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## Gokhale Education Society's N.B. Mehta (Valwada) Science College, Bordi, F.Y. B.Sc. Theory Exam, Sem-I Nov 2023 **Subject:** Algebra I (USMT102) **Date:** 10/11/2023 **Time:** 10:00 to 12:30 PM **Max Marks:** 75 **N.B. All questions are compulsory** ### Q1. **A)** Prove that: Any two non-zero...

## Gokhale Education Society's N.B. Mehta (Valwada) Science College, Bordi, F.Y. B.Sc. Theory Exam, Sem-I Nov 2023 **Subject:** Algebra I (USMT102) **Date:** 10/11/2023 **Time:** 10:00 to 12:30 PM **Max Marks:** 75 **N.B. All questions are compulsory** ### Q1. **A)** Prove that: Any two non-zero integers a and b in Z have a g.c.d and it can be expressed in the form of ma + nb for suitable m, n ∈ Z. **B)** Attempt any **Two** from the following: 1. Find the G.C.D of 2210 and 357 and express it in the form of 2210x + 357y for suitable x, y ∈ N. 2. Show that 281 = 2 (mod 41) using Fermat's Little Theorem. 3. Prove that: If P is prime such that $\frac{P}{ab} $ then $\frac{P}{a}$ or $\frac{P}{b}$. ### Q2. **A)** Prove that: Let f: X→Y, g: Y-Z are the two functions such that gof is bijective and f is surjective then g is injective. **B)** Attempt any **Two** from the following: 1. Show that f: R→R given by f(x) = 2x + 5 is bijective function. 2. On Z define a binary operation as a * b = a + 2b for all a, b ∈ Z. Check whether '*' is commutative and associative. 3. Check whether the following relation are reflexive, symmetric and transitive: > X: Set of all students in college > R: aRb iff a and b are in the same college. ### Q3. **A)** Prove that: $f(x) = a_n x^n + a_{n-1}x^{n-1} +... + a_1x + a_0$ is a polynomial with integer coefficient i.e each a_i ∈ Z. If a rational number $\frac{p}{q}$ where p, q are integers, q ≠ 0 with (p, q) = 1 is root of f(x) then $\frac{p}{a_n}$ and $\frac{q}{a_0}$. **B)** Attempt any **Two** from the following: 1. Find quotient and remainder when x³- 3x² + 4x +8 is divided by x² +2. 2. Find the multiplicity of each root of f(x) = x³ - 4x² + 5x - 2.