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

English

Gujarati

Hindi

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

Share

Similar Questions

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations

$x+y+z=5$ ; $x+2 y+3 z=\mu$ ; $x+3 y+\lambda z=1$

is constructed. If $\mathrm{p}$ is the probability that the system has a unique solution and $\mathrm{q}$ is the probability that the system has no solution, then :

hard

(JEE MAIN-2021)

Let $\lambda $ be a real number for which the system of linear equations $x + y + z = 6$
; $4x + \lambda y – \lambda z = \lambda – 2$ ; $3x + 2y -4z = -5$ Has indefinitely many solutions. Then $\lambda $ is a root of the quadratic equation

hard

(JEE MAIN-2019)

The value of the determinant$\left| {\,\begin{array}{*{20}{c}}{ – 1}&1&1\\1&{ – 1}&1\\1&1&{ – 1}\end{array}\,} \right|$is equal to

normal

Find area of the triangle with vertices at the point given in each of the following: $(2,7),(1,1),(10,8)$

medium

The number of distinct real roots of $\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

hard

(JEE MAIN-2021)