List of top Mathematics Questions on Calculus asked in TS PGECET

Statement-I:
The functions \[ u=x^2+y^2,\qquad v=\tan^{-1}\left(\frac{y}{x}\right) \] are functionally independent.
Statement-II:
The Jacobian \[ \frac{\partial(u,v)}{\partial(x,y)} \] is non-zero.
The correct answer is:

A function satisfying the conditions of Lagrange's mean value theorem on the interval $[1, 5]$ is:

Statement-I: The function $ u = x^2 + y^2 $, $ v = \tan^{-1}\left(\frac{y}{x}\right) $ are functionally independent.
Statement-II: The Jacobian $ \frac{\partial(u,v){\partial(x,y)} $ is non-zero.
The correct answer is}

If $\alpha, \beta$ are two distinct eigenvalues of a matrix $A$, then the corresponding eigenvectors are

$f:\mathbb{R}\rightarrow\mathbb{R}$ is increasing in $(-\infty,0)\cup(1,\infty)$ and decreasing in $(0,1)$. If $f(2)=6$ and the continuous curve $y=f(x)$ passes through $(0,0)$, then