Differentiation Test 38

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Differentiation Test 38

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Weekly Quiz Competition

Question 1
1 / -0

If $$y=y(x)$$ and it follows the relation $$e^{xy^{2}}+y\cos(x^{2})=5$$ then $$y'(0)$$ is equal to

A
$$4$$

B
$$-16$$

C
$$-4$$

D
$$16$$

Solution
Question 2
1 / -0

If $$y=a\ \sin\ x+b\ \cos\ x$$, then $$y^{2}+\left ( \dfrac{dy}{dx} \right )^{2}$$ is

A
function of $$x$$

B
function of $$y$$

C
function of $$x$$ and $$y$$

D
constant

Solution

$$y=a\sin{x}+b\cos{x}$$

$$\dfrac{dy}{dx}=a\cos{x}-b\sin{x}$$

$${y}^{2}+{\left(\dfrac{dy}{dx}\right)}^{2}={\left(a\sin{x}+b\cos{x}\right)}^{2}+{\left(a\cos{x}-b\sin{x}\right)}^{2}$$

$$={a}^{2}{\sin}^{2}{x}+{b}^{2}{\cos}^{2}x+2ab\sin{x}\cos{x}+{a}^{2}{\cos}^{2}{x}+{b}^{2}{\sin}^{2}x-2ab\sin{x}\cos{x}$$

$$={a}^{2}\left({\sin}^{2}{x}+{\cos}^{2}{x}\right)+{b}^{2}\left({\sin}^{2}{x}+{\cos}^{2}{x}\right)$$

$$={a}^{2}+{b}^{2}$$ since $${\sin}^{2}{x}+{\cos}^{2}{x}=1$$

$$=constant$$
Question 3
1 / -0

The total number of ways of arranging the letters $$AAABBBCCDEF$$ in a row such that letters $$C$$ are separated from one another is

A
$$277200$$

B
$$138600$$

C
$$453600$$

D
none of these

Solution
Question 4
1 / -0

The maximum number of points of intersection of $$7$$ staright lines and $$5$$ circles when $$3$$ straight lines are parallel and $$2$$ circles are concentric,is /are

A
$$106$$

B
$$96$$

C
$$90$$

D
$$None\ of\ these$$

Solution
Question 5
1 / -0

Choose the odd one out .

A
KWIJ

B
TPRN

C
CHAF

D
OZMX

Solution
Question 6
1 / -0

If $$f(x+y)=2f(x).f(y)$$ for all $$x,y$$, where $$f'(0)=3$$ and $$f'(4)=2$$ then $$f'(4)=3$$ is equal to

A
6

B
12

C
4

D
3

Solution
Question 7
1 / -0

In how many ways atleast one horse and atleast one dog can be selected out of eight horses and seven dogs.

A
$${2}^{15}-2$$

B
$${2}^{15}-1$$

C
$$({2}^{8}-1)({2}^{7}-1)$$

D
$${ _{ }^{ 15 }{ C } }_{ 2 }$$

Solution

As we can either select or not select any horse; total ways for horses$$=2^r$$ ways to be excluded are when no horse is selected, which is only one way, therefore, favourable ways for horses$$=2^8-1$$
similarly, favourable ways for dogs$$=2^7-1$$
$$\Rightarrow$$ Total ways$$=(2^8-1)(2^7-1)$$.
Question 8
1 / -0

If y = sin $$^{-1} (x. \sqrt{1-x}+ \sqrt{x}\sqrt{1-x^2})$$, then $$\dfrac{dy}{dx} =$$

A
$$\dfrac{-2x}{\sqrt{1+x^2}} + \dfrac{1}{2\sqrt{x-x^2}}$$

B
$$\dfrac{-1}{\sqrt{1+x^2}} - \dfrac{1}{2\sqrt{x-x^2}}$$

C
$$\dfrac{1}{\sqrt{1+x^2}} - \dfrac{1}{2\sqrt{x-x^2}}$$

D
None of these

Solution

$$y=\sin^{-1}(x\sqrt{1-x}+\sqrt{x}\sqrt{1-x^2})\\ \quad=\sin^{-1}x+\sin^{-1}\sqrt x\\ \cfrac{dy}{dx}=\cfrac{1}{\sqrt{1-x^2}}+\cfrac{1}{\sqrt{1-x}}\times\cfrac{1}{2\sqrt{x}}\\ \quad=\cfrac{1}{\sqrt{1-x^2}}+\cfrac{1}{2\sqrt x\sqrt{1-x}}$$
Question 9
1 / -0

If $$64a^2+36b^2=400, ab=4 $$ , then $$8a+6b$$ is:

A
26

B
28

C
22

D
None of the above

Solution

Given $$64a^2+36b^2=400,\quad ab=4$$
We know
$$(a+b)^2=a^2+b^2+2ab$$
therefore,
$$(8a+6b)^2=(8a)^2+(6b)^2+2(8a)(6a)$$
$$=64a^2+36b^2+96ab$$
$$=400+96\times4$$
$$=400+384$$
$$(8a+6b)^2=784$$
$$8a+6b=\sqrt{784}$$
$$8a+6b=28$$
Question 10
1 / -0

The number of positive integral solutions of the equation $$x _ { 1 } x _ { 2 } x _ { 3 } x _ { 4 } x _ { 5 } = 1050$$ is

A
1800

B
1600

C
1400

D
1875

Solution

We have,
$${x_1}{x_2}{x_3}{x_4}{x_5} = 1050 = 2 \times 3 \times {5^2} \times 7$$
Now,
$$2,3$$ and $$7$$ can be put in any boxes $${x_1},{x_2},{x_3},{x_4}$$ and $${x_5}$$.
Also $$5,5$$ can be distributed in $$5$$ boxes in
$$^{2 + 5 - 1}{C_{5 - 1}}{ = ^6}{C_4} = 15$$Ways.
So total no. of positive integral solutions $$=15 \times5 \times5 \times5$$
$$=1875$$
Option $$D$$ is correct answer.