Differentiation Test 39

Result

Differentiation Test 39

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

A Printer number the pages of a book starting with $$1$$ and used $$3193$$ digit in all. How many pages does the book have:

A
$$1074$$

B
$$1120$$

C
$$1595$$

D
None of these

Solution

No. of digits in 1-digit page no. $$\Rightarrow 1\times 9=9$$
No. of digits in 2-digit page no. $$\Rightarrow 2\times 90= 180$$
No. of digits in 3-digit page no. $$\Rightarrow 3\times 900= 2700$$
No. of digits in 4-digit page no.$$ \Rightarrow 300$$
$$\therefore $$ No. of pages with 4- digit page no. $$= (300/4)=75$$
Hence, total no. of pages in the book $$= (999+75)= 1074$$
Question 2
1 / -0

The number of permutations of letters of the word "PARALLAL" atken four at a time must be,

A
$$216$$

B
$$244$$

C
$$286$$

D
$$1680$$

Solution

Permutation of word PARALLAL taken four at a time
Four letter word might be $$LLL$$_
So total of $$16$$ words [As $$4$$ways, $$4$$spots]
For $$AALL$$, $$6$$ different arrangements
$$AA$$_ _, can be $${ 4 }_{ { C }_{ 2 } }$$ ways $$=6$$ ways
Arranged in $$12$$ ways is total of $$72$$ words
And extra words can be formed in $$120$$ ways
$$16+6+144+120=286$$ ways
Question 3
1 / -0

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

A
$$\dfrac{\log (\sin y)+y \tan x}{\log (\cos x)- x \cot y}$$

B
$$\dfrac{\log (\sin y)-y \tan x}{\log (\cos x)+ x \cot y}$$

C
$$\dfrac{\log (\sin y)}{\log (\cos x)}$$

D
$$\dfrac{\log (\cos x)}{\log (\sin y)}$$

Solution

Given,
$$(\cos x)^y=(\sin y)^x$$
Now taking logarithm with respect to the base $$e$$ we get,
$$y\log (\cos x)=x\log(\sin y)$$
Now differentiating both sides with respect to $$x$$ we get,
$$\log(\cos x)\dfrac{dy}{dx}+y.\dfrac{(-\sin x)}{\cos x}$$$$=\log(\sin y)+x.\dfrac{(\cos y)}{\sin y}\dfrac{dy}{dx}$$
or, $$(\log(\cos x)-x\cot y)\dfrac{dy}{dx}=\log (\sin y)+y\tan x$$
or, $$\dfrac{dy}{dx}=\dfrac{\log (\sin y)+y\tan x}{\log(\cos x)-y\cot x}$$
Question 4
1 / -0

7, 11, 23, 51, 103 ?

A
186

B
188

C
185

D
187

E
none of these

Solution

$$7,11,23,51,103,?$$
$$7+4\times 1= 11 $$
$$11+4\times 3=23$$ $$\leftarrow $$ difference of 2
$$23+4\times 7=51$$ $$\leftarrow $$ difference of 4
$$51+4\times 13= 103$$ $$\leftarrow $$ difference of 6
$$103+4\times 21= 187$$ $$\leftarrow $$ difference will be of 8
Question 5
1 / -0

Using all digits $$2,3,4,5,6$$ how many even number can be formed

A
$$24$$

B
$$48$$

C
$$72$$

D
$$120$$

Solution

$$2,3,4,5,6$$
(Ref Image)
$$\Rightarrow 4!\times 3=24\times 3=72$$
Question 6
1 / -0

13, 16, 22, 33, 51 ?

A
89

B
78

C
102

D
69

E
none of these

Solution

REF.Image
$$ 13,16,22,33,51, ?$$
Question 7
1 / -0

655, 439, 314, 250, 223 ?

A
215

B
210

C
195

D
190

E
none of these

Solution

Solution $$\rightarrow $$ Given terms are 655,439,314,250,223,?
As $$a_{1}=655$$
$$a_{2}= 439$$
$$a_{3}=314$$
$$a_{4}= 250$$
$$a_{5}= 223$$
$$a_{6}=?$$
So, $$a_{1}-a_{2}=655- 439= 216=6^{3}$$
$$a_{3}-a_{2}= 439- 314= 125=5^{3}$$
$$a_{3}-a_{4}= -250+314= 64=4^{3}$$
$$a_{4}-a_{5}=-223+250= 27=3^{3}$$
So, we can observe difference b/w terms are cubes
of 6,5,4,3 so, is similar way
$$a_{5}-a_{6}=2^{3}$$
$$223-a_{6}= 8$$
$$a_{6} = 215$$
So, we find next term is 215
Question 8
1 / -0

If $$y = \exp \left\{ {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + ....} \right\}$$ then $$\frac {dy}{dx}=$$

A
$${e^{{{\tan }^2}x}}$$

B
$${e^{{{\tan }^2}x}}{\sec ^2}x$$

C
$$2{e^{{{\tan }^2}x}}\tan x{\sec ^2}x$$

D
none

Solution
Question 9
1 / -0

The differential of $$f(x)=\sqrt{\dfrac{2-x}{2+x}}$$ at $$x=0$$ and $$\delta x=0.15$$ is

A
$$0.07$$

B
$$0.075$$

C
$$-0.075$$

D
$$0.15$$

Solution
Question 10
1 / -0

Six people are going to sit in a row on a bench. $$A$$ and $$B$$ are adjacent, $$C$$ does not want to sit adjacent to $$D.E$$ and $$F$$ can sit anywhere. Number of ways in which these six people can be seated is

A
$$200$$

B
$$144$$

C
$$120$$

D
$$56$$

Solution

A, B, C, D, E, F
Consider AB as group so we have AB, C, D, E, F.
We have totally $$5$$
No. of ways$$(w_1)=5!\times 2$$
$$=240$$
Let CD are adjacent now AB, CD, E, F
No. of ways$$(w_2)=4!2!2!$$
$$=96$$
Total no. of ways
$$W=w_1-w_2$$
$$=240-96$$
$$=144$$.