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Matrices and Determinants question

2022 · Shift 1 · Q35
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  5. /2022 · Shift 1 · Q35

Matrices and Determinants question

2022 · Shift 1 · Q35

JEE AdvancedMathematicsMatrices and DeterminantsMCQ+3 / −1

Let p,q,rp, q, rp,q,r be nonzero real numbers that are, respectively, the 10th ,100th 10^{\text {th }}, 100^{\text {th }}10th ,100th  and 1000th 1000^{\text {th }}1000th  terms of a harmonic progression. Consider the system of linear equations

x+y+z=110x+100y+1000z=0qrx+pry+pqz=0\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}x+y+z=110x+100y+1000z=0qrx+pry+pqz=0​

List-I List-II
(I) If qr=10\frac{q}{r}=10rq​=10, then the system of linear equations has (P) x=0,y=109,z=−19x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}x=0,y=910​,z=−91​ as a solution
(II) If preq100\frac{p}{r} eq 100rp​eq100, then the system of linear equations has (Q) x=109,y=−19,z=0x=\frac{10}{9}, y=-\frac{1}{9}, z=0x=910​,y=−91​,z=0 as a solution
(III) If pqeq10\frac{p}{q} eq 10qp​eq10, then the system of linear equations has (R) infinitely many solutions
(IV) If pq=10\frac{p}{q}=10qp​=10, then the system of linear equations has (S) no solution
(T) at least one solution

The correct option is:

  1. A
    (I) →\rightarrow→(T); (II) →\rightarrow→(R); (III) →\rightarrow→(S); (IV) →\rightarrow→ (T)
  2. B
    (I) →\rightarrow→(Q); (II) →\rightarrow→(S); (III) →\rightarrow→(S); (IV) →\rightarrow→ (R)
  3. C
    (I) →(Q)\rightarrow(\mathrm{Q})→(Q); (II) →\rightarrow→(R); (III) →(P)\rightarrow(\mathrm{P})→(P); (IV) →\rightarrow→ (R)
  4. D
    (I) →\rightarrow→(T); (II) →\rightarrow→(S); (III) →\rightarrow→(P); (IV) →\rightarrow→ (T)
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