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## JAM MA (Mathematics) Question Paper 2019

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SAMPLE QUESTIONS

Q.1. Let 𝐺 be a noncyclic group of order 4. Consider the statements I and II:

I. There is NO injective (one-one) homomorphism from 𝐺 to ℤ8
II. There is NO surjective (onto) homomorphism from ℤ8 to 𝐺

Then

(A) I is true

(B) I is false

(C) II is true

(D) II is false

Q.2. Let 𝐺 be a nonabelian group, 𝑦 ∈ 𝐺, and let the maps 𝑓, 𝑔, ℎ from 𝐺 to itself be defined by

𝑓(𝑥) = 𝑦𝑥𝑦 −1 , 𝑔(𝑥) = 𝑥 −1 and ℎ = 𝑔 ∘ 𝑔.

Then

(A) 𝑔 and ℎ are homomorphisms and 𝑓 is not a homomorphism

(B) ℎ is a homomorphism and 𝑔 is not a homomorphism

(C) 𝑓 is a homomorphism and 𝑔 is not a homomorphism

(D) 𝑓, 𝑔 and ℎ are homomorphisms

Q.3. Let 𝑆 and 𝑇 be linear transformations from a finite dimensional vector space 𝑉 to itself such that 𝑆(𝑇(𝑣)) = 0 for all 𝑣 ∈ 𝑉. Then

(A) rank(𝑇) ≥ nullity(𝑆)

(B) rank(𝑆) ≥ nullity(𝑇)

(C) rank(𝑇) ≤ nullity(𝑆)

(D) rank(𝑆) ≤ nullity(𝑇)

Q.4. Let 𝐹⃗ and 𝐺⃗ be differentiable vector fields and let 𝑔 be a differentiable scalar function. Then

(A) ∇ ⋅ (𝐹⃗ × 𝐺⃗) = 𝐺⃗ ⋅ ∇ × 𝐹⃗ − 𝐹⃗ ⋅ ∇ × 𝐺⃗

(B) ∇ ⋅ (𝐹⃗ × 𝐺⃗) = 𝐺⃗ ⋅ ∇ × 𝐹⃗ + 𝐹⃗ ⋅ ∇ × 𝐺⃗

(C) ∇ ⋅ (𝑔𝐹⃗) = 𝑔∇ ⋅ 𝐹⃗ − ∇g ⋅ 𝐹⃗

(D) ∇ ⋅ (𝑔𝐹⃗) = 𝑔∇ ⋅ 𝐹⃗ + ∇g ⋅ 𝐹⃗

Q.5. Consider the intervals 𝑆 = (0, 2] and 𝑇 = [1, 3). Let 𝑆 and 𝑇 be the sets of interior points of 𝑆 and 𝑇, respectively. Then the set of interior points of 𝑆 \ 𝑇 is equal to

(A) 𝑆 ∖ 𝑇

(B) 𝑆 ∖ 𝑇

(C) 𝑆 ∖ 𝑇

(D) 𝑆 ∖ 𝑇 Click Here to Download