JEE Advance - Mathematics (2022 - Paper 1 Online - No. 17)
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
$$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$$
| List-I | List-II |
|---|---|
| (I) If $$\frac{q}{r}=10$$, then the system of linear equations has | (P) $$x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$$ 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) $$\rightarrow(\mathrm{Q})$$; (II) $$\rightarrow$$ (R); (III) $$\rightarrow(\mathrm{P})$$; (IV) $$\rightarrow$$ (R)
(I) $$\rightarrow$$ (T); (II) $$\rightarrow$$ (S); (III) $$\rightarrow$$ (P); (IV) $$\rightarrow$$ (T)
Explanation
Given:
$$ \begin{gathered} x+y+z=1 ..........(1)\\\\ 10 x+100 y+1000 z=0 .........(2)\\\\ q r x+p r y+p q z=0 ............(3) \end{gathered} $$
Now equation (3) can be re-written as
$$ \frac{x}{p}=\frac{y}{q}=\frac{z}{r}=0 \quad \because\{p, q, r \neq 0\} $$
Now given $p, q$ and $r$ are $10^{\text {th }}, 100^{\text {th }}$ and $1000^{\text {th }}$ term of an. H.P.,
So let $p=\frac{y}{a+9 d}, q=\frac{1}{a+99 d}, r=\frac{1}{a+999 d}$
Now from equation (3)
$$ (a+9 d) x+(a+99 d) y+(a+999 d) z=0 $$
Now from and (1), (2) and (3) we get
$$ \begin{aligned} \Delta & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 10 & 100 & 1000 \\ a+9 d & a+99 d & a+999 d \end{array}\right|=0 \\\\ \Delta x & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 100 & 1000 \\ 0 & a+99 d & a+999 d \end{array}\right|=900(d-a) \\\\ \Delta y & =\left|\begin{array}{ccc} 10 & 0 & 1000 \\ 1+9 d & 0 & a+999 d \end{array}\right|=990(a-d) \\\\ \Delta z & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 10 & 100 & 0 \\ a+9 d & a+99 d & 0 \end{array}\right|=90(d-a) \end{aligned} $$
(I) If $\frac{q}{r}=10 \Rightarrow a=d$
$$ \Delta=\Delta_x=\Delta_y=\Delta_z=0 $$
And equation (1) and equation (2) represents non-parallel plane equation (2) and equation (3) represents same plane
$\Rightarrow$ Infinitely many solutions.
Now finding solution by taking $z=\lambda$ then
$$ \begin{aligned} & x+y =1-\lambda \\\\ & x+10 y =-100 \lambda \\\\ & x =\frac{10}{9}+100 \lambda \\\\ & y =\frac{-1}{9}-11 \lambda \\\\ & \Rightarrow (x, y, z) \in\left(\frac{10}{9}+10 \lambda, \frac{-1}{9}-11 \lambda, \lambda\right) \end{aligned} $$
So $\mathrm{P}$ is not valid for any value of $\lambda \rightarrow \mathrm{Q}$
(II) $\frac{p}{r} \neq 100 \Rightarrow a \neq d$
$$ \Delta=0 \,\text{and}\, \Delta_{x^{\prime}} \Delta_{y^{\prime}} \Delta_z \neq 0 $$
So no solution.
$$ \text { II } \rightarrow \mathrm{S} $$
(III) If $$ \frac{p}{q} \neq 10 \Rightarrow a \neq d \text { then } \Delta_z \neq 0 $$
So no solution.
$$ \text { III } \rightarrow \mathrm{S} $$
(IV) If $$ \Rightarrow a=d \text { then } \Delta_z=0 \Rightarrow \Delta_x=\Delta_y=0 $$
So infinitely many solutions.
$$ \mathrm{IV} \rightarrow \mathrm{R} $$
$$ \begin{gathered} x+y+z=1 ..........(1)\\\\ 10 x+100 y+1000 z=0 .........(2)\\\\ q r x+p r y+p q z=0 ............(3) \end{gathered} $$
Now equation (3) can be re-written as
$$ \frac{x}{p}=\frac{y}{q}=\frac{z}{r}=0 \quad \because\{p, q, r \neq 0\} $$
Now given $p, q$ and $r$ are $10^{\text {th }}, 100^{\text {th }}$ and $1000^{\text {th }}$ term of an. H.P.,
So let $p=\frac{y}{a+9 d}, q=\frac{1}{a+99 d}, r=\frac{1}{a+999 d}$
Now from equation (3)
$$ (a+9 d) x+(a+99 d) y+(a+999 d) z=0 $$
Now from and (1), (2) and (3) we get
$$ \begin{aligned} \Delta & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 10 & 100 & 1000 \\ a+9 d & a+99 d & a+999 d \end{array}\right|=0 \\\\ \Delta x & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 100 & 1000 \\ 0 & a+99 d & a+999 d \end{array}\right|=900(d-a) \\\\ \Delta y & =\left|\begin{array}{ccc} 10 & 0 & 1000 \\ 1+9 d & 0 & a+999 d \end{array}\right|=990(a-d) \\\\ \Delta z & =\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 10 & 100 & 0 \\ a+9 d & a+99 d & 0 \end{array}\right|=90(d-a) \end{aligned} $$
(I) If $\frac{q}{r}=10 \Rightarrow a=d$
$$ \Delta=\Delta_x=\Delta_y=\Delta_z=0 $$
And equation (1) and equation (2) represents non-parallel plane equation (2) and equation (3) represents same plane
$\Rightarrow$ Infinitely many solutions.
Now finding solution by taking $z=\lambda$ then
$$ \begin{aligned} & x+y =1-\lambda \\\\ & x+10 y =-100 \lambda \\\\ & x =\frac{10}{9}+100 \lambda \\\\ & y =\frac{-1}{9}-11 \lambda \\\\ & \Rightarrow (x, y, z) \in\left(\frac{10}{9}+10 \lambda, \frac{-1}{9}-11 \lambda, \lambda\right) \end{aligned} $$
So $\mathrm{P}$ is not valid for any value of $\lambda \rightarrow \mathrm{Q}$
(II) $\frac{p}{r} \neq 100 \Rightarrow a \neq d$
$$ \Delta=0 \,\text{and}\, \Delta_{x^{\prime}} \Delta_{y^{\prime}} \Delta_z \neq 0 $$
So no solution.
$$ \text { II } \rightarrow \mathrm{S} $$
(III) If $$ \frac{p}{q} \neq 10 \Rightarrow a \neq d \text { then } \Delta_z \neq 0 $$
So no solution.
$$ \text { III } \rightarrow \mathrm{S} $$
(IV) If $$ \Rightarrow a=d \text { then } \Delta_z=0 \Rightarrow \Delta_x=\Delta_y=0 $$
So infinitely many solutions.
$$ \mathrm{IV} \rightarrow \mathrm{R} $$
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