What is the dimension of the vector space of all \( n \times n \) real symmetric matrices?

• (A) \( n^2 \)
• (B) \( \frac{n(n+1)}{2} \)
• (C) \( \frac{n(n-1)}{2} \)
• (D) \( 2n \)

If a function \( f(x) \) is continuous on a closed interval \( [a,b] \), is it necessarily uniformly continuous?

• (A) Yes, always uniformly continuous
• (B) No, never uniformly continuous
• (C) Only if differentiable
• (D) Only if bounded

What is the value of the limit \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \)?

• (A) \(1\)
• (B) \(0\)
• (C) \(e\)
• (D) \(\infty\)

How many elements of order 5 are there in a cyclic group of order 25?

• (A) \(1\)
• (B) \(4\)
• (C) \(5\)
• (D) \(10\)

If \( A \) is a \(3 \times 3\) matrix with eigenvalues \(1, 2, 3\), what is the determinant of \(A^2\)?

• (A) \(6\)
• (B) \(12\)
• (C) \(18\)
• (D) \(36\)

Which theorem states that every bounded sequence in \( \mathbb{R}^n \) has a convergent subsequence?

• (A) Mean Value Theorem
• (B) Bolzano–Weierstrass Theorem
• (C) Rolle’s Theorem
• (D) Taylor’s Theorem

What is the radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \)?

• (A) \(0\)
• (B) \(1\)
• (C) \(\infty\)
• (D) \(e\)

Is the set of all rational numbers \( \mathbb{Q} \) a countable or uncountable set?

• (A) Finite set
• (B) Countable set
• (C) Uncountable set
• (D) Empty set

If \( T: V \to W \) is a linear transformation, what is the relationship between rank(\(T\)), nullity(\(T\)), and dim(\(V\))?

• (A) rank(\(T\)) + nullity(\(T\)) = dim(\(W\))
• (B) rank(\(T\)) \(\times\) nullity(\(T\)) = dim(\(V\))
• (C) rank(\(T\)) + nullity(\(T\)) = dim(\(V\))
• (D) rank(\(T\)) = nullity(\(T\))

What is the condition for a group \( G \) to be Abelian based on the commutator subgroup?

• (A) Commutator subgroup is equal to \(G\)
• (B) Commutator subgroup is trivial
• (C) Commutator subgroup is infinite
• (D) Commutator subgroup is cyclic

What is the value of the integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \)?

• (A) \(0\)
• (B) \(1\)
• (C) \(\sqrt{\pi}\)
• (D) \(\pi\)

In a metric space, is every Cauchy sequence necessarily a convergent sequence?

• (A) Yes, always
• (B) No, not always
• (C) Only in finite spaces
• (D) Only for bounded sequences

What are the possible values for the rank of a \(4 \times 3\) matrix?

• (A) \(0,1,2,3,4\)
• (B) \(1,2,3,4\)
• (C) \(0,1,2,3\)
• (D) Only \(3\)

What is the order of the group of permutations \( S_3 \)?

• (A) \(3\)
• (B) \(6\)
• (C) \(9\)
• (D) \(12\)

Which partial differential equation represents the Laplace equation in two dimensions?

• (A) \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0 \)
• (B) \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \)
• (C) \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \)
• (D) \( \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \)