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2025
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12 maths questions

Q5 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
Consider the real matrix
$$A = \left( \begin{array} { l l } \lambda & 2 \\ 3 & 5 \end{array} \right)$$
Assume that $-1$ is an eigenvalue of $A$. Which of the following are true?
(A) The other eigenvalue is in $\mathbb { C } \backslash \mathbb { R }$.
(B) $A + I _ { 2 }$ is singular.
(C) $\lambda = 1$.
(D) Trace of $A$ is 5.
Q6 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Let $a _ { n } , n \geq 1$, be a sequence of positive real numbers such that $a _ { n } \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are true?
(A) There exists a natural number $M$ such that
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( a _ { n } \right) ^ { M } } \in \mathbb { R }$$
(B)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n ^ { 2 } a _ { n } \right) } \in \mathbb { R } .$$
(C)
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \left( n a _ { n } \right) } \in \mathbb { R }$$
(D) For all positive real numbers $R$,
$$\sum _ { n = 1 } ^ { \infty } \frac { R ^ { n } } { \left( a _ { n } \right) ^ { n } } \in \mathbb { R } .$$
Q8 Groups Group Order and Structure Theorems View
Which of the following groups are cyclic?
(A) $\mathbb { Z } / 2 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(B) $\mathbb { Z } / 3 \mathbb { Z } \oplus \mathbb { Z } / 9 \mathbb { Z }$
(C) Every group of order 18.
(D) $\left( \mathbb { Q } ^ { \times } , \cdot \right)$
Q11 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let $f : \mathbb { R } _ { \geq 0 } \longrightarrow \mathbb { R }$ be the function
$$f ( x ) = \begin{cases} 1 , & x = 0 \\ x ^ { - x } , & x > 0 \end{cases}$$
Determine whether the following statement is true:
$$\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \sum _ { i = 0 } ^ { \infty } n ^ { - n }$$
Q12 10 marks Groups Subgroup and Normal Subgroup Properties View
(A) (3 marks) Let $G$ be a group such that $| G | = p ^ { a } d$ with $a \geq 1$ and $( p , d ) = 1$. Let $P$ be a Sylow $p$-subgroup and let $Q$ be any $p$-subgroup of $G$ such that $Q$ is not a subgroup of $P$. Show that $P Q$ is not a subgroup of $G$.
(B) (7 marks) Let $\Gamma$ be a group that is the direct product of its Sylow subgroups. Show that every subgroup of $\Gamma$ also satisfies the same property.
Q13 10 marks Matrices Linear Transformation and Endomorphism Properties View
(A) (5 marks) Let $n \geq 2$ be an integer. Let $V$ be the $\mathbb { R }$-vector-space of homogeneous real polynomials in three variables $X , Y , Z$ of degree $n$. Let $p = ( 1,0,0 )$. Let
$$W = \left\{ f \in V \left\lvert \, f ( p ) = \frac { \partial f } { \partial X } ( p ) \right. \right\}$$
Determine the dimension of $V / W$.
(B) (5 marks) A linear transformation $T : \mathbb { R } ^ { 9 } \longrightarrow \mathbb { R } ^ { 9 }$ is defined on the standard basis $e _ { 1 } , \ldots , e _ { 9 }$ by
$$\begin{aligned} & T e _ { i } = e _ { i - 1 } , \quad i = 3 , \ldots , 9 \\ & T e _ { 2 } = e _ { 3 } \\ & T e _ { 1 } = e _ { 1 } + e _ { 3 } + e _ { 8 } . \end{aligned}$$
Determine the nullity of $T$.
Q14 10 marks Number Theory Algebraic Structures in Number Theory View
Let $F$ be a field and $R$ a subring of $F$ that is not a field. Let $x$ be a variable. Let $S = \left\{ a _ { 0 } + a _ { 1 } x + \cdots + a _ { n } x ^ { n } \mid n \geq 0 \text{ and } a _ { 0 } \in R , a _ { 1 } , \cdots a _ { n } \in F \right\}$.
(A) (2 marks) Show that, with the natural operations of addition and multiplication of polynomials, $S$ is an integral domain.
(B) (4 marks) Let $I = \{ f ( x ) \in S \mid f ( 0 ) = 0 \}$. Determine whether $I$ is a prime ideal.
(C) (4 marks) Determine whether $S$ is a PID.
Q15 10 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
(A) (6 marks) Let $f , g : [ 0,1 ] \mapsto \mathbb { R }$ be monotonically increasing continuous functions. Show that
$$\left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right) \leq \int _ { 0 } ^ { 1 } f ( x ) g ( x ) d x$$
(Hint: try double integrals.)
(B) (4 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be an infinitely differentiable function such that $f ( 1 ) = f ( 0 ) = 0$. Also, suppose that for some $n > 0$, the first $n$ derivatives of $f$ vanish at zero. Then prove that for the $( n + 1 )$ th derivative of $f$, $f ^ { ( n + 1 ) } ( x ) = 0$ for some $x \in ( 0,1 )$.
Q16 10 marks Vector Product and Surfaces View
(A) (5 marks) Consider the euclidean space $\mathbb { R } ^ { n }$ with the usual norm and dot product. Let $\mathrm { x } , \mathrm { y } \in \mathbb { R } ^ { n }$ be such that
$$\| \mathrm { x } + t \mathrm { y } \| \geq \| \mathrm { x } \| , \text { for all } t \in \mathbb { R }$$
Show that $\mathbf { x } \cdot \mathbf { y } = 0$.
(B) (5 marks) Consider the vector field $\vec { v } = \left( v _ { x } , v _ { y } \right)$ (with components $\left( v _ { x } , v _ { y } \right)$ ) on $\mathbb { R } ^ { 2 }$:
$$v _ { x } ( x , y ) = x - y , v _ { y } ( x , y ) = y + x$$
Compute the line integral of $\vec { v }$ along the unit circle (counterclockwise). Is there a function $f$ such that $\vec { v } = \operatorname { grad } f$?
Q17 Matrices Linear Transformation and Endomorphism Properties View
Denote by $V$ the $\mathbb { Q }$-vector-space $\mathbb { Q } [ X ]$ (polynomial ring in one variable $X$ ). Show that $V ^ { * }$ is not isomorphic to $V$.
Q19 Groups Symplectic and Orthogonal Group Properties View
Let $U ( n )$ be the group of $n \times n$ unitary complex matrices. Let $P \subset U ( n )$ be the set of all finite order elements of $U ( n )$, that is, $P = \left\{ X \in U ( n ) \mid X ^ { m } = 1 \text{ for some } m \geq 1 \right\}$. Show that $P$ is dense in $U ( n )$.
Q20 Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $A$ be a non-trivial subgroup of $\mathbb { R }$ generated by finitely many elements. Let $r$ be a real number such that $x \longrightarrow r x$ is an automorphism of $A$. Show that $r$ and $r ^ { - 1 }$ are zeros of monic polynomials with integer coefficients.