List of practice Questions

The solution of the differential equation \(x^2\,dx+y^2\,dy=0\) is
Evaluate \((\vec j-2\vec i)\cdot(\vec k+3\vec i-\vec j)\).
Solve \(e^{3x}\,dx+e^{4y}\,dy=0\).
The solution of \(\dfrac{dx}{x}+\dfrac{dy}{y}=0\) is
The integrating factor of the linear DE \(\dfrac{dy}{dx}-y\sin x=\cot x\) is
\([\,{-}1\,]\,[\,1\ \ {-}1\,]=\ \ ?\)
\(\begin{bmatrix}2&3 \\ 5&7\end{bmatrix}\!\begin{bmatrix}2 \\ 5\end{bmatrix}=\ \ ?\)
\(\begin{bmatrix}13&15 \\ -1&4\end{bmatrix}\begin{bmatrix}2&0 \\ 0&2\end{bmatrix}=\ \ ?\)

\(3\!\begin{bmatrix}7&-2 \\ 8&0\end{bmatrix}=\ \ ?\)

The maximum value of \(Z=6x+3y\) subject to \(x+y\le 25,\ x\ge0,\ y\ge0\) is
The maximum value of \(Z=x-3y\) subject to \(x+y\le13,\ x\ge0,\ y\ge0\) is
The minimum value of \(Z=7x+8y\) subject to \(3x+4y\le24,\ x\ge0,\ y\ge0\) is
Compute \((2\vec i-3\vec j)\cdot(\vec i+\vec k)\).
For \(|x|\le1\), \(2\tan^{-1}x=\ \ ?\)
For \(x\in\mathbb R\), \(\cot^{-1}x=\ \ ?\)
\(\tan^{-1}\!\left(\dfrac{x+y}{1-xy}\right)=\ \ ?\)
\(\sin^{-1}\!\big(\sin \tfrac{2\pi}{3}\big)=\ \ ?\)
\(\displaystyle \int_{\pi/6}^{\pi/4}\tan\theta\,d\theta=\ \ ?\)
\(\displaystyle \int \sin^{3}\theta\ \csc^{2}\theta\,d\theta=\ \ ?\)
\(\displaystyle \int \big(\cos\theta\,\csc^{2}\theta-\cos\theta\,\cot^{2}\theta\big)\,d\theta=\ \ ?\)
\(\displaystyle \int (4\cos x-5\sin x)\,dx=\ \ ?\)
\(\displaystyle \int \frac{3\cos x-2\sin x}{2\cos x+3\sin x}\,dx=\ \ ?\)
\(\displaystyle \int \frac{3x^{2}+2}{x^{3}+2x}\,dx=\ \ ?\)
\(\displaystyle \int \frac{dx}{x^{2}+5}=\ \ ?\)

\(\displaystyle \int_{-1}^{1}\log\!\left(\frac{3+x}{\,3-x\,}\right)dx=\ \ ?\)