बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।. 

सदिशों और के बीच का कोण ज्ञात कीजिये।. 

Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.

(a) Every subgroup of S₃ is normal.

(b) Every abelian group is cyclic.

(c) In a ring with unity, the sum of any two units is a unit.

(d) If a field has characteristic p, p a prime, the field is finite.

(e) If every element in group has finite order, the group is finite.

Let R be a ring (not necessarily commutative) and I and J be ideal of R. Show that I ∩ J and I + J = {a + bla ∈ I, b ∈ J} are ideals of R.

Check whether or not (3) is a maximal ideal in Z₉.

Show that (x, 5) is not a principal ideal in Z[x].

Let S¹ = {z ∈ C* | |z| = 1} and Un = {z ∈ C* | z" = 1} for n ∈ N. Check that U_n ⊆ S¹. Further, show that U_n ≤ S¹.

Let F be a field and let f(x) ∈ F[x] be irreducible in F[x]. Show that the ideal (f(x)) is a maximal ideal in F[x]. Use this to deduce that Q[x]/(x² + 6x3 +12) is a field.

Find the order of each of the elements in U(15). Is U (15) cyclic? Justify your answer.

If H and K are normal abelian subgroups of a group, and if H∩ K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.

Find the gcd of the polynomials x4 + 3x + 2 and x³ + 3x² + 5x + 3.

Let   . Check whether Check whether R is a subring of M₂(R). Is R an ideal of M₂(R)? Justify your answer.

Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.

Calculate the following:

(i) (3x² + 4x + 1) + (3x3 + 4x² + 2x + 3) in Z₅[x].

(ii) (3x²+2x+6). (3x3+4x+5) in Z₇[x].

Define an integral domain. Give an example of an integral domain which is not a field.

Let R be a ring in which a² = a for all a ∈ R. Show that a = -a and R is commutative.

Let R = Z₂₀

(i) Give, with justificaiton, a nilpotent element in R.

(ii) Give, with justification, a zero divisor in R which is not nilpotent.

(iii) What is the order of U(R)?

If F is a field, show that U(F[x]) = F*.

Let a = (125), β = (1432) ∈ Ss. Compute σ = α β-¹. Write as a product of transpositons. What is the signature of σ?

Let A be a 3 x 4 real matrix, B be a 4 × 2 real matrix and C be a 2 x 3 real matrix. Which of the following operations are defined?

(i) CA + B¹

(ii) AB + C¹

For those operations that are defined, what is the order of the resulting matrix?