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FUNCTION OF SEVERAL VARIABLES – MCQ – LEVEL – 1

FUNCTION OF SEVERAL VARIABLES – MCQ – LEVEL – 1

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Question 1 of 10

1. Question
The value of $\underset{(x, y)\rightarrow(0, 0)}\lim\left (\dfrac{x^2 - y^2}{x^2 + y^2}\right)^2$
- $0$
- $1$
- $\infty$
- $\dfrac{1}{2}$
Correct

Incorrect
Question 2 of 10

2. Question
The limit $\underset{(x, y)\rightarrow(\pi, \pi)}\lim \left[x\sin\left( \dfrac{x+y}{4} \right) \right]$
- does not exists
- exists and equals to $\pi$
- exists infinitely
- exists and equals to $\dfrac{\pi}{2}$
Correct

Incorrect
Question 3 of 10

3. Question
If $f(x, y) = \dfrac{x^3 + y^3}{x^2 + y^2},$ when $(x, y) \rightarrow(0, 0)$
- limits exists and equals to 0
- limit does not exist
- limit exists and equals to 1
- cannot be determined
Correct

Incorrect
Question 4 of 10

4. Question
If $f''=0$ at a stationary point, then we have :
- minimum
- minimum or maximum
- maximum
- minimum or maximum or inflection
Correct

Incorrect
Question 5 of 10

5. Question
The limit of the function $f(x, y) = -\dfrac{xy}{x^2 + y^2}$ along the line $y = x$ is :
- $1$
- $\dfrac{1}{2}$
- $-\dfrac{1}{2}$
- $-1$
Correct

Incorrect
Question 6 of 10

6. Question
The function $f(x, y) = \dfrac{xy^2}{x^2+y^4}$ has
- limit as $(x, y) \rightarrow(0, 0)$
- does not have a limit
- limit exists infinitely
- cannot be determined
Correct

Incorrect
Question 7 of 10

7. Question
If $g(x,y) \leq f(x,y) \leq h(x,y) \forall (x,y)\neq(x_{0}, y_{0})$ and $\underset{(x,y)\rightarrow(x_0, y_0)}\lim g(x,y) = \underset{(x,y)\rightarrow(x_0, y_0)}\lim h(x,y) = L,$ then :
- limit $f(x, y)$ does not exit
- limit $f(x, y)$ does not depends on $L$
- limit $f(x, y) = L$
- limit $f(x, y) = \dfrac{L}{2}$
Correct

Incorrect
Question 8 of 10

8. Question
The function $f(x, y ) = \dfrac{x^3}{x^2+y^2}$ has :
- limit as $(x, y)\rightarrow(0, 0)$
- limit exists infinitely
- cannot be determined
- does not have a limit
Correct

Incorrect
Question 9 of 10

9. Question
Let $f(x, y, z) = z^3 + 3xz^2 + y^2z,$ then, $\dfrac{\partial z}{\partial x}\ \textrm{and} \ \dfrac{\partial z}{\partial y}$ at the point (1, 1, 1) be :
- $\dfrac{\partial z}{\partial x}=\dfrac{1}{10}, ~\dfrac{\partial z}{\partial y}=\dfrac{3}{10}$
- $\dfrac{\partial z}{\partial x}=\dfrac{3}{10} , \dfrac{\partial z}{\partial y}=\dfrac{1}{10}$
- $\dfrac{\partial z}{\partial x}={10} ,\dfrac{\partial z}{\partial y}=\dfrac{10}{3}$
- $\dfrac{\partial z}{\partial x} = \dfrac{10}{3} , \dfrac{\partial z}{\partial y} = {10}$
Correct

Incorrect
Question 10 of 10

10. Question
$\underset{(x, y)\rightarrow(0, 0)}\lim \ \dfrac{x^2}{x^2+y^2}$
- exists and equals to zero
- exists and equals to one
- does not exists
- exists infinitely
Correct

Incorrect

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