#### 1600 satellites were sent up by a country for several purposes. The purposes are classified as broadcasting (B), communication (C), surveillance (S), and others (O). A satellite can serve multiple purposes; however a satellite serving either B, or C, or S does not serve O. The following facts are known about the satellites: 1. The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2: 1:1. 2. The number of satellites serving all three of B, C, and S is 100. 3. The number of satellites exclusively serving C is the same as the number of satellites exclusively serving S. This number is 30% of the number of satellites exclusively serving B. 4. The number of satellites serving O is the same as the number of satellites serving both C and S but not B. Question: If the number of satellites serving at least two among B, C, and S is 1200, which of the following MUST be FALSE? Option: 1 The number of satellites serving C cannot be uniquely determined   Option: 2 The number of satellites serving B is more than 1000   Option: 3 All 1600 satellites serve B or C or S   Option: 4 The number of satellites serving B exclusively is exactly 250

Given data:

1. Number of satellites serving B, C, and S (though maybe not exclusively) are in the ratio $2:1:1.\$

2. Number of satellites serving all three of B, C, and S is 100

3. Number of satellites exclusively serving C is the same as the number of satellites exclusively serving S (= 30% of the number of satellites exclusively serving B)

4. Number of satellites serving O is the same as the number of satellites serving both C and S but not B

Since there are satellites serving a single or multiple purposes, Venn Diagram is a good method to solve.

For ease of calculation, let the number of satellites exclusively serving B = 10x.

$\Rightarrow$ the number of satellites exclusively serving C and S
$\mathrm{= 0.30\times10x }$
$\mathrm{= 3x }$

Now, let the number of satellites serving others(O) by y.

Lastly, let the number of satellites serving B, C but not S be z.

Since the numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio $\mathrm{2:1:1. }$

$\Rightarrow$ number of satellites serving B, S but not C = z.

It is given that the total number of satellites =1600 :
\mathrm{\begin{aligned} & \Rightarrow 10 x+2 z+2 y+6 x=1600 \\ & \Rightarrow 8 x+z+y=750 \ldots \text { (1) } \end{aligned}}
The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2:1:1
\mathrm{ \begin{aligned} 10 x+2 z+100 \quad \ldots(2) \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow z+100+3 x+y=1 \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow 10 x+2 z+100=2(z+100+3 x+y) \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow 4 x=100+2 y \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow 2 x=50+y \\ \end{aligned} }

\mathrm{ \begin{aligned} \Rightarrow y=2 x-50 \end{aligned} }
Solving the equations (substitute this in equation 1):
\mathrm{ \begin{aligned} \Rightarrow 8 x+z+2 x-50 & =750 \\ z=800-10 x & \ldots(3) \end{aligned} }

Using the boundary condition for x,

$\mathrm{\Rightarrow 2x-50\geq 0}$

$\mathrm{\Rightarrow x\geq 25}$

Also,

$\mathrm{800-10x\geq 0}$

$\mathrm{\Rightarrow x\leq 80}$

Therefore, number of satellites in each of the category can be determined.

Hence, option C is definitely false.

Therefore, we can say that option C is incorrect.