Question

If a young man rides his motorcycle at $25 \,km / h$, he has to spend ₹ 2 per $km$ in petrol, and if he rides it at $40\, km / h$, the petrol cost rises to ₹ 5 per $km$. He has ₹ 100 to spend on petrol and wishes to find the maximum distance, he can travel within one hour. If $x$ and $y$ denote the distance travelled by him (in $km$ ) at $25\, km / h$ and $40 km / h$, respectively.
The solution of the inequalities representing the data is

• A. unique
• B. intinite
• C. No solution
• D. None of the above

Answer 1

Answer: (B)

The inequalities represent the data are
$2 x+5 y \leq 100$....(i)
$\frac{x}{25}+\frac{y}{40} \leq 1$....(ii)
$x \geq 0 $....(iii)
$y \geq 0$....(iv)
The line corresponding to (i) and (ii) are
$ 2 x+5 y=100 $....(v)
$\frac{x}{25}+\frac{y}{40}=1$....(vi)
Line $(v)$ cuts $X$-axis at $(50,0)$ and $Y$-axis at $(0,20)$. Line (vi) cuts $X$-axis at $(25,0)$ and $Y$-axis at $(0,40)$. Since, $(0,0)$ satisfies both (i) and (ii). So, both inequality represent the region which contains the origin.
The common region of (i), (ii), (iii), (vi) is the shaded region in the figure.

Hence, there are infinite point which satisfy the inequalities (i), (ii), (iii) and (vi).