# NCERT Class 12-Mathematics: Chapter – 6 Application of Derivatives Part 9 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 4**:

Two men A and B start with velocities *v* at the same time from the junction of two roads inclined at to each other. If they travel by different roads, find the rate at which they are being separated.

**Answer**:

Given: two men A and B start with velocities v at the same time from the junction of the two roads inclined at 45° to each other

To find the rate at which they are being separated

Explanation:

Let A and B move a distance of x on different roads as shown above, there distance at any time t will be same as they have same velocity.

Hence

Now consider ΔAOB, applying the cosine rule, we get

Now multiplying and dividing by , we get

Now applying the derivative with respect to , we get

Taking out the constant terms, we get

Substituting the value from equation (i) , we get

Hence this is the rate at which the two roads are being separated

**Question 5**:

Find an angle which increases twice as fast as its sine.

**Answer**:

Given: a condition

To find the angle such that it increases twice as fast as its sine.

Explanation: Let

On differentiating with respect to , we get

Applying the derivative, we get

But it is given that

Substituting this value in equation (i) , we get

Now cancelling the like terms, we get

But given this is possible only when

Hence the angle is .