Question

Raman Lal runs a stationery shop in Pune. The analysis of his sales, expenditures and profits showed that for x number of notebooks sold, the weekly profit (in ₹) was P(x) = –2x2 + 88x – 680. Raman Lal found that:  He has a loss if he does not sell any notebook in a week. There is no profit no loss for a certain value x0 of x. The profit goes on increasing with an increase in x i.e. the number of notebooks sold. But he gets a maximum profit at a sale of 22 notebooks in a week.

Now answer the following questions:

1. What will be Raman Lal’s profit if he sold 20 notebooks in a week?

1. ₹ 144
2. ₹ 280
3. ₹ 340
4. ₹ 560

2. What is the maximum profit that Raman Lal can earn in a week?

1. ₹ 144
2. ₹ 288
3. ₹ 340
4. ₹ 680

3. What is Raman Lal’s loss if he does not sell any notebooks in a particular week?

1. ₹ 0
2. ₹ 340
3. ₹ 680
4. ₹ 960

4. Write a quadratic equation for the condition when Raman Lal does not have any profit or loss during a week.

1. 2x² – 44x + 340 = 0
2. x² + 44x – 340 = 0
3. x² – 88x + 340 = 0
4. x² – 44x + 340 = 0

5. What is the minimum number of notebooks x that Raman Lal should sell in a week so that he does not incur any loss?

1. 0
2. 10
3. 11
4. 12

Answer 1

1. Given,

⇒ P(x) = –2x² + 88x – 680

Books sold by Raman lal is 20.

Therefore, x = 20

⇒ P(20) = –2(20)² + 88(20) – 680

⇒ P(20) = –2(400) + 1760 – 680

⇒ P(20) = –800 + 1080

⇒ P(20) = 280

Hence, option (b) is the correct option.

2. Given,

⇒ P(x) = –2x² + 88x – 680

Maximum profit occurs when 22 books are sold, thus x = 22.

⇒ P(22) = –2(22)² + 88(22) – 680

= –2(484) + 1936 – 680

= –968 + 1256

= ₹ 288

Hence, option (b) is the correct option.

3. Given,

⇒ Number of notebooks sold in a particular week = 0

Thus, x = 0

The weekly profit P(x) = –2x² + 88x – 680.

Raman Lal’s loss if no notebooks sold in a week,

⇒ P(0) = –2(0)² + 88(0) – 680

= –680

= Loss of ₹ 680

Hence, option (c) is the correct option.

4. Given,

⇒ P(x) = –2x² + 88x – 680

If there is no profit/loss, then profit = ₹ 0.

⇒ P(x) = 0

⇒ –2x² + 88x – 680 = 0

⇒ –2(x² – 44x + 340) = 0

⇒ x² – 44x + 340 = 0

Hence, option (d) is the correct option.

5. When Raman lal has no profit/loss we get the equation x² – 44x + 340 = 0.

⇒ x² – 44x + 340 = 0

⇒ x² – 10x – 34x + 340 = 0

⇒ x(x – 10) – 34(x – 10) = 0

⇒ (x – 10)(x – 34) = 0

⇒ (x – 10) = 0 or (x – 34) = 0   ...[Using zero-product rule]

⇒ x = 10 or x = 34

The minimum value of x with no loss is x = 10. Raman lal has to sell 10 books so that he does not incur any loss.

Hence, option (b) is the correct option.