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JEE Main 2-April-2025 Paper - Shift 1 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 2-April-2025 Paper - Shift 14 Marks
Question
Three distinct numbers are selected randomly from the set $\{1,2,3, \ldots \ldots, 40\}$. If the probability, that the selected numbers are in an increasing G.P. is $\frac{ m }{ n }$, $\operatorname{gcd}( m , n )=1$, then $m + n$ is equal to __________.
✓

Answer

(4949)
Explanation: $1 \leq a < ar < ar ^2 \leq 40$
$($ If $r \in N)$
If r=2
$1 \leq a<2 a<4 a \leq 40$
$a \in\{1, \ldots \ldots, 10\} \quad(10 GP )$
If r=3
$1 \leq a<3 a<9 a \leq 40$
$a \in\{1,2,3,4\} \quad(4 GP )$
If r=4
$1 \leq a<4 a<16 a \leq 40$
$a \in\{1,2\} \quad(2 GP )$
If r=5
$1 \leq a<5 a<25 a \leq 40$
$a \in\{1\} \quad(1 GP )$
If r=6
$1 \leq a<6 a<36 a \leq 40$
$a \in\{1\} \quad(1 GP)$
$\left( P =\frac{18}{9880}=\frac{9}{4940}\right)$ as per NTA for $r \in N$
$m + n =4949$
If $r \notin N$ (also possible)
$r=\frac{3}{2}$
$ar ^2=\frac{9 a }{4} ; a =4 k$
$\left.\begin{array}{l}(4,6,9) \\ (8,12,18) \\ (12,18,27) \\ (16,24,36)\end{array}\right\} 4 GP$
$r =\frac{5}{2} \quad ar ^2=\frac{25 a }{4} ; a =4 k$
$(4,10,25)$........(1) GP
$r =\frac{4}{3} \quad ar ^2=\frac{16 a }{9} \rightarrow a =9 k$
$(9,12,16),(18,24,32) \ldots \ldots . .(2)$ GP
$r =\frac{5}{3} \quad ar ^2=\frac{25 a }{9} ; a =9 k$
$(9,15,25) \ldots . . . . .(1)$ GP
$r =\frac{5}{4} \quad ar ^2=\frac{25 a }{16} ; a =16 k$
$(16,20,25) \ldots . . . . . . . .(1) GP$
$r =\frac{6}{5} \quad ar ^2=\frac{36 a }{25} ; a =25 k$
$(25,30,36)$.............(1) GP
Total $=18+10=28$
$P =\frac{28}{{ }^{40} C _3}=\frac{28}{9880}=\frac{7}{2470}$
$m + n =2477$

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