Number of integers x satisfying -3 x^4+operatorname{det}left[begin{array}{ccc}1 & x & x^2 1

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3 and 4 .Determinants and Matrices

hard

The number of integers $x$ satisfying $-3 x^4+\operatorname{det}\left[\begin{array}{ccc}1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6\end{array}\right]=0$ is equal to

A

$1$

B

$2$

C

$5$

D

$8$

(KVPY-2019)

Solution

(b)

Given, $-3 x^4+\operatorname{det}\left[\begin{array}{ccc}1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6\end{array}\right]=0$

$x^8+x^5+x^5-x^4-x^7-x^7=3 x^4$

$x^8-2 x^7+2 x^5-4 x^4 =0$

$x^4\left[x^4-2 x^3+2 x-4\right] =0$

$x^4\left[x^3(x-2)+2(x-2)\right]=0$

$x^4\left(x^3+2\right)(x-2)=0$

$\because x$ is an integer, so $x=0,2$.

Standard 12

Mathematics

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