- Home
- Standard 12
- Mathematics
Let $p, q, r$ be nonzero real numbers that are, respectively, the $10^{\text {th }}, 100^{\text {th }}$ and $1000^{\text {th }}$ terms of a harmonic progression. Consider the system of linear equations
$x+y+z=1$
$10 x+100 y+1000 z=0$
$q r x+p r y+p q z=0$.
| $List-I$ | $List-II$ |
| ($I$) If $\frac{q}{r}=10$, then the system of linear equations has | ($P$) $x=0, y=\frac{10}{9}, z=-\frac{1}{9}$ as a solution |
| ($II$) If $\frac{ p }{ r } \neq 100$, then the system of linear equations has | ($Q$) $x =\frac{10}{9}, y =-\frac{1}{9}, z =0$ as a solution |
| ($III$) If $\frac{p}{q} \neq 10$, then the system of linear equations has | ($R$) infinitely many solutions |
| ($IV$) If $\frac{p}{q}=10$, then the system of linear equations has | ($S$) no solution |
| ($T$) at least one solution |
The correct option is:
$(I) \rightarrow (T); (II) \rightarrow (R); (III) \rightarrow (S); (IV) \rightarrow (T)$
$(I) \rightarrow (Q); (II) \rightarrow (S); (III) \rightarrow (S); (IV) \rightarrow (R)$
$(I) \rightarrow (Q); (II) \rightarrow (R); (III) \rightarrow (P); (IV) \rightarrow (R)$
$(I) \rightarrow (T); (II) \rightarrow (S); (III) \rightarrow (P); (IV) \rightarrow (T)$
Solution
If $\frac{ q }{ r }=10 \Rightarrow A = D \Rightarrow D _{ x }= D _{ y }= D _{ z }=0$
So, there are infinitely many solutions
Look of infinitely many solutions can be given as
$x+y+z=1$
$10 x+100 y+1000 z=0 \Rightarrow x+10 y+100 z=0$
Let $z=\lambda$
then $x+y=1-\lambda$
and $x+10 y=-100 \lambda$
$\Rightarrow x=\frac{10}{9}+10 \lambda ; y=\frac{-1}{9}-11 \lambda$
i.e., $(x, y, z)=\left(\frac{10}{9}+10 \lambda, \frac{-1}{9}-11 \lambda, \lambda\right)$
$Q\left(\frac{10}{9}, \frac{-1}{9}, 0\right)$ valid for $\lambda=0$
$P\left(0, \frac{10}{9}, \frac{-1}{9}\right)$ not valid for any $\lambda$.
$(I)$ $\rightarrow$ $Q,R,T$
$(II)$ If $\frac{p}{I} \neq 100$, then $D_y \neq 0$
So no solution
$(II)$ $\rightarrow$ $(S)$
$(III)$ If $\frac{ P }{ q } \neq 10$, then $D _z \neq 0$ so, no solution
$(III)$ $\rightarrow$ $(S)$
$(IV)$ If $\frac{ p }{ q }=10 \Rightarrow D _z=0 \Rightarrow D _{ x }= D _y=0$
so infinitely many solution
$(IV)$ $\rightarrow$ $Q.R.T$
