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Mathematics Question on integral

The only integral root of the equation ∣2−y23 25−y6 3410−y∣=0\begin{vmatrix}2-y&2&3\\\ 2&5-y&6\\\ 3&4&10-y\end{vmatrix} =0 is

A

y=3y = 3

B

y=2y = 2

C

y=1y = 1

D

None of theseNone\, of\, these

Answer

y=1y = 1

Explanation

Solution

We have, ∣2−y23 25−y6 3410−y∣=0\begin{vmatrix}2-y&2&3\\\ 2&5-y&6\\\ 3&4&10-y\end{vmatrix} = 0 ⇒(2−r)[(5−y)(10−y)−24]−2[2(10−y)−18]\Rightarrow\left(2 - r\right)\left[\left(5-y\right)\left(10-y\right)-24\right]-2\left[2\left(10-y\right)-18\right]
+3[8−3(5−y)]=0+3\left[8 - 3\left(5-y\right)\right] = 0
⇒(2−y)[50−15y+y2−24]−2[20−2y−18]\Rightarrow\left(2 -y\right)\left[50 - 15y + y^{2} -24\right]-2\left[20 - 2y -18\right]
+3[8−15+3y]=0+ 3\left[8-15 + 3y\right] = 0
⇒(2−y)[y2−15y+26]\Rightarrow\left(2 -y\right)\left[y^{2} -15y +26\right]
−2[2−2y]+3[3y−7]=0-2\left[2-2y\right]+ 3\left[3y-7\right] = 0
⇒2y2−30y+52−y3+15y2−26y−4+4y\Rightarrow 2y^{2} -30 y + 52 -y^{3} + 15 y^{2} - 26 y - 4 + 4y
+9y−21=0+ 9 y - 21 = 0
⇒−y3+17y2−43y+27=0\Rightarrow -y^{3} + 17 y^{2} - 43 y + 27 = 0
⇒y3−17y2+43y−27=0\Rightarrow y^{3} - 17y^{2} + 43y -27 = 0
⇒(y−1)(y2−16y+27)=0\Rightarrow\left(y-1\right)\left(y^{2}-16y+27\right) = 0
⇒y−1=0\Rightarrow y -1 = 0 or y2−16y+27=0y^{2 } - 16 y + 27 = 0
⇒y=1\Rightarrow y = 1