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Properties of Determinants

Let D1 = ...

Question

Let $D_{1}$ = $∣ ∣ ∣ ∣ \begin{matrix} a & b & a + b c & d & c + d a & b & a - b \end{matrix} ∣ ∣ ∣ ∣$ and $D_{2}$ = $∣ ∣ ∣ ∣ \begin{matrix} a & c & a + c b & d & b + d a & c & a + b + c \end{matrix} ∣ ∣ ∣ ∣$ then the value of $\frac{D_{1}}{D_{2}}$ where $b \neq 0$ and $a d \neq b c,$ is

A

- 2

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B

0

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C

- 2 b

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D

2 b

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Solution

The correct option is B $- 2$
$D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a & b & a + b c & d & c + d a & b & a - b \end{matrix} ∣ ∣ ∣ ∣ C_{2} \to C_{1} + C_{2} C_{3} \to C_{2} - C_{3} + C_{1} D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a & a + b & 0 c & c + d & 0 a & a + b & 2 b \end{matrix} ∣ ∣ ∣ ∣ = 2 b (a c + a d - a c - c b) = 2 b (a d - b c) D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} a & c & a + c b & d & b + d a & c & a + b + c \end{matrix} ∣ ∣ ∣ ∣ C_{2} \to C_{1} + C_{2} C_{3} \to C_{1} + C_{2} - C_{3} D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} a & a + c & 0 b & b + d & 0 a & a + c & - b \end{matrix} ∣ ∣ ∣ ∣ = - b (a b + a d - a b - a c) = - b (a d - b c) ∴ \frac{D_{1}}{D_{2}} = \frac{2 b (a d - b c)}{- b (a d - b c)} = - 2$

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Similar questions

D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a & b & a + b c & d & c + d a & b & a - b \end{matrix} ∣ ∣ ∣ ∣, D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} a & c & a + c b & d & b + d a & c & a + b + c \end{matrix} ∣ ∣ ∣ ∣

, then the value of

∣ ∣ ∣ \frac{D_{1}}{D_{2}} ∣ ∣ ∣

b \neq 0

a d \neq b c

, is ............

D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} a & b & a + b c & d & c + d a & b & a - b \end{matrix} ∣ ∣ ∣ ∣

D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} a & c & a + c b & d & b + d a & c & a + b + c \end{matrix} ∣ ∣ ∣ ∣

then the value of

\frac{D_{1}}{D_{2}}

b \neq 0

a d \neq b c

Q. Equal masses of two liquids having densities

d_{1}

d_{2}

are mixed together. The density of the resulting mixture is

D_{1} = ∣ ∣ ∣ ∣ \begin{matrix} x & a & b - 1 & 0 & x x & 2 & 1 \end{matrix} ∣ ∣ ∣ ∣

D_{2} = ∣ ∣ ∣ ∣ \begin{matrix} c x^{2} & 2 a & - b x & 2 & 1 - 1 & 0 & x \end{matrix} ∣ ∣ ∣ ∣ .

If all the roots of

(x^{2} - 4 x - 7) (x^{2} - 2 x - 3) = 0

satisfies the equation

D_{1} + D_{2} = 0,

then the value of

a + 4 b + c

A line meets the co-ordinate axes in A & B, a circle is circumscribed about the triangle OAB. If $d_{1} a n d d_{2}$ are the distances of the tangent to the circle at the origin O from the points A and B respectively the diameter of the circle is:

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