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3 and 4 .Determinants and Matrices
hard
અહી $p$ અને $p+2$ એ અવિભાજ્ય સંખ્યા છે અને $\Delta=\left|\begin{array}{ccc}p ! & (p+1) ! & (p+2) ! \\ (p+1) ! & (p+2) ! & (p+3) ! \\ (p+2) ! & (p+3) ! & (p+4) !\end{array}\right|$ હોય તો $\alpha$ અને $\beta$ ની મહતમ કિમંતોનો સરવાળો મેળવો કે જેથી $p ^{\alpha}$ અને $( p +2)^{\beta}$ એ $\Delta$ ને વિભાજે .
A
$4$
B
$3$
C
$2$
D
$1$
(JEE MAIN-2022)
Solution
$\Delta=\left|\begin{array}{ccc} P ! & ( P +1) ! & ( P +2) ! \\( P +1) ! & ( P +2) ! & ( P +3) ! \\( P +2) ! & ( P+3) ! & ( P +4) !\end{array}\right|$
$\Delta= P !( P +1) !( P +2) !\left|\begin{array}{ccc}1 & 1 & 1 \\P +1 & P +2 & P +3 \\( P +2)( P +1) & ( P +3)( P +2) & ( P +4)( P +3)\end{array}\right|$
$\Delta=2 P !( P +1) !( P +2) !$
Which is divisible by $P ^{\alpha}\,and\,( P +2)^{\beta}$
$\therefore \alpha=3, \beta=1$
Standard 12
Mathematics
