Differentiation Test 33

Result

Differentiation Test 33

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

Number of arrangements of the letter $$HOLLYWOOD$$ in which all $$Os$$ are separated.

A
$$360\ ^{ 7 }{ C }_{ 3 }$$

B
$$15\ ^{ 7 }{ P }_{ 4 }$$

C
$$15\ ^{ 7 }{ C }_{ 4 }$$

D
$$30\ ^{ 7 }{ P }_{ 4 }$$
Question 2
1 / -0

If $$\dfrac { ^{ n }{ P }_{ r-1 } }{ a } =\dfrac { ^{ n }{ P }_{ r } }{ b } =\dfrac { ^{ n }{ P }_{ r+1 } }{ c } $$, then which of the following holds good:

A
$$c^{2}=a(b+c)$$

B
$$a^{2}=c(a+b)$$

C
$$b^{2}=c(a-b)$$

D
$$\dfrac {1}{a}+\dfrac {1}{b}+\dfrac {1}{c}=1$$

Solution

$$\dfrac{^{n} {P}_{r-1}}{a} = \dfrac{^{n}{P}_{r}}{b} = \dfrac{^{n}{P}_{r+1}}{c} $$

$$\implies \dfrac{n!}{(n - r+ 1)!a} = \dfrac{n!}{(n-r)!b} = \dfrac{n!}{(n-r-1)!c} $$

$$\implies \dfrac{n!}{(n - r+ 1)!a} = \dfrac{n!}{(n-r)(n-r+1)!b} = \dfrac{n!}{(n-r-1)(n-r)(n-r+1)!c} $$

$$\implies \dfrac{1}{a} = \dfrac{1}{(n-r)b} = \dfrac{1}{(n-r-1)(n-r)c} $$

$$\implies n-r = \dfrac{a}{b}$$ and $$n-r-1 = \dfrac{b}{c}$$

$$\implies n-r - 1 = \dfrac{a}{b} - 1 $$

$$\implies \dfrac{b}{c} = \dfrac{a}{b} - 1$$

$$\implies \dfrac{b}{c} - \dfrac{a}{b} = - 1$$

$$\implies b^2 - ac = - bc $$

$$\implies b^2 = ac - bc$$

$$\therefore b^2 = c(a - b)$$ (Ans)
Question 3
1 / -0

The number of permutations of the letters of the word $$HONOLULU$$ taken $$4$$ at a time is

A
$$ 354$$

B
$$ 314$$

C
$$ 210$$

D
$$ 124$$

Solution

$$H-1$$
$$N-1$$
$$O-2$$
$$L-2$$
$$U-2$$
Number of permutation $$=\dfrac {n!}{r_1\times r_2\times....\times r_n} =\dfrac { 8! }{ { (2! })^{ 3 }4! } =210$$
Question 4
1 / -0

Function f(x)=$$\left| {x - 2} \right| - 2\left| {x - 4} \right|\,is$$ discontinous at:

A
$$x=2,4$$

B
$$x=2$$

C
No where

D
Except $$x=2$$

Solution

$$f(x)=\mid x-2 \mid -2\mid x-4 \mid$$
$$\therefore f\left(x \right)=\begin{Bmatrix} -(x-2)+2(x-4),\qquad x<2\\ (x-2)+2(x-4), \qquad 2\le x \le 4\\ (x-2)-2(x-4) \qquad x\gt 4 \\ \end{Bmatrix}$$
At $$x=2$$
L.H.L(Left Hand Limit)
$$\displaystyle\lim_{x\to\ 2^{-}}=-(2-2)+2(2-4) = -4$$
R.H.LRight Hand Limit/0
$$\displaystyle\lim_{x\to\ 2^{+}}=(2-2)+2(2-4)=-4$$
$$\therefore \displaystyle\lim_{x\to\ 2^{-}}f(x) = \displaystyle\lim_{x\to\ 2^{+}}f(x)$$
At $$x=4$$
L.H.L(Left Hand Limit)
$$\displaystyle\lim_{x\to\ 4^{-}}=(4-2)+2(4-4)=2$$
R.H.L(Right Hand Limit)
$$\displaystyle\lim_{x\to\ 4^{+}}=(4-2)-2(4-4)=2$$
$$\therefore \displaystyle\lim_{x\to\ 4^{-}}f(x) = \displaystyle\lim_{x\to\ 4^{+}}f(x)$$
$$\therefore \ f(x)$$ is continuous at everywhere.
Question 5
1 / -0

Differentiate with respect to $$x$$.
$${x^{\cos x}} + \sin {x^{\tan x}}$$

A
$$x^{\cos x}\left[\dfrac {\cos x}{x}-\sin x\right]+\sin x^{\tan x}[1+\sec^2x.\log \sin x]$$

B
$$x^{\cos x}\left[\dfrac {\cos x}{x}-\sin x.\log x\right]+\sin x^{\tan x}[1+\sec^2x.\log \sin x]$$

C
$$x^{\cos x}\left[ {\cos x}-\sin x.\log x\right]+\sin x^{\tan x}[1+\sec x.\log \sin x]$$

D
None

Solution

let $$y=x^{\cos x} +\sin x ^{\tan x}$$
let $$h=x^{\cos x}$$ $$\dfrac{d}{dx}(f(x)g(x))=f'(x)g(x)+f'(x)f(x)$$
applying $$log$$ on both sides we get
$$\log h=\cos x\log x$$
differentiate on both sides wrt $$x$$
$$\dfrac{1}{h}\dfrac{dh}{dx}=-\sin x\log x+\dfrac{\cos x}{x}$$
$$\therefore \dfrac{dh}{dx}=x^{\cos x}\left(\dfrac{\cos x}{x}-\sin x \log x\right)$$
let $$t=\sin x^{\tan x}$$
applying $$\log $$ on both sides
$$\dfrac{1}{t}\dfrac{dt}{dx}=\sec ^2x\log {(\sin x)}+\dfrac{\tan x}{\sin x}(\cos x)$$
$$\therefore \dfrac{dt}{dx}=\sin x^{\tan x}\left(1+\sec ^2x \log (\sin x)\right)$$
$$y=h+t$$
$$\implies \dfrac{dy}{dx}=\dfrac{dh}{dx}+\dfrac{dt}{dx}$$
$$\therefore \dfrac{dy}{dx}=x^{\cos x}\left(\dfrac{\cos x}{x}-\sin x\log x\right)+\sin x^{\tan x}\left(1+\sec ^2x\log (\sin x)\right)$$
Question 6
1 / -0

Differentiate $${x^{\tan x}} + {{\mathop{\rm tanx}\nolimits} ^x}$$ with respect to $$x$$

A
$$x^{\tan x}(\sec^2x \log x+\dfrac{\tan x}{x})+\tan x^x(\log \tan x+\dfrac{2}{\sin 2x})$$

B
$$x^{\tan x}(\sec^2x \log \sec x+\dfrac{\tan x}{x})+\tan x^x(\log \tan x+\dfrac{2}{\sin 2x})$$

C
$$x^{\tan x}(\sec^2x \log x+\dfrac{\tan x}{x})-\tan x^x(\log \tan x+\dfrac{2}{\sin 2x})$$

D
$$x^{\tan x}(\sec^2x \log x+\dfrac{\tan x}{x})+\tan x^x(\log \tan x-\dfrac{2}{\sin 2x})$$

Solution
Question 7
1 / -0

If $$y = {\left( {\sin \,x} \right)^x}$$, then $$\dfrac{{dy}}{{dx}} = $$

A
$$(\sin x)^x(\ln (\sin x)+x\cot x)$$

B
$$(\ln (\sin x)+x\cot x)$$

C
$$(\sin x)^x(\ln (\sin x)+x\tan x)$$

D
$$(\sin x)^x(\ln (\sin x)-\cot x)$$

Solution

given $$y=(\sin x)^x$$
applying $$\ln$$ on both sides we get
$$\ln y=x\ln(\sin x)$$
differentiating both sides wrt $$x$$
$$\dfrac{1}{y}\dfrac{dy}{dx}=\dfrac{d}{dx}\left(x\ln(\sin x)\right)$$
$$\dfrac{1}{y}\dfrac{dy}{dx}=\ln(\sin x)+x\dfrac{1}{\sin x}\cos x$$
$$\dfrac{dy}{dx}=y\left(\ln(\sin x)+x\cot x\right)$$
$$\therefore \dfrac{dy}{dx}=(\sin x)^x\left(x\cot x\right) + (\sin x)^x \ln(\sin x)$$
Question 8
1 / -0

Number of identical terms in the sequence $$2, 5, 8, 11,$$___ upto $$100$$ terms and $$3, 5, 7, 9, 11$$____ upto $$100$$ terms are

A
$$17$$

B
$$33$$

C
$$50$$

D
$$147$$

Solution

Fierst series: $$2, 5, 8, 11.........$$
$$a_1 =2;\ d=3;\ n=100$$
$$l=20(100-1)^*5=2+99^*3=2+297=299$$
so, series $$=2, 5, 8, 11,.....,197, 197,200,203,....,299$$
second series; $$3, 5, 9, 11,.....$$
$$a_1=3;\ d=2;\ n=100$$
$$l=3+(100-1)^*2=3+99^*2=3+198=201$$
so series $$=3, 5, 7, 9, 11, ...., 201$$
series having similar terms; $$5, 11, 17, ....., 197$$
$$a_1=5;\ d=6;\ l=197$$
$$l=a_1+(n-1)d$$
$$197=5+(n-1)6$$
$$192/6=n-1$$
$$32+1=n$$
$$n=33$$
Question 9
1 / -0

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?

A
$$50000$$

B
$$50100$$

C
$$50300$$

D
$$50400$$

Solution

$$2$$ vowels can be chosen in $$5{c}_{2}$$ ways.
$$2$$ consonants can be chosen in $$21{c}_{2}$$ ways.
$$4$$ letter can be arranged in $$4$$ ways.
$$\therefore$$ The no of words containing $$2$$ vowels and $$2$$ consonants $$=5{c}_{2}\times 21{c}_{2}\times 4!=10\times 210\times 24=50400$$
Question 10
1 / -0

$$67,84,95,.,133,158$$

A
$$116$$

B
$$129$$

C
$$108$$

D
$$111$$

Solution

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Differentiation Test 33

Result

Differentiation Test 33

Score

-

Rank

-

SHARING IS CARING

Question 1

$$360\ ^{ 7 }{ C }_{ 3 }$$

$$15\ ^{ 7 }{ P }_{ 4 }$$

$$15\ ^{ 7 }{ C }_{ 4 }$$

$$30\ ^{ 7 }{ P }_{ 4 }$$

Question 2

$$c^{2}=a(b+c)$$

$$a^{2}=c(a+b)$$

$$b^{2}=c(a-b)$$

$$\dfrac {1}{a}+\dfrac {1}{b}+\dfrac {1}{c}=1$$

Solution

Question 3

$$ 354$$

$$ 314$$

$$ 210$$

$$ 124$$

Solution

Question 4

$$x=2,4$$

$$x=2$$

No where

Except $$x=2$$

Solution

Question 5

$$x^{\cos x}\left[\dfrac {\cos x}{x}-\sin x\right]+\sin x^{\tan x}[1+\sec^2x.\log \sin x]$$

$$x^{\cos x}\left[\dfrac {\cos x}{x}-\sin x.\log x\right]+\sin x^{\tan x}[1+\sec^2x.\log \sin x]$$

$$x^{\cos x}\left[ {\cos x}-\sin x.\log x\right]+\sin x^{\tan x}[1+\sec x.\log \sin x]$$

None

Solution

Question 6

$$x^{\tan x}(\sec^2x \log x+\dfrac{\tan x}{x})+\tan x^x(\log \tan x+\dfrac{2}{\sin 2x})$$

$$x^{\tan x}(\sec^2x \log \sec x+\dfrac{\tan x}{x})+\tan x^x(\log \tan x+\dfrac{2}{\sin 2x})$$

$$x^{\tan x}(\sec^2x \log x+\dfrac{\tan x}{x})-\tan x^x(\log \tan x+\dfrac{2}{\sin 2x})$$

$$x^{\tan x}(\sec^2x \log x+\dfrac{\tan x}{x})+\tan x^x(\log \tan x-\dfrac{2}{\sin 2x})$$

Solution

Question 7

$$(\sin x)^x(\ln (\sin x)+x\cot x)$$

$$(\ln (\sin x)+x\cot x)$$

$$(\sin x)^x(\ln (\sin x)+x\tan x)$$

$$(\sin x)^x(\ln (\sin x)-\cot x)$$

Solution

Question 8

$$17$$

$$33$$

$$50$$

$$147$$

Solution

Question 9

$$50000$$

$$50100$$

$$50300$$

$$50400$$

Solution

Question 10

$$116$$

$$129$$

$$108$$

$$111$$

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.