List of top Mathematics Questions

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:

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.

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

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

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 for the ideal \(I=\langle x^2+5\rangle\) in the ring \(\mathbb{Q}[x]\)?

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.

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/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 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.

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 \(A_3\)

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 \(G\)?

For an entire function \(f\), which one of the following statements 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.

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.

In the dihedral group \(D_5=\{r,s:\; r^5=e,\; s^2=e,\; sr=r^{-1}s\}\) under composition as the binary operation, which of the following options is NOT true?

Which one of the options given below is false?

If \(\displaystyle\sum_{n=-\infty}^{\infty}a_nz^n\) is the Laurent series expansion of the function \(f(z)=\dfrac{1}{2z^2-13z+15}\) in the annulus \(\{z\in\mathbb{C}:3/2<|z|<5\}\), then \(a_1/a_2=\) ____.

If \(C\) is a simple closed curve not passing through the points \(-1\), \(0\) and \(1\) in the complex plane, then the set of all possible value(s) of the integral \(\int_C \dfrac{dz}{z(1-z^2)}\) is ____.

Which of the statements given below is/are false?
(I) \(f(z)=x^2+y^2+2ixy\) (\(z=x+iy\)) is differentiable only at the points that lie on the x-axis.
(II) \(f(z)=|z-1|^2\) is differentiable at \(z=1\), but not analytic at \(z=1\).
(III) There exists an analytic function in \(\mathbb{C}\) whose imaginary part is \(x^2\).

Which one of the following options is true?
(A) If \(\operatorname{Log}(z)\) denotes the principal value of the logarithm, then \(\operatorname{Log}(zw)=\operatorname{Log}(z)+\operatorname{Log}(w)\) for all \(z,w\in\mathbb{C}\setminus\{0\}\).
(B) The solution set of the equation \(e^{iz}=-1\) is \(\{(2k+1)\pi:k\in\mathbb{Z}\}\).
(C) \(\cos(z)\) is bounded in the whole complex plane.
(D) A Mobius transformation with three fixed points is a constant.

The radius of convergence of the power series \(\sum_{n=1}^\infty\left(\frac{2^n}{n}+\frac{3^n}{n^2}\right)z^n\) is ____.

Consider the following statements:
(I) The function \(f(z)=\dfrac{x^2-y^2+2ixy}{x^2+y^2}\) (\(z=x+iy\)) has a limit as \(z\to0\).
(II) The function \(f(z)=\begin{cases}\operatorname{Re}(z)/|z|, & z\ne0\\ 1, & z=0\end{cases}\) is continuous at \(z=0\).
Choose the correct answer:

Consider the following assertions:
(I) \(\displaystyle\iint_S(x\hat i+y\hat j+z\hat k)\cdot\hat n\,dS=2V\), where \(\hat n\) is the unit outward surface normal and S is a closed surface enclosing a volume V.
(II) \(\displaystyle\int_C(yz\hat i+xz\hat j+xy\hat k)\cdot d\vec r=3\), where C is the boundary of a surface S.
Choose the correct option: