बिन्दु (1,2) पर फलन का तृतीय टेलर बहुपद ज्ञात कीजिए।

निम्नलिखित सीमा ज्ञात कीजिए:

(i)  

(ii)   

बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए।

(i) 

(ii) तीन घरों वाला एक वास्तविक मान फलन, जो सर्वत्र संतत है, अवकलनीय होता है।

(iii)F(x, y) = ( y + ,2 x + y) से परिभाषित फलन  किसी भी बिन्दु (x, y) ∈R²पर स्थानिकतः व्युत्क्रमणीय होता है।

(iv)    है फ(एक्स,य)= 10, यदि वाइ परिमेय नहीं है. से परिभाषित फलन   समाकलनीय होता है।

(v)  से परिभाषित फलन  का  (0,0)पर एक चरम मान होता है।

Show that there are infinitely many values of a for which x² +15x²-30x + a is irreducible in Q[x].

Let R = [√2] and M = (a+b√2∈ R 51a and 51b).

i) Show that M is an ideal of R.

ii) Show that if 5 Ja or 5]b, then 5] (a²+b²), for a, b∈ Z..

iii) Hence show that if N is an ideal of R properly containing M, then N = R.

iv) Show that R/M is a field, and give two distinct non-zero elements of this  field.

Let . Check whether D is a UFD or not.

Let R be a commutative ring with unity and re R. Prove that  using

the Fundamental Theorem of Homomorphism.

Hence show that .

Find all the units of Z₁₂.

Prove that   as rings.

Check whether  is a subring of the ring M₂(Z) or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.

Let G be a group such that Aut G is cyclic. Prove that G is abelian.

Find a group G, and a homomorphism Ø of G, so that Ø (G)=S3 and Ker Ø = A4. Is G abelian? Give reasons for your answer.

Let G be a group of order 56. What are all its Sylow p-subgroups? Show that G is not simple, i.e., G must have a proper normal non-trivial subgroup.

Explicitly give the elements and structure of the group S_n/A_n, n ≥5. 

Consider the map f_ab: R→R: f_ab(x)=ax+b. Let B = {f_ab, Ia, b ∈ R, a≠0}. Then B is a group with respect to the composition of functions. Check whether or not A={f_ab I a ∈ Q⁺, b∈ R} is a normal subgroup of B. 

Let G be a group of order n ≥ 2, with only two subgroups - {e} and itself. Find a minimal generating set for G. Also, find out whether n is a prime or a composite number, or can be either.

Let (G,.) be a finite abelian group and me N. Prove that S={g∈Gl(o(g), m) = 1}

Check whether or not A = {z∈ C^*I IzI∈Q} is a subgroup of

i) (C,.),       ii)  (C, +)

Give an example, with justification, of a commutative subgroup of a non- commutative group.

Consider the set X = R \{-1}. Define * on X by

X₁ * X2 = X + X2 + X₁X2 X1, X2 Ε Χ.

i) Check whether (X, *) is a group or not.

ii) Prove that x * x*x*...*x (n times) = (1+x)" -1∀ n∈ N and x ∈ X.