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**Question 1.** Which of the following is the correct limit definition of the derivative of a
function \(f\) at a point \(a\)?
A) \(\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\)
B) \(\displaystyle \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\)
C) Both A and B are equivalent
D) Neither A nor B
Answer: C
Explanation: Both expressions are algebraically identical and define \(f'(a)\).
**Question 2.** The series \(\sum_{n=1}^{\infty}\frac{1}{n^2}\) converges to which of the
following values?
A) \(\pi\)
B) \(\frac{\pi^2}{6}\)
C) \(\frac{\pi^2}{8}\)
D) \(\frac{\pi}{2}\)
Answer: B
Explanation: The Basel problem result states \(\sum_{n=1}^{\infty}1/n^2=\pi^2/6\).
**Question 3.** In a vector space, which property guarantees that scalar multiplication
distributes over vector addition?
A) Associativity of addition
B) Existence of a zero vector
C) Distributive law: \(c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}\)
D) Commutativity of scalar multiplication
Answer: C
Explanation: The distributive law is one of the vector space axioms.
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**Question 4.** Which of the following matrices is invertible?
A) \(\begin{pmatrix}1&2\\2&4\end{pmatrix}\)
B) \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\)
C) \(\begin{pmatrix}3&6\\9&12\end{pmatrix}\)
D) \(\begin{pmatrix}5&5\\5&5\end{pmatrix}\)
Answer: B
Explanation: Its determinant is \(1\), non‑zero, so the matrix is invertible.
**Question 5.** In group theory, a subgroup \(H\) of a group \(G\) is called normal if:
A) \(|H|=|G|\)
B) \(ghg^{-1}\in H\) for all \(g\in G\) and \(h\in H\)
C) \(H\) contains the identity element only
D) \(H\) is a cyclic subgroup
Answer: B
Explanation: Normality requires closure under conjugation by any element of \(G\).
**Question 6.** The Green’s theorem relates a line integral around a simple closed curve \(C\)
to a double integral over the region \(D\) it encloses. Which of the following expressions is
correct?
A) \(\displaystyle \oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x}-\frac{\partial
P}{\partial y}\right)\,dA\)
B) \(\displaystyle \oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial P}{\partial x}+\frac{\partial
Q}{\partial y}\right)\,dA\)
C) \(\displaystyle \oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial P}{\partial y}-\frac{\partial
Q}{\partial x}\right)\,dA\)
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D) None of the above
Answer: A
Explanation: This is the standard statement of Green’s theorem in the plane.
**Question 7.** Which of the following functions is uniformly continuous on \((0,1)\)?
A) \(f(x)=\frac{1}{x}\)
B) \(f(x)=\sqrt{x}\)
C) \(f(x)=\tan(\pi x/2)\)
D) \(f(x)=\ln x\)
Answer: B
Explanation: \(\sqrt{x}\) has bounded derivative on \((0,1)\) and thus is uniformly continuous;
the others blow up near an endpoint.
**Question 8.** The divergence theorem (Gauss’s theorem) in three dimensions states that:
A) \(\displaystyle \oint_S \mathbf{F}\cdot d\mathbf{r} = \iiint_V (\nabla\times\mathbf{F})\cdot
dV\)
B) \(\displaystyle \oint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_V (\nabla\cdot\mathbf{F})\,dV\)
C) \(\displaystyle \oint_S (\nabla\cdot\mathbf{F})\,dS = \iiint_V \mathbf{F}\cdot dV\)
D) None of the above
Answer: B
Explanation: The flux of \(\mathbf{F}\) through a closed surface equals the volume integral of its
divergence.
**Question 9.** In a linear transformation \(T:\mathbb{R}^n\to\mathbb{R}^m\), the rank–
nullity theorem states:
A) \(\text{rank}(T)+\text{nullity}(T)=n\)
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B) \(\text{rank}(T)=\text{nullity}(T)\)
C) \(\text{rank}(T)+\text{nullity}(T)=m\)
D) \(\text{rank}(T)\times\text{nullity}(T)=n\)
Answer: A
Explanation: The theorem relates dimensions of the image and kernel to the domain dimension.
**Question 10.** Which of the following statements about the harmonic series
\(\sum_{n=1}^{\infty}\frac{1}{n}\) is true?
A) It converges conditionally.
B) It converges absolutely.
C) It diverges.
D) It converges to \(\ln 2\).
Answer: C
Explanation: The harmonic series diverges (integral test).
**Question 11.** The eigenvalues of a \(2\times2\) matrix
\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) satisfy the characteristic equation:
A) \(\lambda^2-(a+d)\lambda+(ad-bc)=0\)
B) \(\lambda^2+(a+d)\lambda+(ad+bc)=0\)
C) \(\lambda^2-(ad-bc)\lambda+(a+d)=0\)
D) None of the above
Answer: A
Explanation: Determinant of \(\lambda I-A\) yields that quadratic.
**Question 12.** Which of the following is a necessary condition for a function
\(f:\mathbb{R}\to\mathbb{R}\) to be Riemann integrable on \([a,b]\)?
Exam Preparation
**Question 1.** Which of the following is the correct limit definition of the derivative of a
function \(f\) at a point \(a\)?
A) \(\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\)
B) \(\displaystyle \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\)
C) Both A and B are equivalent
D) Neither A nor B
Answer: C
Explanation: Both expressions are algebraically identical and define \(f'(a)\).
**Question 2.** The series \(\sum_{n=1}^{\infty}\frac{1}{n^2}\) converges to which of the
following values?
A) \(\pi\)
B) \(\frac{\pi^2}{6}\)
C) \(\frac{\pi^2}{8}\)
D) \(\frac{\pi}{2}\)
Answer: B
Explanation: The Basel problem result states \(\sum_{n=1}^{\infty}1/n^2=\pi^2/6\).
**Question 3.** In a vector space, which property guarantees that scalar multiplication
distributes over vector addition?
A) Associativity of addition
B) Existence of a zero vector
C) Distributive law: \(c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}\)
D) Commutativity of scalar multiplication
Answer: C
Explanation: The distributive law is one of the vector space axioms.
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**Question 4.** Which of the following matrices is invertible?
A) \(\begin{pmatrix}1&2\\2&4\end{pmatrix}\)
B) \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\)
C) \(\begin{pmatrix}3&6\\9&12\end{pmatrix}\)
D) \(\begin{pmatrix}5&5\\5&5\end{pmatrix}\)
Answer: B
Explanation: Its determinant is \(1\), non‑zero, so the matrix is invertible.
**Question 5.** In group theory, a subgroup \(H\) of a group \(G\) is called normal if:
A) \(|H|=|G|\)
B) \(ghg^{-1}\in H\) for all \(g\in G\) and \(h\in H\)
C) \(H\) contains the identity element only
D) \(H\) is a cyclic subgroup
Answer: B
Explanation: Normality requires closure under conjugation by any element of \(G\).
**Question 6.** The Green’s theorem relates a line integral around a simple closed curve \(C\)
to a double integral over the region \(D\) it encloses. Which of the following expressions is
correct?
A) \(\displaystyle \oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x}-\frac{\partial
P}{\partial y}\right)\,dA\)
B) \(\displaystyle \oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial P}{\partial x}+\frac{\partial
Q}{\partial y}\right)\,dA\)
C) \(\displaystyle \oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial P}{\partial y}-\frac{\partial
Q}{\partial x}\right)\,dA\)
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D) None of the above
Answer: A
Explanation: This is the standard statement of Green’s theorem in the plane.
**Question 7.** Which of the following functions is uniformly continuous on \((0,1)\)?
A) \(f(x)=\frac{1}{x}\)
B) \(f(x)=\sqrt{x}\)
C) \(f(x)=\tan(\pi x/2)\)
D) \(f(x)=\ln x\)
Answer: B
Explanation: \(\sqrt{x}\) has bounded derivative on \((0,1)\) and thus is uniformly continuous;
the others blow up near an endpoint.
**Question 8.** The divergence theorem (Gauss’s theorem) in three dimensions states that:
A) \(\displaystyle \oint_S \mathbf{F}\cdot d\mathbf{r} = \iiint_V (\nabla\times\mathbf{F})\cdot
dV\)
B) \(\displaystyle \oint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_V (\nabla\cdot\mathbf{F})\,dV\)
C) \(\displaystyle \oint_S (\nabla\cdot\mathbf{F})\,dS = \iiint_V \mathbf{F}\cdot dV\)
D) None of the above
Answer: B
Explanation: The flux of \(\mathbf{F}\) through a closed surface equals the volume integral of its
divergence.
**Question 9.** In a linear transformation \(T:\mathbb{R}^n\to\mathbb{R}^m\), the rank–
nullity theorem states:
A) \(\text{rank}(T)+\text{nullity}(T)=n\)
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B) \(\text{rank}(T)=\text{nullity}(T)\)
C) \(\text{rank}(T)+\text{nullity}(T)=m\)
D) \(\text{rank}(T)\times\text{nullity}(T)=n\)
Answer: A
Explanation: The theorem relates dimensions of the image and kernel to the domain dimension.
**Question 10.** Which of the following statements about the harmonic series
\(\sum_{n=1}^{\infty}\frac{1}{n}\) is true?
A) It converges conditionally.
B) It converges absolutely.
C) It diverges.
D) It converges to \(\ln 2\).
Answer: C
Explanation: The harmonic series diverges (integral test).
**Question 11.** The eigenvalues of a \(2\times2\) matrix
\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) satisfy the characteristic equation:
A) \(\lambda^2-(a+d)\lambda+(ad-bc)=0\)
B) \(\lambda^2+(a+d)\lambda+(ad+bc)=0\)
C) \(\lambda^2-(ad-bc)\lambda+(a+d)=0\)
D) None of the above
Answer: A
Explanation: Determinant of \(\lambda I-A\) yields that quadratic.
**Question 12.** Which of the following is a necessary condition for a function
\(f:\mathbb{R}\to\mathbb{R}\) to be Riemann integrable on \([a,b]\)?
