JEE MAIN - Mathematics (2023 - 25th January Evening Shift - No. 19)
A triangle is formed by X-axis, Y-axis and the line $$3x+4y=60$$. Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____________.
Answer
31
Explanation
If x = 1, y = $57 \over 4 $ = 14.25
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$(1,1)(1,2)-(1,14) \Rightarrow 14$ pts.
If $x=2, y=\frac{27}{2}=13.5$
$(2,2)(2,4) \ldots(2,12) \quad \Rightarrow 6$ pts.
If $\mathrm{x}=3, \mathrm{y}=\frac{51}{4}=12.75$
$(3,3)(3,6)-(3,12) \Rightarrow 4$ pts.
If $x=4, y=12$
$(4,4)(4,8) \quad \Rightarrow 2$ pts.
If $x=5 . y=\frac{45}{4}=11.25$
$(5,5),(5,10) \Rightarrow 2$ pts.
If $\mathrm{x}=6, \mathrm{y}=\frac{21}{2}=10.5$
$(6,6) \quad \Rightarrow 1 \mathrm{pt}$.
If $x=7, y=\frac{39}{4}=9.75$
$(7,7) \Rightarrow 1 \mathrm{pt}$.
If $x=8, y=9$
$(8,8) \quad \Rightarrow 1$ pt.
If $\mathrm{x}=9 \mathrm{y}=\frac{33}{4}=8.25 \Rightarrow$ no $\mathrm{pt}$.
Total $=31$ pts.
$(1,1)(1,2)-(1,14) \Rightarrow 14$ pts.
If $x=2, y=\frac{27}{2}=13.5$
$(2,2)(2,4) \ldots(2,12) \quad \Rightarrow 6$ pts.
If $\mathrm{x}=3, \mathrm{y}=\frac{51}{4}=12.75$
$(3,3)(3,6)-(3,12) \Rightarrow 4$ pts.
If $x=4, y=12$
$(4,4)(4,8) \quad \Rightarrow 2$ pts.
If $x=5 . y=\frac{45}{4}=11.25$
$(5,5),(5,10) \Rightarrow 2$ pts.
If $\mathrm{x}=6, \mathrm{y}=\frac{21}{2}=10.5$
$(6,6) \quad \Rightarrow 1 \mathrm{pt}$.
If $x=7, y=\frac{39}{4}=9.75$
$(7,7) \Rightarrow 1 \mathrm{pt}$.
If $x=8, y=9$
$(8,8) \quad \Rightarrow 1$ pt.
If $\mathrm{x}=9 \mathrm{y}=\frac{33}{4}=8.25 \Rightarrow$ no $\mathrm{pt}$.
Total $=31$ pts.
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