List of top Engineering Mathematics Questions on Calculus asked in TS PGECET

Let $ f(x) = \log x $. The number $ C $ strictly between $ e^2 $ and $ e^3 $ such that its reciprocal is equal to $ \frac{f(e^3) - f(e^2)}{e^3 - e^2} $ is

The value of the integral $ \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy $ where $ C $ is the boundary of the region enclosed by $ y = x^2 $ and $ y^2 = x $, described in the positive sense, is

Let $ \vec{F} $ be an irrotational vector function. If $ C $ is the closed curve which is the boundary of an open surface $ S $, then: \[ \oint_C \vec{F} \cdot d\vec{R} = ? \]

Consider the statements:
I. If a series of positive terms is not convergent, then it is either divergent or oscillatory.
II. In an alternating series, if $lim⁡n→∞∣un∣≠0\lim_{n \to \infty} |u_n| \neq 0$, then it is convergent.
III. In a series of positive terms, if $lim⁡n→∞un=0\lim_{n \to \infty} u_n = 0$, then it may be convergent or divergent.
IV. If $∑∣un∣\sum |u_n|$ is divergent and $∑un\sum u_n$ is convergent, then $∑un\sum u_n$ is conditionally convergent.

Which of the above statement(s) is(are) correct?

Let $ \vec{F} $ be a vector point function defined inside and on the surface (S) of the sphere $ x^2+y^2+z^2=1 $. Then $ \oint_S (\text{Curl} \vec{F}) \cdot \hat{n} ds = $

$ \int_{0}^{2} \int_{0}^{\sqrt{y}} e^{y/x} dx dy = $

The value of $ \int_0^1 \int_y^{\sqrt{y}} yx \, dxdy $ is

Compute the Jacobian $ \frac{\partial(x,y)}{\partial(u,v)} $ of the transformation: $$ x = \frac{u+v}{1 - uv}, \quad y = \tan^{-1} u + \tan^{-1} v $$

The value of $$ \int_0^1 x^4 (1 - x)^2 \, dx $$ is:

If $ E $ is the region bounded by a closed curve $ C $ in the $ xy $-plane, then $ \iint_E dx\,dy = $

The maximum value of $ f(x, y) = x^2 y^3 (1 - x - y) $ is:

Let $\phi(x,y,z) = (x^2+y^2+z^2)^{n/2}$ and $f(x,y,z)=(xyz)^{-n/2}$. Then div Curl grad$\phi$ + (Curl grad $f$) $\cdot$ grad$\phi =$

If $f(x)$ is a twice differentiable function such that $f(0) = f(1) = f'(0) = 0$, then

If Rolle’s theorem is applied for $ f(x) = x^3 - x $ in $ [-1,1] $, then the values of ‘c’ of this theorem are:

The value of $\int_0^\infty e^{-x^2} \, dx$ is: