Let S be the set of all integer solutions,(x, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The given system of equations is:$x - 2y + 5z = 0$  $(1)$$-2x + 4y + z = 0$  $(2)$$-7x + 14y + 9z = 0$  $(3)$First,we calculate the determinant of

Vedclass · Accepted Answer

$8$

Vedclass · Answer

(D) The given system of equations is:$x - 2y + 5z = 0$  $(1)$$-2x + 4y + z = 0$  $(2)$$-7x + 14y + 9z = 0$  $(3)$First,we calculate the determinant of the coefficient matrix:$\Delta = \begin{vmatrix} 1 & -2 & 5 \ -2 & 4 & 1 \ -7 & 14 & 9 \end{vmatrix} = 1(36 - 14) - (-2)(-18 + 7) + 5(-28 + 28) = 1(22) + 2(-11) + 0 = 22 - 22 = 0$.Since $\Delta = 0$,the system has infinitely many solutions.From $(1)$ and $(2)$,we have:$x - 2y = -5z$$-2x + 4y = -z$Multiplying the first equation by $2$,we get $2x - 4y = -10z$. Adding this to the second equation gives $0 = -11z$,so $z = 0$.Substituting $z = 0$ into $(1)$,we get $x - 2y = 0$,which implies $x = 2y$.Let $y = k$,where $k$ is an integer. Then $x = 2k$ and $z = 0$.The condition $15 \leq x^{2} + y^{2} + z^{2} \leq 150$ becomes:$15 \leq (2k)^{2} + k^{2} + 0^{2} \leq 150$$15 \leq 5k^{2} \leq 150$$3 \leq k^{2} \leq 30$.Since $k$ is an integer,$k^{2}$ can be $4, 9, 16, 25$.Thus,$k \in \{ \pm 2, \pm 3, \pm 4, \pm 5 \}$.There are $8$ possible values for $k$,each corresponding to a unique solution $(x, y, z)$.Therefore,the number of elements in $S$ is $8$.

Let $S$ be the set of all integer solutions,$(x, y, z)$,of the system of equations
$x-2y+5z=0$
$-2x+4y+z=0$
$-7x+14y+9z=0$
such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then,the number of elements in the set $S$ is equal to

Similar Questions

Solve the system of linear equations using the matrix method: $x-y+z=4$,$2x+y-3z=0$,and $x+y+z=2$.

If the system of linear equations $x + ky + 3z = 0$,$3x + ky - 2z = 0$,and $2x + 4y - 3z = 0$ has a non-zero solution $(x, y, z)$,then $\frac{xz}{y^2} = \dots$

If the system of linear equations $x + y + z = 5$,$x + 2y + 2z = 6$,and $x + 3y + \lambda z = \mu$ (where $\lambda, \mu \in \mathbb{R}$) has infinitely many solutions,then the value of $\lambda + \mu$ is:

If the system of equations $2x + 9y + 5z = 8$,$2x + 3y - z = -4$,$x - 2z = -5$ has an infinite number of solutions $x = -5 + at$,$y = 2 + bt$,$z = ct$,$t \in R$,then $a$,$b$,$c$ respectively are

If the system of equations $(\lambda-1) x+(\lambda-4) y+\lambda z=5$,$\lambda x+(\lambda-1) y+(\lambda-4) z=7$,and $(\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9$ has infinitely many solutions,then $\lambda^2+\lambda$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $S$ be the set of all integer solutions,$(x, y, z)$,of the system of equations$x-2y+5z=0$$-2x+4y+z=0$$-7x+14y+9z=0$such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then,the number of elements in the set $S$ is equal to