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

If \[ f(x,y,z)=e^{\,1-y\cos x}+yze^{\frac{-1}{1+x^2}}, \] then \[ \frac{\partial f}{\partial y} \] at \((0,1,e)\) is

Let \[ \vec{F}=x^2yz\,\hat{i}+y^2z\,\hat{j}+zx^3\,\hat{k} \] be a vector point function. Let \(S\) be the surface of the sphere \[ x^2+y^2+z^2=a^2. \] Then \[ \iint_{S}(\nabla\times\vec{F})\cdot\vec{N}\,dS \] is equal to (\(\vec{N}\) is any unit outward normal vector to \(S\)).

If \(y_{1}=x^{2}\) is a known solution of \[ x^{2}y^{\prime\prime}-3xy^{\prime}+4y=0, \] then the second independent solution \(y_{2}\) is

\[ \lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}= \]

If \[ \vec{v}=(x+y+1)\hat{i}+\hat{j}+(-x-y)\hat{k}, \] then the value of \[ \vec{v}\cdot(\nabla\times\vec{v}) \] is

\[ \lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}= \]

If \[ \vec{v}=(x+y+1)\hat{i}+\hat{j}+(-x-y)\hat{k}, \] then the value of \[ \vec{v}\cdot(\nabla\times\vec{v}) \] is

For the function \(f(x)=\log x\), the number \(c\) strictly between \(e^2\) and \(e^3\) that satisfies \(f'(c)=\dfrac{f(e^3)-f(e^2)}{e^3-e^2}\) is