GATE Mechanical 2017 Set 2 | GA | Question: 10

Question

In the graph below, the concentration of a particular pollutant in a lake is plotted over (alternate) days of a month in winter (average temperature $10^{\circ} C$) and a month in summer (average temperature $30^{\circ} C$).

Consider the following statements based on the data shown above:

1. Over the given months, the difference between the maximum and the minimum pollutant concentrations is the same in both winter and summer
2. There are atleast four days in the summer month such that the pollutant concentrations on those days are within $1$ ppm of the pollutant concentrations on the corresponding days in the winter month.

Which one of the following options is correct?

• A. Only i
• B. Only ii
• C. Both i and ii
• D. Neither i nor ii

Comments

Got it. I misread it. Poor doubt

---

option B) seems correct.

(i) (max-min)_summer = 9 , (max-min)_winter = 8

(ii) 4 days where ppm difference is within 1 ppm are 8,10,12,14

---

I too got the same and answer is (b). Just wanted to confirm points for (b).

Answer 1

• Maximum pollutant concentration in the summer month $= 10.5$
• Maximum pollutant concentration in the winter month $= 8$
• Minimum pollutant concentration in the summer month $= 1.5$
• Minimum pollutant concentration in the winter month $= 0$

So, the difference between the maximum and minimum pollutant concentration in summer $ = 10.5 - 1.5 = 9.$

The difference between the maximum and minimum pollutant concentration in winter $ = 8-0= 8.$

So, $(i)$ is FALSE.

If we see days $10-14$ the pollutant concentration between the summer and winter months are within $0.5$ ppm. So, $(ii)$ is TRUE.

Correct Option: B.

Comments

within p means  $\leq p$  or it would mean  $< p$.

Anyone ??

---

within $X \implies \leq X$

---

Thanks Sir

---

If someone having confusion that when we don't have the data of 11th and 13th day of month then how can we assume that they are also within $\leq 1$ ppm range ?

that is because graph is monotonically increasing. so for ex. if at 10th day the values are: $(3, 3)$ for summer and winter respectively and at 13th day the values are: $(4, 4.5)$ for summer and winter respectively then at 12th day the values can't be out of this range $(3-4, 3-4.5)$ for summer and winter respectively and because graph follows straight line path in b/w 2 alternate days then we can be sure that the values at 11th day will be: $\Big(\dfrac{3 + 4}{2}, \dfrac{3 + 4.5}{2}\Big) = (3.5, 3.75)$ for summer and winter respectively and they are within $\leq 1$ ppm range. we can think similarly for 13th day also.

Answer 2

Let's justify the $1^{st}$ statement

In winter:

The maximum concentration of pollutant = $8$ ppm

The minimum concentration of pollutant = $0$ ppm

Difference between the maximum and the minimum pollutant concentrations(in winter) = $8-0$ ppm = $8$ ppm

In Summer:

The maximum concentration of pollutant = $10.5$ ppm

The minimum concentration of pollutant = $1.5$ ppm

Difference between the maximum and the minimum pollutant concentrations(in summer) = $10.5-1.5$ ppm = $9$ ppm

∴ We can easily see that the difference between the maximum and the minimum pollutant concentrations are not same in both winter and summer.

So, statement i) is false.

Now, justify the justify the $2^{nd}$ statement

It says that- There are at least four days in the summer month such that the pollutant concentrations on those days are within 1 ppm of the pollutant concentrations on the corresponding days in the winter month.

So, we can see that statement ii) is true.

Hence, the correct option is $\textbf{B. only ii)}$.

Comments

I was doing mistakes because I thought the question ask us to find 4 days of winter in which pollutant concentration are within 1 ppm

---

In objective exams it is very important to avoid these mistakes.

---

Part ii) asks for "at least four days in the summer month " which implies there should be atleast 4 distinct days in the summer month such that on these 4 days the ppm is either +1 or -1 than the ppm of the corresponding days in winter month. In simple terms there should be 4 distinct pair of points such that absolute value of diff between ppm of each pair of points should be less than or equal to 1. Days 6,8,10,12,14 qualify this criteria but according to Arjun sir only 2 such data points are enough. Where did I go wrong ?