SATHEE: Matrices and Determinants 2 Question 33

Matrices and Determinants 2 Question 33

37. Find the value of the determinant $| \begin{matrix} b c & c a & a b \\ p & q & r \\ 1 & 1 & 1 \end{matrix} |$ , where $a, b$ and $c$ are respectively the $p$ th, $q$ th and $r$ th terms of a harmonic progression.

(1997C, 2M)

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Answer:

Correct Answer: 37. (0)

Solution:

Since, $a, b, c$ are $p$ th, $q$ th and $r$ th terms of HP.

$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c} are in an AP.$

$\frac{1}{a} = A + (p - 1) D$

$\frac{1}{b} = A + (q - 1) D$

$\frac{1}{c} = A + (r - 1) D$ ….(i)

Let $Δ = | \begin{matrix} b c & c a & a b \\ p & q & r \\ 1 & 1 & 1 \end{matrix} | = a b c | \begin{matrix} \frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\ p & q & r \\ 1 & 1 & 1 \end{matrix} |$

$= a b c | \begin{matrix} A + (p - 1) D & A + (q - 1) D & A + (r - 1) D \\ p & q & r \\ 1 & 1 & 1 \end{matrix} |$

Applying $R_{1} \to R_{1} - (A - D) R_{3} - D R_{2}$

$= a b c | \begin{matrix} 0 & 0 & 0 \\ p & q & r \\ 1 & 1 & 1 \end{matrix} | \Rightarrow 0 \Rightarrow | \begin{matrix} b c & c a & a b \\ p & q & r \\ 1 & 1 & 1 \end{matrix} | = 0$

Matrices and Determinants 2 Question 33

37. Find the value of the determinant $| \begin{matrix} b c & c a & a b \\ p & q & r \\ 1 & 1 & 1 \end{matrix} |$ , where $a, b$ and $c$ are respectively the $p$ th, $q$ th and $r$ th terms of a harmonic progression.

Answer:

Solution:

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Matrices and Determinants 2 Question 33

37. Find the value of the determinant |bccaabpqr111|, where a,b and c are respectively the p th, q th and r th terms of a harmonic progression.

Answer:

Solution:

Engineering Preparation

Medical Preparation

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Important Resources

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Syllabus

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Sample Mock Test

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NCERT Exercise Video Solution

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37. Find the value of the determinant $| \begin{matrix} b c & c a & a b \\ p & q & r \\ 1 & 1 & 1 \end{matrix} |$ , where $a, b$ and $c$ are respectively the $p$ th, $q$ th and $r$ th terms of a harmonic progression.