Differentiation Test 34

Result

Differentiation Test 34

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

How many committee of five persons with a chairperson can be selected from $$12$$ persons.

A
$$924$$

B
$$825$$

C
$$736$$

D
$$643$$

Solution

No of Persons $$=5+1$$(Chairperson)

$$\therefore$$ No of committees that can be selected $$={}^{12}C_6=924$$
Question 2
1 / -0

Find the value of ?.

A
$$125$$

B
$$25$$

C
$$625$$

D
$$156$$

Solution

$${\left( {3 + 1} \right)^2} = {\left( 4 \right)^2} = 16$$

$${\left( {15 + 6} \right)^2} = {\left( {21} \right)^2} = 441$$

$${\left( {10 + 5} \right)^2} = {\left( {15} \right)^2} = 225$$

$$so,\,{\left( {12 + 13} \right)^2} = {\left( {25} \right)^2} = 625$$

therefore the answer is $$625$$
Question 3
1 / -0

The shortest distance between line $$y-x=1$$ and curve $$x={y}^{2}$$ is

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

B
$$\dfrac {3\sqrt {2}}{8}$$

C
$$\dfrac {8}{2\sqrt {2}}$$

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

Solution

Line is $$y-x=1$$ and curve $$x=y^2$$
$$y^2=x$$
$$2y\dfrac{dy}{dx}=1$$
$$\dfrac{dy}{dx}=\dfrac{1}{2y}=1$$
$$\therefore$$ $$y=\dfrac{1}{2}$$
$$x=\dfrac{1}{4}$$
Tangent at $$\left(\dfrac{1}{4},\,\dfrac{1}{2}\right)$$
$$\dfrac{1}{2}y=\dfrac{1}{2}\left(x+\dfrac{1}{4}\right)$$
$$y=x+\dfrac{1}{4}$$
Distance $$=\left|\dfrac{1-\dfrac{1}{4}}{\sqrt{2}}\right|=\dfrac{3}{4\sqrt{2}}=\dfrac{3\sqrt{2}}{8}$$
Question 4
1 / -0

How many $$3$$-digit even numbers can be formed from the digits $$1, 2, 3, 4, 5, 6$$ if the digits can be repeated?

A
108

B
98

C
72

D
112

Solution

Only $$3$$ numbers are possible at units place $$(2,4,6)$$ as we need even numbers. But at $$10's$$ place and $$100's$$ place all $$6$$ are possible.
No. of digit even no's$$=3\times 6\times 6=108$$
Question 5
1 / -0

Let $$y=a\cos t+b\sin t$$ then $$\dfrac{d^2y}{dt^2}=$$

A
$$y$$

B
$$-y$$

C
$$2y$$

D
none of these

Solution

Given,
$$y=a\cos t+b\sin t$$......(1).
Now differentiating both sides with respect to $$t$$ we get,
$$\dfrac{dy}{dt}=-a\sin t+b\cos t$$
Again differentiating both sides with respect to $$t$$ we get,
$$\dfrac{d^2y}{dt^2}=-a\cos t-b\sin t$$
or, $$\dfrac{d^2y}{dt^2}=-y$$. [ Using (1)]
Question 6
1 / -0

Find the area of the triangle whose vertices are $$(-5, -1), (3, -5), (5, 2)$$

A
$$31 sq$$ $$units$$

B
$$32sq$$ $$units$$

C
$$33 sq$$ $$units$$

D
$$\text{no triangle can be formed}$$

Solution

Area of a triangle whose vertices are $$A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)$$ is given as,
$$Area=(\dfrac{1}{2})[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)] $$
Let $$A$$ be the required area
$$A=(\dfrac{1}{2})[(-5)(-5-2)+3(2+1)+5(-1+5)]\\ (\dfrac{1}{2})[35+9+20]\\=32sq\>unit$$
So, the correct option is (B)
Question 7
1 / -0

Let $$f(x)=\dfrac{1}{ax+b}$$ then $$f''(0)=$$

A
$$\dfrac{2a^3}{b^2}$$

B
$$\dfrac{2a^2}{b^3}$$

C
$$\dfrac{2a^3}{b^3}$$

D
none of these

Solution

Given,
$$f(x)=\dfrac{1}{ax+b}$$
Now differentiating both sides with respect to $$x$$ we get,
$$f'(x)=-\dfrac{a}{(ax+b)^2}$$
Again differentiating both sides with respect to $$x$$ we get,
$$f''(x)=\dfrac{2a^2}{(ax+b)^3}$$
So $$f''(0)=\dfrac{2a^2}{b^3}$$.
Question 8
1 / -0

Ten students of the physics department decided to go on a educational trip.They hired a mini bus for the trip, but the bus can only carry eight students at a time and each student goes at least once. Find the minimum number of trips the bus has to make so that each students can go for equal number of trips.

A
$$5$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

Ten students $$\longrightarrow$$ Eight students in bus $$2$$ students are excluded/not taken on a trip so we have exclude every students once, so that no. of trips made by each students are equal so no. of trips $$= \dfrac{10}{2} =5$$
$$A$$ is correct.
Question 9
1 / -0

The sides AB, BC, CA of a triangle ABC have $$3,4$$ and $$5$$ interior points respectively on them. The number of triangles that can be constructed using these points as vertices is-

A
$$205$$

B
$$210$$

C
$$315$$

D
$$216$$

Solution

$$\rightarrow $$Suppose we had $$12$$ points, none of them collinear, number of possible triangles $$=^{ 12 }{ C }_{ 3 }$$
$$\rightarrow $$If out of these $$3,4$$ and $$5$$ points were chosen separately to lie on each side of a triangle ABC, then number of possible triangles
$$=^{ 12 }{ C }_{ 3 }-^{ 3 }{ C }_{ 3 }-^{ 5 }{ C }_{ 3 }-^{ 4 }{ C }_{ 3 }$$
$$=220-1-10-4$$
$$=205$$
Question 10
1 / -0

If $$y=(x^{x})^{x}$$ then $$\dfrac {dy}{dx}=$$

A
$$(x^{x})^{x}(1+2\log x)$$

B
$$(x^{x})^{x}(1-2\log x)$$

C
$$x(x^{x})^{x}(1+2\log x)$$

D
$$x(x^{x})^{x}(1-2\log x)$$

Solution

We have,

$$ y={{\left( {{x}^{x}} \right)}^{x}} $$

$$ y={{x}^{{{x}^{2}}}} $$

On taking $$\log $$ both sides, we get

$$\log y={{x}^{2}}\log x$$ …… (1)

On differentiating w.r.t $$x$$, we get

$$ \dfrac{1}{y}\dfrac{dy}{dx}=\dfrac{{{x}^{2}}}{x}+\log x\left( 2x \right) $$

$$ \dfrac{1}{y}\dfrac{dy}{dx}=x+2x\log x $$

$$ \dfrac{dy}{dx}=y\left( x+2x\log x \right) $$

$$ \dfrac{dy}{dx}={{\left( {{x}^{x}} \right)}^{x}}\left( x+2x\log x \right) $$

$$ \dfrac{dy}{dx}=x{{\left( {{x}^{x}} \right)}^{x}}\left( 1+2\log x \right) $$

Hence, this is the answer.

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Differentiation Test 34

Result

Differentiation Test 34

Score

-

Rank

-

SHARING IS CARING

Question 1

$$924$$

$$825$$

$$736$$

$$643$$

Solution

Question 2

$$125$$

$$25$$

$$625$$

$$156$$

Solution

Question 3

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

$$\dfrac {3\sqrt {2}}{8}$$

$$\dfrac {8}{2\sqrt {2}}$$

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

Solution

Question 4

108

98

72

112

Solution

Question 5

$$y$$

$$-y$$

$$2y$$

none of these

Solution

Question 6

$$31 sq$$ $$units$$

$$32sq$$ $$units$$

$$33 sq$$ $$units$$

$$\text{no triangle can be formed}$$

Solution

Question 7

$$\dfrac{2a^3}{b^2}$$

$$\dfrac{2a^2}{b^3}$$

$$\dfrac{2a^3}{b^3}$$

none of these

Solution

Question 8

$$5$$

$$4$$

$$6$$

$$8$$

Solution

Question 9

$$205$$

$$210$$

$$315$$

$$216$$

Solution

Question 10

$$(x^{x})^{x}(1+2\log x)$$

$$(x^{x})^{x}(1-2\log x)$$

$$x(x^{x})^{x}(1+2\log x)$$

$$x(x^{x})^{x}(1-2\log x)$$

Solution

User

Question Analysis

My Perfomance

Score

-

Rank

-

Results Announcement on: coming Monday

Stay Tuned: We’ll update you on your result via email or SMS.