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# NCERT Solutions class 12 Maths Exercise 6.4

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## NCERT Solutions for Class 12 Maths Application of Derivatives

1. Using differentials, find the approximate value of each of the following up to 3 places of decimal:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

Ans. (i)

Let  ……….(i)

=

……….(ii)

#### Now, from eq. (i),

=

Here,  and , then

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

= 0.03

Therefore, approximately value of  is 5 + 0.03 = 5.03.

(ii)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and , then

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

= 0.0357

Therefore, approximately value of  is 7 + 0.0357 = 7.0357.

(iii)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and , then

=

#### Since,  and  is approximately equal to  and  respectively.

From eq. (ii),   =

Therefore, approximately value of  is 0.8 – 0.025 = 0.775.

(iv)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and ,

then

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

=  = 0.0083

Therefore, approximately value of  is 0.2 + 0.0083 = 0.2083.

(v)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=  =  ……….(iii)

Here  and

Then

=

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

#### Therefore, approximate value of  is 1 – 0.0001 = 0.9999.

(vi)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

= =  ……….(iii)

Here  and

Then

=

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

Therefore, approximate value of  is  = 1.96875.

(vii)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and ,

then  =

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),   =

Therefore, approximately value of  is  = 2.9629.

(viii)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

= =  ……….(iii)

Here  and

Then

=

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

Therefore, approximate value of  is  = 3.9961.

(ix)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),  = = ……….(iii)

Here  and

Then

=

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

Therefore, approximate value of  is  = 3.0092.

(x)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and , then

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),   =

Therefore, approximately value of  is  = 20.025.

(xi)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and , then

=

### Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

=

Therefore, approximately value of  is 0.06 +  = 0.060833.

(xi)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

=

Here,  and ,

then

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

=  = 0.0159

#### Therefore, approximately value of  is 3 – 0.0159 = 2.9841.

(xii)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

= =   ……….(iii)

Here  and

Then

=

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

= 0.00462

Therefore, approximate value of  is 3 + 0.00462= 3.00462.

(xiv)

Let  ……….(i)

……….(ii)

Now, from eq. (i),

=

Here,  and , then

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

=

#### Therefore, approximately value of  is 8 – 0.096 = 7.904.

(xv)

Let  ……….(i)

=

……….(ii)

Now, from eq. (i),

= =   ……….(iii)

Here  and

Then

=

=

Since,  and  is approximately equal to  and  respectively.

From eq. (ii),

= 0.001875

Therefore, approximate value of  is 2 + 0.001875= 2.001875.

### 2. Find the approximate value of  where

Ans. Let   ……….(i)

=   ……….(ii)

Changing  to  and  to  in eq. (i),

=   ……….(iii)

Here,  and

From eq. (iii),

Since,  and  is approximately equal to  and  respectively.

From eq. (i) and (ii),

= 28.21

Therefore, approximate value of  is 28.21.

### 3. Find the approximate value of  where

Ans. Let   ……….(i)

……….(ii)

Changing  to  and  to  in eq. (i),

=   ……….(iii)

Here,  and

From eq. (iii),

Since,  and  is approximately equal to  and  respectively.

From eq. (i) and (ii),

=  =

Therefore, approximate value of  is .

### 4. Find the approximate change in the volume of a cube of side  meters caused by increasing the side by 1%.

Ans. Since Volume (V) =   ……….(i)

……….(ii)

It is given that increase in side = 1% =

……….(iii)

Since approximate change in volume V of cube =  =

=    cubic meters

### 5. Find the approximate change in the surface area of a cube of side  meters caused by decreasing the side by 1%.

Ans. Since Surface area (S) =

It is given that decrease in side =  of

Since approximate change in surface area S of cube =  =

=    square meters (decreasing)

### NCERT Solutions class 12 Maths Exercise 6.4

#### 6. If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.

Ans. Let  be the radius of the sphere and  be the error in measuring the radius.

Then, according to the question,  = 7 m and  = 0.02 m

Volume of sphere (V) =

Approximate error in calculating the volume = Approximate value of

=  =

=

=

=  = 12.32 m3

Therefore, the approximate error in calculating volume is 12.32 m3.

### 7. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.

Ans. Let  be the radius of the sphere.

Surface area of the sphere (S) =

·

=  square meters

### 8. If  then the approximate value of  is:

(A) 47.66

(B) 57.66

(C) 67.66

(D) 77.66

Ans. Let   ……….(i)

……….(ii)

Changing  to  and  to  in eq. (i),

=   ……….(iii)

Here,  and

From eq. (iii),

Since,  and  is approximately equal to  and  respectively.

From eq. (i) and (ii),

=  =

Therefore, option (D) is correct.

### 9. The approximate change in the volume of a cube of side  meters caused by increasing the side by 3% is:

(A) 0.06  m3

(B) 0.6  m3

(C) 0.09  m3

(D) 0.9  m3

Ans. Since Volume (V) =    ……….(i)

……….(ii)

It is given that increase in side = 3% =

……….(iii)

Since approximate change in volume V of cube =  =

=    cubic meters

Therefore, option (C) is correct.

## NCERT Solutions class 12 Maths Exercise 6.4

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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