90

D

95

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14

2011 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2011

Three friends, R, S and T shared toffee from a bowl. R took 1/3^rd of the toffees, but returned four to the bowl. S took 1/4^th of what was left but returned three toffees to the bowl. T took half of the remainder but returned two back into the bowl. If the bowl had 17 toffees left, how many toffees were originally there in the bowl?

A

38

B

31

C

48

D

41

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15

2011 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2011

The sum of n terms of the series 4+44+444+.... is

A

(4/81) [10ⁿ⁺¹ - 9n - 1]

B

(4/81) [10^n-1 - 9n - 1]

C

(4/81) [10ⁿ⁺¹ - 9n - 10]

D

(4/81) [10ⁿ - 9n - 10]

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16

2012 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2012

Let \(R=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}\) and \(I=\mathbb{Z}\times\mathbb{Z}\times\{0\}\). Then which of the following statement is correct?

A

\(I\) is a maximal ideal but not a prime ideal of \(R\).

B

\(I\) is a prime ideal but not a maximal ideal of \(R\).

C

\(I\) is both maximal ideal as well as a prime ideal of \(R\).

D

\(I\) is neither a maximal ideal nor a prime ideal of \(R\).

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17

2012 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2012

Raju has 14 currency notes in his pocket consisting of only Rs. 20 notes and Rs. 10 notes. The total money value of the notes is Rs. 230. The number of Rs. 10 notes that Raju has is

A

5

B

6

C

9

D

10

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18

2013 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2013

Suppose that \(R\) is a unique factorization domain and that \(a, b \in R\) are distinct irreducible elements. Which of the following statements is TRUE?

A

The ideal \((1 + a)\) is a prime ideal

B

The ideal \((a + b)\) is a prime ideal

C

The ideal \((1 + ab)\) is a prime ideal

D

The ideal \((a)\) is not necessarily a maximal ideal

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19

2013 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2013

Which of the following is a field?

A

\(\mathbb{C}[x]/(x^2 + 2)\)

B

\(\mathbb{Z}[x]/(x^2 + 2)\)

C

\(\mathbb{Q}[x]/(x^2 - 2)\)

D

\(\mathbb{R}[x]/(x^2 - 2)\)

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20

2013 · Mathematics · Algebra · Rings and Polynomial Domains

Mathematics (MA) 2013

Select the pair that best expresses a relationship similar to that expressed in the pair:
Medicine: Health

A

Science: Experiment

B

Wealth: Peace

C

Education: Knowledge

D

Money: Happiness

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