Let the numbers 2- b - c be in an A-P- and A - begin bmatrix 181 - 1112 -b -cll4 - b- 2 - c- 2lend bmatrix - If det A -2-16- then c lies in the interval -A- -4-6-B- -3-2-23 - 4-C- -2-3-D- -2-23 - 4- 4-

The correct option is A [4,6] nbsp;|A|= 1 amp;1 amp;1 2 amp;b amp;c 4 amp;b^2 amp;c^2 C_2 → C_2 C_1, C_3 → C_3 C_1 nbsp; |A|= 1 ...

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Byju's Answer

Standard XII

Mathematics

Arithmetic Progression

Let the numbe...

Question

Let the numbers $2, b, c$ be in an A.P. and $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 2 & b & c 4 & b^{2} & c^{2} \end{matrix} ⎤ ⎥ ⎦$ . If $det (A) \in [2, 16]$ , then $c$ lies in the interval :

[3, 2 + 2^{3 / 4}]

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[4, 6]

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[2, 3]

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[2 + 2^{3 / 4}, 4]

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Solution

The correct option is B $[4, 6]$
$| A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 2 & b & c 4 & b^{2} & c^{2} \end{matrix} ∣ ∣ ∣ ∣$

$C_{2} \to C_{2} - C_{1}, C_{3} \to C_{3} - C_{1}$
$| A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 2 & b - 2 & c - 2 4 & b^{2} - 4 & c^{2} - 4 \end{matrix} ∣ ∣ ∣ ∣$

$\Rightarrow | A | = (b - 2) (c - b) (c - 2)$

$2, b, c$ are in A.P.
Let the common difference be $k$ .
Then $| A | = k \cdot k \cdot 2 k = 2 k^{3}$
$\Rightarrow 2 \leq 2 k^{3} \leq 16$
$\Rightarrow 1 \leq k \leq 2$
$\Rightarrow 1 \leq \frac{c - 2}{2} \leq 2$
$\Rightarrow 4 \leq c \leq 6$

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Let the numbers 2,b,c be in an A.P. and A=⎡⎢⎣1112bc4b2c2⎤⎥⎦. If det(A)∈[2,16], then c lies in the interval :

Let the numbers $2, b, c$ be in an A.P. and $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 2 & b & c 4 & b^{2} & c^{2} \end{matrix} ⎤ ⎥ ⎦$ . If $det (A) \in [2, 16]$ , then $c$ lies in the interval :