JEE Advanced 2022 · Paper 1 · Mathematics Q17

← JEE Advanced 2022 · Paper 1

MathematicsList match+3 / -1Question 17 of 54

Let $p, q, r$ be nonzero real numbers that are, respectively, the $10^{th}$ , $100^{th}$ and $1000^{th}$ terms of a harmonic progression. Consider the system of linear equations
$\begin{aligned} x + y + z &= 1\\ 10x + 100y + 1000z &= 0\\ qr\,x + pr\,y + pq\,z &= 0\end{aligned}$

The correct option is:

List-I

I
If $\dfrac qr = 10$ , then the system of linear equations has
II
If $\dfrac pr \neq 100$ , then the system of linear equations has
III
If $\dfrac pq \neq 10$ , then the system of linear equations has
IV
If $\dfrac pq = 10$ , then the system of linear equations has

List-II

P
$x = 0$ , $y = \frac{10}9$ , $z = -\frac19$ as a solution
Q
$x = \frac{10}9$ , $y = -\frac19$ , $z = 0$ as a solution
R
infinitely many solutions
S
no solution
T
at least one solution

(I) → (T); (II) → (R); (III) → (S); (IV) → (T)

(I) → (Q); (II) → (S); (III) → (S); (IV) → (R)

(I) → (Q); (II) → (R); (III) → (P); (IV) → (R)

(I) → (T); (II) → (S); (III) → (P); (IV) → (T)

Answer and worked solution

Official answer: B

HP to AP. The reciprocals of an HP form an AP. Write
$\frac1p=a+9d,\quad\frac1q=a+99d,\quad\frac1r=a+999d.$

Third equation. Divide it by $pqr$ :
$\frac xp+\frac yq+\frac zr=a(x+y+z)+d(9x+99y+999z)=0.$
The first two equations give $x+y+z=1$ and $9x+99y+999z=-(x+y+z)=-1$ . So the third equation becomes $a-d=0$ .

Consequence. The system is consistent (with infinitely many solutions) iff $a=d$ . That is equivalent to each of:
$\frac pq=10,\qquad\frac qr=10,\qquad\frac pr=100.$

Solutions of the first two equations. Both $(0,\frac{10}9,-\frac19)$ and $(\frac{10}9,-\frac19,0)$ satisfy them.

Matching.

(I) $\frac qr=10$ : consistent, and Q is a solution → Q.
(II) $\frac pr\ne100$ : no solution → S.
(III) $\frac pq\ne10$ : no solution → S.
(IV) $\frac pq=10$ : infinitely many solutions → R.

Topics:Systems of linear equations Harmonic progression

← Previous question Next question →

JEE Advanced 2022 · Paper 1, Mathematics question 17