MCQs of Partial Derivatives (Mathematics - I - 3110014)

Mathematics - I (3110014) MCQs

MCQs of Partial Derivatives

Showing 11 to 20 out of 43 Questions

11.

If $f (x, y) = c$ be an implicit function, then $q^{3} \frac{d^{2} y}{{dx}^{2}} =$ _____.

(a)	$2 pqs - p^{2} t - q^{2} r$
(b)	$- [\frac{q^{2} r - 2 pqs + p^{2} t}{q^{3}}]$
(c)	$[\frac{q^{2} r - 2 pqs + p^{2} t}{q^{3}}]$
(d)	$p^{2} t - 2 pqs + q^{2} r$

12.

The equation of tangent plane to the surface $z = xy - 2$ at point $P (1, 1, - 1)$ is _____.

(a)	$x + y + z - 3 = 0$
(b)	$x + y - z - 3 = 0$
(c)	$- x - y - z - 3 = 0$
(d)	$x - y - z + 3 = 0$

13.

If $u = e^{xyz}$ , then $\frac{\partial u}{\partial z} =$ _____.

(a)	$e^{xyz}$
(b)	$z e^{xyz}$
(c)	$xy e^{xyz}$
(d)	$xyz e^{xyz}$

14.

The equation of normal line to the surface $x^{2} + y^{2} + z^{2} = 3$ at the point $(1, 1, 1)$ is _____.

(a)	$\frac{x - 1}{2} = \frac{y - 1}{2} = \frac{z + 1}{2}$
(b)	$x + 1 = y + 1 = z - 1$
(c)	$x = 2 y = 3 z$
(d)	$x = y = z$

15.

The another representation of $f_{xyxz}$ is _____.

(a)	$\frac{\partial^{4} f}{\partial x \partial y \partial x \partial z}$
(b)	$\frac{\partial^{4} f}{\partial x^{2} \partial y \partial z}$
(c)	$\frac{\partial^{4} f}{\partial z \partial x \partial y \partial x}$
(d)	$\frac{\partial^{4} f}{\partial z \partial x^{2} \partial y}$

16.

For the stationary point $(- 1, 2)$ , $f (x, y) = x^{3} + y^{3} - 3 x - 12 y + 20$ . Then

(a)	$f$ has local minimum at $(- 1, 2)$ .
(b)	$f$ has local maximum at $(- 1, 2)$ .
(c)	$(- 1, 2)$ is a saddle point.
(d)	No conclusion at $(- 1, 2)$ .

17.

Which of the following is not a stationary point of $f (x, y) = x + y + \frac{1}{x} + \frac{1}{y}$ ?

(a)	$(1, 0)$
(b)	$(1, 1)$
(c)	$(- 1, 1)$
(d)	$(1, - 1)$

18.

If $x + y + z = 12$ , then $\max (xyz)$ is _____.

(a)	12
(b)	64
(c)	4
(d)	16

19.

Find $$\frac{\operatorname d\mathrm u}{\operatorname d\mathrm t}$$, if $$\mathrm u=4\mathrm x+\mathrm y^2,\;\mathrm x=\cos\left(2\mathrm t\right)\;\&\;\mathrm y=\sin\left(2\mathrm t\right).$$

(a)	$$2\sin\left(2\mathrm t\right)-8\sin\left(4\mathrm t\right)$$
(b)	$$-8\sin\left(2\mathrm t\right)-2\sin\left(4\mathrm t\right)$$
(c)	$$-8\sin\left(4\mathrm t\right)+2\sin\left(4\mathrm t\right)$$
(d)	$$2\sin\left(4\mathrm t\right)-8\sin\left(2\mathrm t\right)$$

20.

If $u = f (\frac{x}{y}, \frac{y}{z}, \frac{z}{x}),$ where $r = \frac{x}{y}, s = \frac{y}{z} & t = \frac{z}{x} .$ Which of the following is true?

(a)	$\frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial r} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial r}$
(b)	$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r} \cdot \frac{\partial r}{\partial x} + \frac{\partial u}{\partial s} \cdot \frac{\partial s}{\partial x} + \frac{\partial u}{\partial t} \cdot \frac{\partial t}{\partial x}$
(c)	$\frac{\partial u}{\partial s} = \frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial s}$
(d)	$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial u}{\partial y} \cdot \frac{\partial y}{\partial t} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial t}$

Showing 11 to 20 out of 43 Questions