*JAM Question Paper 2019-20 MA (Mathematics).*

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**JAM Question Paper 2019-20 MA (Mathematics).**

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**01. If {v _{1}, v_{2}, v_{3}} is a linearly independent set of vectors in a vector space over ℝ, then which one of the following sets is also linearly independent**?

- (A) {
*v*_{1}+*v*_{2 }–*v*_{3}, 2*v*_{1}+*v*_{2 }+ 3*v*_{3}, 5*v*_{1}+ 4*v*_{2}} - (B) {
*v*_{1}–*v*_{2},*v*_{2 }–*v*_{3},*v*_{3}–*v*_{1}} - (C) {
*v*_{1}+*v*_{2 }–*v*_{3},*v*_{2 }+*v*_{3 }–*v*_{1},*v*_{3 }–*v*_{1 }+*v*_{2},*v*_{1}+*v*_{2 }+*v*_{3}} - (D) {
*v*_{1}+*v*_{2},*v*_{2}+ 2*v*_{3},*v*_{3 }+ 3*v*_{1}}

**02. Let U, V and W be finite dimensional real vector spaces, T:U → V, S:V → W and P:W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is TRUE?**

- (A) nullity of
*T*= nullity of*S* - (B) dimension U ≠ of dimension of
*W* - (C) If dimension of
*V*= 3, dimension of*U*= 4, then*P*is not identically zero - (D) If dimension of
*V*= 4, dimension of*U*= 3, and*T*is one-one, then*P*is identically zero

**03. Let G be a group satisfying the property that f:G → ℤ_{221 }is a homomorphism implies f (g) = 0, ∀g ∈ G. Then a possible group G is**

- (A) ℤ
_{21} - (B) ℤ
_{51} - (C) ℤ
_{91} - (D) ℤ
_{119}

**04. Let H be the quotient group ℚ/ ℤ. Consider the following statements**

**I.** Every cyclic subgroup of *H* is finite.

**II.** Every finite cyclic group is isomorphic to a subgroup of *H*.

Which one of the following holds?

- (A) I is TRUE but II is FALSE
- (B) II is TRUE but I is FALSE
- (C) both I and II are TRUE
- (D) neither I nor II is TRUE

**05. Let P and Q be two non-empty disjoint subsets of ℝ. Which of the following is (are) FALSE?**

- (A) If
*P*and*Q*are compact, then*P*∪*Q*is also compact - (B) If
*P*and*Q*are not connected, then*P*∪*Q*is also not connected - (C) If
*P*∪*Q*and*P*are closed, then*Q*is closed - (D) If
*P*∪*Q*and*P*are open, then*Q*is open

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