List of top Questions asked in CPET

Which one of the following matrices has \(p(x) = x^3 - 8x^2 + 5x + 7\) as the minimal polynomial?

For a linear transformation \(L:V\to W\) with \(\dim(V)=10\) and \(\dim(W)=8\), which one of the following options is correct?

Let \(L\) be a linear transformation defined on the vector space \(\mathbb{R}^6\) over the field \(\mathbb{R}\). If \(\operatorname{Rank}(L)=4\) and \(\operatorname{Nullity}(L^2)=3\), then \(\operatorname{Rank}(L^2)\) is ____ (\(L^2\) denotes the composition of \(L\) with itself).

On the vector space \(\mathbb{R}^3\) over the field \(\mathbb{R}\) with the standard basis \(\{e_1,e_2,e_3\}\), what is the annihilator \((S^0)\) of the subspace \(S=\text{span}\{e_1,e_2\}\)?

Let V be a 7-dimensional vector space. If U, W are subspaces of V with dimensions 4 and 5 respectively, then which of the following is NOT a possible value of dimension of \(U \cap W\)?

Which one of the following options is incorrect?

Which of the statements given below is/are true?
(I) \(\mathbb{Z}[i]\) is a Principal ideal domain and Euclidean domain.
(II) \(\mathbb{Z}[\sqrt{-5}]\) is neither a Principal ideal domain nor a Unique factorization domain.

Let V be the real vector space consisting of all polynomials in one real variable with real coefficients of degree at most 6, including the zero polynomial. Then which one of the following options is true?

Consider the following statements:
(I) The kernel of the ring homomorphism \(f:\mathbb{Z}[x]\to\mathbb{Z}\) given by \(f(p(x))=p(1)\) is \(\{(x-1)q(x): q(x)\in\mathbb{Z}[x]\}\).
(II) The ring \(\mathbb{Z}_4\) is isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\).
Choose the correct answer:

For a real \(4 \times 3\) matrix \(M\) and the standard basis \(\{e_1,e_2,e_3\}\) of \(\mathbb{R}^3\), which of the following statements is/are true?
(I) If \(\text{rank}(M)=1\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).
(II) If \(\text{rank}(M)=2\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).
(III) If \(\text{rank}(M)=3\), then \(\{Me_1,Me_2\}\) is a linearly independent set in \(\mathbb{R}^4\).

Which one of the following options is correct for the ideal \(I=\langle x^2+5\rangle\) in the ring \(\mathbb{Q}[x]\)?

If \(G\) is a group of order 30, then the number of Sylow 5-subgroups in \(G\) must be ____.

On \(R=\left\{\begin{pmatrix}a&b\\0&c\end{pmatrix}: a,b,c\in\mathbb{R}\right\}\) with the usual addition and multiplication of matrices, which of the following statements is true?

Which of the following statements is false?
(I) \(\mathbb{Z}_5\times\mathbb{Z}_7\) is a cyclic group.
(II) The order of the element \((2,5)\in\mathbb{Z}_5\times\mathbb{Z}_7\) is 10.
(III) \(\text{Aut}(\mathbb{Z}_5\times\mathbb{Z}_7)\) is isomorphic to \(\mathbb{Z}_{35}\).

Which one of the following options is correct? (\(p\) is a prime number.)

Which of the following statements is/are false?
(I) \(I=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}: a,c,d\in\mathbb{R}\right\}\) is both a subring and an ideal of \(M_2(\mathbb{R})\).
(II) \(\mathbb{Z}[\sqrt2]=\{a+b\sqrt2 : a,b\in\mathbb{Z}\}\) is an integral domain, but not a field.

Which of the following statements is/are correct?
(I) In \(GL_2(\mathbb{R})\), matrices with the same determinant always belong to the same conjugacy class.
(II) The class equation of \(A_5\) (the alternating group on 5 elements) is \(60=1+6+10+15+28\).

Which one of the following statements is incorrect?
(I) The group \(\mathbb{Z}_{15}\) is isomorphic to \(\mathbb{Z}_3\times\mathbb{Z}_5\)
(II) \(\text{Inn}(A_3)\) is isomorphic to \(A_3\)

Which of the following statements is/are true?
(I) The maximum possible order of an element in the group \(S_5\) (with composition \(\circ\)) is 6.
(II) If \(f:(S_3,\circ)\to(\mathbb{Z}_6,+_6)\) is a group homomorphism, then the order of \(f(S_3)\) is 1, 2 or 3.

In a group \(G\), let \(x^5=e\) (\(e\): identity element of \(G\)) and \(xyx^{-1}=y^2\) for \(x,y\in G\). If \(y\ne e\), then the order of \(y\) is ____.

Let \(G\) be a finite Abelian group. If the subgroups \(H\) and \(K\) are of index 3 each in \(G\), then what is the index of the subgroup \(H\cap K\) in \(G\)?

Which of the following statements is/are true?
(I) The function \(f(z)=\sin\left(\frac{1}{\cos(1/z)}\right)\) has an isolated singularity at \(z=0\).
(II) The value of the integral \(\oint_{|z|=1}\frac{dz}{z\sin(z)}\) is zero.

Which one of the options given below is false?

Consider the following statements:
(I) Residue of \(f(z)=e^z\) at \(z=\infty\) is \(-1\).
(II) If a meromorphic function \(f\) has 5 simple zeros and 2 simple poles inside \(|z|=1\), then \(\oint_{|z|=1}\frac{f'(z)}{f(z)}dz = 6i\pi\).
Pick out the correct option.