Differentiation Test 3

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Differentiation Test 3

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Weekly Quiz Competition

Question 1
1 / -0

Find the differential equations of all parabolas each having latus rectum $$4a$$ and whose axes are parallel to the x-axis.

A
$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=a$$

B
$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=-a$$

C
$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=2a$$

D
$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=-2a$$

Solution

The equation of the parabola is $$\displaystyle \left ( y-k \right )^{2}=4ax \cdots \left ( 1 \right )$$
$$\displaystyle \therefore 2\left( y-k \right )y_{1}=4a$$
or $$ y-k=\cfrac{2a}{y_{1}}$$
Putting in (1), we get
$$\displaystyle \frac{4a^{2}}{y^{2}_{1}}=4ax$$ or $$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=a$$
Question 2
1 / -0

$$\displaystyle \frac{d}{dx}(\tan ^{-1}\frac{\cos x-\sin x}{\cos x+\sin x})$$

A
$$-1$$

B
$$-2$$

C
$$1$$

D
$$x$$

Solution

Let $$\displaystyle y = \tan ^{-1}\frac{\cos x-\sin x}{\cos x+\sin x}=\tan ^{-1}\frac{1-\tan x}{1+\tan x}$$
$$\displaystyle \Rightarrow y =\tan ^{-1}\frac{\tan\frac{\pi}{4}-\tan x}{1+\tan\frac{\pi}{4}\tan x}=\tan^{-1}\tan(\frac{\pi}{4}-x)=\frac{\pi}{4}-x$$
$$\therefore \cfrac{dy}{dx}=-1$$
Question 3
1 / -0

$$\displaystyle \frac{d}{dx}(\tan^{-1}\sqrt{\left ( \frac{1-\cos x}{1+\cos x} \right )})$$

A
$$\displaystyle \frac{1}{2}$$

B
$$\displaystyle \frac{1}{4}$$

C
$$\displaystyle \frac{1}{\sqrt2}$$

D
$$\displaystyle \frac{-1}{2}$$

Solution

Let $$\displaystyle y = \tan^{-1}\sqrt{\left ( \frac{1-\cos x}{1+\cos x} \right )}$$
$$\Rightarrow \displaystyle y = \tan^{-1}\sqrt{\left ( \frac{2\sin^2\frac{x}{2} }{2\cos^2 \frac{x}{2}} \right )}$$
$$\Rightarrow \displaystyle y= \tan^{-1}\left [ \tan\left ( \frac{x}{2} \right ) \right ]=\frac{x}{2}$$
$$\displaystyle \therefore \frac{dy}{dx}= \frac{1}{2}$$
Question 4
1 / -0

CE, GI, KM, OQ, ....

A
TW

B
TV

C
SU

D
RT

E
UW

Solution

The letters of each term are alternate Also the last letter of each term and the first letter of the next term are alternate
Question 5
1 / -0

How many combinations of two-digit numbers having 8 can be made from the following numbers?
8, 5, 2, 1, 7, 6

A
$$9$$

B
$$10$$

C
$$11$$

D
$$12$$

Solution

The possible two-digit number are
88, 85, 82, 81, 87, 86, 58, 28, 18, 78, 68
These are 11 in number
Question 6
1 / -0

Which letter will be the fifth from the right if the first and second, the third and fourth and so on are interchanged among each other in the word "COMPANIONATE"?

A
A

B
I

C
N

D
O

Solution

When we interchange the first and the second, third and the fourth, and so on letters in the word COMPANIONATE, the new word becomes OCPMNAOIANET.
Therefore, 5th letter from right is I
Answer is Option B
Question 7
1 / -0

There are $$5$$ roads leading to a town from a village. The number of different ways in which a villager can go to the town and return back, is

A
$$25$$

B
$$20$$

C
$$10$$

D
$$5$$

Solution

Permutation : - A Permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement.

For example, Suppose we have a set of three letters :A, B and C..... each possible arrangement would be an example of permutation.

Here in this question, there are $$5$$roads leading to a town from a village.

He can go by $$5$$ ways and can return by $$4$$ ways

So the number of different ways in which a villager can go to the town and return back is $$5 \times 4 = 20$$
Question 8
1 / -0

BDF, CFI, DHL, ....

A
CJM

B
EIM

C
EJO

D
EMI

Solution

First, 2nd, 3rd letters are moving +1, +2, +3 steps
Question 9
1 / -0

The spread sheet on the right contains 20 cells. A cell In a spread sheet can be identified fIrst by the column letter and then by the row number. For example,. the number 10 is found in cell C4. If the number in cell A3 =B4-3(E2 +D4) then which of the following must be the number in cell E2?

A
-21

B
-15

C
-4

D
-12

Solution

Its given
A3=B4-3(E2+D4)
Substituting the values from the grid
$$18=-6-3(E2+4)$$
$$24=-3(E2+4)$$
$$-8-4=E2$$
$$E2=-12$$
Answer is Option D
Question 10
1 / -0

JE, LH, OL, SQ, ....

A
WV

B
WX

C
VW

D
VX

E
XW

Solution

First letter: +2, +3, +4, +5,.... Second letter: +3, +4, +5, +6, ....

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Differentiation Test 3

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Differentiation Test 3

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Question 1

$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=a$$

$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=-a$$

$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=2a$$

$$\displaystyle x\left ( \frac{dy}{dx} \right )^{2}=-2a$$

Solution

Question 2

$$-1$$

$$-2$$

$$1$$

$$x$$

Solution

Question 3

$$\displaystyle \frac{1}{2}$$

$$\displaystyle \frac{1}{4}$$

$$\displaystyle \frac{1}{\sqrt2}$$

$$\displaystyle \frac{-1}{2}$$

Solution

Question 4

TW

TV

SU

RT

UW

Solution

Question 5

$$9$$

$$10$$

$$11$$

$$12$$

Solution

Question 6

A

I

N

O

Solution

Question 7

$$25$$

$$20$$

$$10$$

$$5$$

Solution

Question 8

CJM

EIM

EJO

EMI

Solution

Question 9

-21

-15

-4

-12

Solution

Question 10

WV

WX

VW

VX

XW

Solution

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