List of top Questions asked in CPET

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?

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.

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 \(x^2\).

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 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:

A unit vector that maximizes the directional derivative of the function \(f(x,y)=g(2x+y)\), where \(g'(3)=3\), at the point (1,1) is ____.

What is the value of the integral \(\iint_D dA\), where D is the region bounded by the curve \(y=x^2\) and the line \(y=4\)?

Which of the following statements is/are true?
(I) The vector field \(\vec{F}(x,y,z)=(yz,\,zx,\,xy)\) is solenoidal.
(II) The vector field \(\vec{F}(x,y,z)=(x^2-y^2,\,2xy,\,z^3)\) is conservative.

The iterated integral \[\int_0^2\int_0^{4-y^2}\int_0^{2-x} dz\,dy\,dx\] represents the volume of which one of the following solid regions?

For the function \(f(x,y)=x^3+y^3-3xy\), consider the following statements:
(I) f has 3 critical points.
(II) f has two saddle points and one point for local minimum.
(III) f has only one saddle point, a point for local minimum and a point for local maximum.
Choose the correct answer.

Consider the metric spaces \((C[0,1],d_1)\) and \((C[0,1],d_\infty)\), under the metrics \(d_1(f,g)=\int_0^1 |f(x)-g(x)|\,dx\) and \(d_\infty(f,g)=\sup_{x\in[0,1]}\{|f(x)-g(x)|\}\), for \(f,g\in C[0,1]\). Then which one of the options given below is correct?

Consider the function \[f(x,y)=\begin{cases}(x^2+y^2)\sin\dfrac{1}{x^2+y^2}, & (x,y)\neq(0,0)\\ 0, & (x,y)=(0,0)\end{cases}\] Which one of the following statements is correct for this function?

Which one of the following options is necessarily true?

Let \((X,d)\) be a metric space. If \(S\) and \(T\) are subsets of \(X\), then which one of the options given below is NOT necessarily true? (For any \(Y\subseteq X\), \(\text{Int}(Y)\) denotes the interior of \(Y\) and \(\text{Clos}(Y)\) denotes the closure of \(Y\).)

Consider the following statements:
(I) \((0,1)\) and \(\mathbb{R}\) are homeomorphic.
(II) \((0,1)\) and \((0,1]\) are homeomorphic.
(III) \((0,1)\) and \([0,1]\) are homeomorphic.
Choose the correct option.

Consider the following statements for a function \(f:[a,b]\to\mathbb{R}\):
(I) \(f\) is Riemann integrable if and only if it is bounded on \([a,b]\).
(II) \(f\) is Riemann integrable if and only if it is monotonic on \([a,b]\).
(III) \(f\) is Riemann integrable if and only if it is continuous on \([a,b]\).
Which one of the following options is correct?

Which of the following series does not converge uniformly on \((0,1)\)?
(I) \(\displaystyle\sum_{n=1}^\infty \frac{x^n}{n+x}\)
(II) \(\displaystyle\sum_{n=1}^\infty \frac{x}{n^2+x}\)
(III) \(\displaystyle\sum_{n=1}^\infty \frac{x^n}{1+x^n}\)

Consider the following statements:
(I) The sequence \(\{f_n\}_{n\ge1}\) of functions defined by \(f_n(x)=x^n\) is both point-wise and uniformly convergent on \([0,1]\).
(II) The sequence \(\{f_n\}_{n\ge1}\) of functions defined by \(f_n(x)=\dfrac{\ln(1+nx)}{n}\) is both point-wise and uniformly convergent on \([0,1]\).
Choose the correct answer.

Which one of the following is not a valid metric on \(\mathbb{R}\)?

The coefficient of \(x^5\) in the Taylor series expansion of \(f(x)=\tan(x)\) about the point \(x=0\) is ____.

Evaluate: \(\displaystyle\int_0^2 \left([x^2]-[x]^2\right)dx\), where \([\cdot]\) denotes the greatest integer (floor) function.