Differentiation Test 2

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Differentiation Test 2

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Weekly Quiz Competition

Question 1
1 / -0

Each combination corresponds to many permutations.

A
True

B
False

C
Either

D
Neither

Solution

In combination
Each combination can be considered as a set of selection an order
Each selection has a defined order
They can be considered as a permutation
Each cpmbination corresponds to many permutations
Hence the above statement is true.
Question 2
1 / -0

Factorial of negative numbers is always greater than 1.

A
True

B
False

C
Either

D
Neither

Solution

Factorial : product of an integer with integer less than it.
Factorial can be interpolated using gamma function and gamma function and
gamma function is not defined for negative integer.
Factorial is not defined for negative integer
Question 3
1 / -0

Choose the correct option for the following.
$$n!=n(n-1)(n-2).....3.2.1$$

A
True

B
False

C
Ambiguous

D
Data insufficient

Solution

Factorial : the product of an integer and all integer less than that
$$\therefore n!=n\times (n-1)\times (n-2)............3\times 2\times 1$$

$$\therefore $$ The given statement
$$n!=n\times (n-1)\times (n-2)............3\times 2\times 1$$ is True
Question 4
1 / -0

The positions of the first and the fifth digits in the number 83256479 are interchanged. Similarly the positions of the second and the sixth digits are interchanged and so on. Which of the following will be the third to the right of the seventh digit from the right end after rearrangement?

A
3

B
4

C
7

D
8

Solution

In the given number 83256479, the first and the fifth digits are interchanged.The number becomes 63258479.
Then second and the sixth digit are interchanged. The number becomes 64258379.
The third and the seventh digit are interchanged. The number becomes 64758329.
The fourth and the eighth digit are interchanged. The number becomes 64798325.
The seventh digit from the right end after rearrangement is 4.The third digit to the right of 4 is 8.
Answer is Option D
Question 5
1 / -0

Differentiate with respect to x $$\displaystyle x^{4}+3x^{2}-2x$$

A
$$\displaystyle 4x^{3}+6x-2$$

B
$$\displaystyle 4x^{3}+6x-3$$

C
$$\displaystyle 4x^{4}+6x-2$$

D
None of the above

Solution

Given, $$y=\displaystyle x^{4}+3x^{2}-2x$$

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

$$\dfrac{dy}{dx}=\dfrac{d(x^4)}{dx} + \dfrac{d(3x^2)}{dx}-\dfrac{d(2x)}{dx}$$

$$\dfrac{dy}{dx}=4{ x }^{ 3 }+6x-2$$
Question 6
1 / -0

The area of a triangle whose vertices are $$(a, c+a), (a, c) $$ and $$(-a, c-a) $$ are

A
$$\displaystyle a^{2}$$

B
$$\displaystyle b^{2}$$

C
$$\displaystyle c^{2}$$

D
$$\displaystyle a^{2}+c^{2}$$

Solution

Area of a triangle with vertices $$({ x }_{ 1 },{ y }_{ 1 })$$ ; $$({ x }_{ 2 },{ y

}_{ 2 })$$ and $$({ x }_{ 3 },{ y }_{ 3 })$$ is $$ \left| \frac { {

x }_{ 1 }({ y }_{ 2 }-{ y }_{ 3 })+{ x }_{ 2 }({ y }_{ 3 }-{ y }_{ 1 })+{ x }_{

3 }({ y }_{ 1 }-{ y }_{ 2 }) }{ 2 } \right| $$
Hence, area of the triangle with given vertices is $$ \left| \frac { { a }(c-c+a)+a(c-a-c-a)-a(c+a-c) }{ 2 } \right| = \left| \frac { { a }^{ 2 }-2{ a }^{ 2 }-{ a }^{ 2 } }{ 2 } \right| = { a }^{ 2 } $$
Question 7
1 / -0

The greatest number that can be formed by the digits $$7,0,9,8,6,3$$

A
$$9,87,360$$

B
$$9,87,063$$

C
$$9,87,630$$

D
$$9,87,603$$

Solution

The greatest number that can be formed by the digits $$7,0,9,8,6,3$$ is $$9 8 7 6 3 0$$
To achieve this arrange the given numbers in descending order.
Question 8
1 / -0

In the series given below. count the number of 9s, each of which Is not immediately preceded by 5 but is immediately followed by either 2 or 3. How many such 9s are there?
1 9 2 6 5 9 3 8 3 9 3 2 5 9 2 9 3 4 8 2 6 9 8

A
One

B
Three

C
Five

D
Six

Solution

There are 3 such 9s that are not immediately preceded by 5 and immediately followed by 2 or 3 in the given series. They are marked in bold.
1 9 2 6 5 9 3 8 3 9 3 2 5 9 2 9 3 4 8 2 6 9 8
Answer is Option B
Question 9
1 / -0

If $$f(x) = \displaystyle \log \left | 2x \right |, x\neq 0 $$ then $$f'(x)$$ is equal to-

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

B
$$\displaystyle -\frac{1}{x}$$

C
$$\displaystyle \frac{1}{\left | x \right |}$$

D
None of these

Solution

$$f(x) = \displaystyle \log \left | 2x \right |$$ $$\displaystyle x\neq 0$$
$$\displaystyle \log \left | x \right |$$ is defined for $$| x | > 0$$
$$f(x) = \displaystyle \log 2x = \log 2+\log x $$
$$\displaystyle f'\left ( x \right )=0+\frac{1}{x}=\frac{1}{x}$$
Question 10
1 / -0

Differentiation of $$\displaystyle x^{3}+5x^{2}-2$$ with respect to $$x$$ is

A
$$3x^{2}+10x$$

B
$$3x^{2}+10$$

C
$$3x^{2}-2$$

D
$$3x^{2}+10x-2$$

Solution

Given, $$x^3+5 x^2-2$$

Differentiating with respect to $$x$$, we get

$$\displaystyle \frac{d}{dx}\left ( x^{3}+5x^{2}-2 \right )=\frac{d}{dx}\left ( x^{3} \right )+5\frac{d}{dx}\left ( x^{2} \right )-\frac{d}{dx}\left ( 2 \right )$$

$$=\displaystyle 3x^{2}+5\left ( 2x \right )-0$$

$$=3x^{2}+10x$$

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Re-Attempt Weekly Quiz Competition

Differentiation Test 2

Result

Differentiation Test 2

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SHARING IS CARING

Question 1

True

False

Either

Neither

Solution

Question 2

True

False

Either

Neither

Solution

Question 3

True

False

Ambiguous

Data insufficient

Solution

Question 4

3

4

7

8

Solution

Question 5

$$\displaystyle 4x^{3}+6x-2$$

$$\displaystyle 4x^{3}+6x-3$$

$$\displaystyle 4x^{4}+6x-2$$

None of the above

Solution

Question 6

$$\displaystyle a^{2}$$

$$\displaystyle b^{2}$$

$$\displaystyle c^{2}$$

$$\displaystyle a^{2}+c^{2}$$

Solution

Question 7

$$9,87,360$$

$$9,87,063$$

$$9,87,630$$

$$9,87,603$$

Solution

Question 8

One

Three

Five

Six

Solution

Question 9

$$\displaystyle \frac{1}{x}$$

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

$$\displaystyle \frac{1}{\left | x \right |}$$

None of these

Solution

Question 10

$$3x^{2}+10x$$

$$3x^{2}+10$$

$$3x^{2}-2$$

$$3x^{2}+10x-2$$

Solution

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