Application of Derivatives Test

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Application of Derivatives Test - 65

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following statements is/are correct ?

A
x + sinx is increasing function

B
tanx is an increasing function

C
x + sinx is decreasing function

D
sec x is an increasing function

Solution
Question 2
1 / -0

$$f(x)$$ is differentiable function satisfying the relation $$f(x)=x^{2}+\displaystyle \int^{x}_{0}e^{-t}f(x-t)dt$$, then $$\displaystyle \sum^{9}_{k=1}f(k)$$ equals

A
$$1060$$

B
$$1260$$

C
$$960$$

D
$$1224$$

Solution
Question 3
1 / -0

$$f(x)=\frac{x}{log x}-\frac{log}{x}$$ is increasing in

A
$$(e,\infty )$$

B
$$(0,1)\epsilon (1,e)$$

C
(0,1)

D
(1,e)

Solution
Question 4
1 / -0

If $$y = \log _ { \sin x } ( \tan x ) ,$$ then $$\frac { d y } { d x }$$ at $$x = \frac { \pi } { 4 }$$ is:

A
$$\frac { 4 } { \log 2 }$$

B
$$-\frac { 4 } { \log 2 }$$

C
$$\frac { 1 } { \log 2 }$$

D
none of these

Solution
Question 5
1 / -0

If $$\theta$$ is angle of intersection between $$y=10-x^{2}$$ and $$y=4+x^{2}$$ then $$|\tan \theta|$$ is-

A
$$\dfrac {5\sqrt {3}}{11}$$

B
$$\dfrac {7\sqrt {3}}{15}$$

C
$$\dfrac {4\sqrt {3}}{11}$$

D
$$none$$

Solution

Given:$$y=10-{x}^{2}$$ ......$$(1)$$
$$y=4+{x}^{2}$$ ......$$(2)$$
On solving,
$$10-{x}^{2}=4+{x}^{2}$$
$$\Rightarrow 2{x}^{2}=6$$
$$\Rightarrow {x}^{2}=\dfrac{6}{2}=3$$
$$\therefore x=\pm \sqrt{3}$$ from $$(1)$$
$$\Rightarrow y=10-{\left(\pm\sqrt{3}\right)}^{2}=10-3=7$$
Consider $$y=10-{x}^{2}$$
$$\dfrac{dy}{dx}=-2x$$
$${m}_{1}=\left[\dfrac{dy}{dx}\right]_{\left(\sqrt{3},7\right)}=-2\times\sqrt{3}=-2\sqrt{3}$$
Consider $$y=4+{x}^{2}$$
$$\dfrac{dy}{dx}=2x$$
$${m}_{2}=\left[\dfrac{dy}{dx}\right]_{\left(\sqrt{3},7\right)}=2\times\sqrt{3}=2\sqrt{3}$$
$$=\left|\dfrac{{m}_{1}-{m}_{2}}{1+{m}_{1}{m}_{2}}\right|$$
$$\tan{\theta}=\left|\dfrac{-2\sqrt{3}-2\sqrt{3}}{1+\left(-2\sqrt{3}\right)\left(2\sqrt{3}\right)}\right|$$
$$=\left|\dfrac{-4\sqrt{3}}{1-12}\right|$$
$$=\dfrac{4\sqrt{3}}{11}$$
$$\therefore \tan{\theta}=\dfrac{4\sqrt{3}}{11}$$
Question 6
1 / -0

The function $$f(x)=\sqrt{25-4x^{2}}$$ is increasing in

A
(-3,0)

B
(4,0)

C
(-5/20)

D
R
Question 7
1 / -0

The function log (log x) increases in

A
$$(1,\infty )$$

B
$$(0,\infty )$$

C
$$\infty $$

D
R
Question 8
1 / -0

The function f defined by $$f(x)=(x+2)e^{-x}$$ si

A
decreasing for all x

B
decreasing in $$(-\infty ,-1)$$ and increasing in $$(1,\infty )$$

C
increasing for all x

D
decreasing in $$(-1,-\infty )$$ and increasing in $$(-\infty ,-1)$$
Question 9
1 / -0

If $$f:R\rightarrow R$$ is the function defined by $$f\left( x \right) =\frac { { e }^{ { x }^{ 2 } }-{ e }^{ -x^{ 2 } } }{ { e }^{ x^{ 2 } }+{ e }^{ { -x }^{ 2 } } } ,$$ then

A
$$f(x) $$ is an increasing function

B
$$f(x) $$ is an decreasing function

C
$$f(x) $$ is onto (surjective)

D
Into function
Question 10
1 / -0

Let $$f(x)=\left\{\begin{matrix} max \{|x|, x^2\}, & |x|\leq 2\\ 8-2|x|, & 2 < |x|\leq 4\end{matrix}\right.$$
Let S be the set of points in the interval $$(-4, 4)$$ at which f is not differentiable. Then S?

A
Is an empty set

B
Equals $$\{-2, -1, 1, 2\}$$

C
Equals $$\{-2, -1, 0, 1, 2\}$$

D
Equals $$\{-2, 2\}$$

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Re-Attempt Weekly Quiz Competition

Application of Derivatives Test - 65

Result

Application of Derivatives Test - 65

Score

-

Rank

-

SHARING IS CARING

Question 1

x + sinx is increasing function

tanx is an increasing function

x + sinx is decreasing function

sec x is an increasing function

Solution

Question 2

$$1060$$

$$1260$$

$$960$$

$$1224$$

Solution

Question 3

$$(e,\infty )$$

$$(0,1)\epsilon (1,e)$$

(0,1)

(1,e)

Solution

Question 4

$$\frac { 4 } { \log 2 }$$

$$-\frac { 4 } { \log 2 }$$

$$\frac { 1 } { \log 2 }$$

none of these

Solution

Question 5

$$\dfrac {5\sqrt {3}}{11}$$

$$\dfrac {7\sqrt {3}}{15}$$

$$\dfrac {4\sqrt {3}}{11}$$

$$none$$

Solution

Question 6

(-3,0)

(4,0)

(-5/20)

R

Question 7

$$(1,\infty )$$

$$(0,\infty )$$

$$\infty $$

R

Question 8

decreasing for all x

decreasing in $$(-\infty ,-1)$$ and increasing in $$(1,\infty )$$

increasing for all x

decreasing in $$(-1,-\infty )$$ and increasing in $$(-\infty ,-1)$$

Question 9

$$f(x) $$ is an increasing function

$$f(x) $$ is an decreasing function

$$f(x) $$ is onto (surjective)

Into function

Question 10

Is an empty set

Equals $$\{-2, -1, 1, 2\}$$

Equals $$\{-2, -1, 0, 1, 2\}$$

Equals $$\{-2, 2\}$$

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