Problem 2
If \(x=u v, y=u / v\), then \(\frac{\partial(x, y)}{\partial(u, v)}\) is \(\begin{array}{llll}(a)-2 u / v & \text { (b) }-2 x / u & \text { (c) } 0 & \text { (d) } 1 .\end{array}\)
Problem 3
\(J_{1}=\frac{\partial(u, v)}{\partial(x, y)}\) and \(J_{2}=\frac{\partial(x, y)}{\partial(u, v)}\), then \(J_{1} J_{2}=\ldots\)
Problem 4
Select the correct ansuer or fill up the blanks in each of the following problems: If \(u=f(y / x)\), then \((a) x \frac{\partial u}{\partial x}-y \frac{\partial u}{\partial y}=0\) (b) \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=0\) (c) \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=2 u\) (d) \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=1\).
Problem 6
Select the correct ansuer or fill up the blanks in each of the following problems: If \(x=r \cos \theta, y=r \sin \theta\), then (a) \(\frac{d x}{\partial r}=1 / \frac{\partial r}{\partial x}\) (b) \(\frac{\partial x}{\partial r}=\frac{\partial r}{d x}\) (c) \(\frac{\partial x}{\partial r}=0\).
Problem 7
Select the correct ansuer or fill up the blanks in each of the following problems: If \(u=x^{y}\), then dulcy is \((a) y x^{y-1}\) (b) 0 (c) \(x^{y} \log x\).
Problem 8
Select the correct ansuer or fill up the blanks in each of the following problems: If \(u=x^{3}+y^{2}\), then \(\frac{\partial^{2} u}{\partial x \partial y}\) is equal to \(\begin{array}{llll}\text { (a) }-3 & \text { (b) } 3 & \text { (c) } 0 & \text { (d) } 3 x+3 y\end{array}\)
Problem 9
Select the correct ansuer or fill up the blanks in each of the following problems: If \(u=x^{2}+2 x y+y^{2}+x+y\), then \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}\) is equal to (a) \(2 u\) (b) \(\underline{b}\) (c) 0 (d) none of these.
Problem 10
Select the correct ansuer or fill up the blanks in each of the following problems: If \(u=\log \frac{x^{2}}{y}\), then \(x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}\) is equal to (a) \(2 u\) (b) \(3 u\) (c) \(u\) (d) 1 .
Problem 11
Select the correct ansuer or fill up the blanks in each of the following problems: If \(x=r\) cos \(\theta, y=r \sin \theta\), thèn \(\frac{\partial(x, y)}{\partial(r, \theta)}\) is equal to (a) 1 (b) \(c\) (c) \(\mathbf{1} / \boldsymbol{r}\) (d) 0 .
Problem 15
If an error of \(1 \%\) is made in measuring ite length and breadth, the percentnge error in the area of a reetangle in (a) \(0.24\) \(\begin{array}{lllll}\text { (b) } 0.02 \% & \text { (c) } 2 \% & (\text { d }) 18 . & \text { 18. } & \text { (V.T.U, 2010) }\end{array}\)
