If | , * 20 c a&b& a - b &c& b - c 2&1&0 , | = 0 and 1 2 , then - Vidyadip

MCQ

If $\left| {\,\begin{array}{*{20}{c}}a&b&{a\alpha - b}\\b&c&{b\alpha - c}\\2&1&0\end{array}\,} \right| = 0$ and $\alpha \ne \frac{1}{2},$ then

A
$a,b,c$ are in $A. P.$
✓
$a,b,c$ are in $G. P.$
C
$a,b,c$ are in $H. P.$
D
None of these

✓

Answer

Correct option: B.

$a,b,c$ are in $G. P.$

b
(b) $\left| {\,\begin{array}{*{20}{c}}a&b&{a\alpha - b}\\b&c&{b\alpha - c}\\2&1&0\end{array}\,} \right| = 0$

==> $a[ - (b\alpha - c)] - b[ - 2(b\alpha - c)] + [a\alpha - b)(b - 2c)] = 0$

==>$ - ab\alpha + ac + 2{b^2}\alpha - 2bc + ab\alpha - 2ac\alpha - {b^2} + 2bc = 0$

==> $ac + 2{b^2}\alpha - 2ac\alpha - {b^2} = 0$

==> $(ac - {b^2}) - 2\alpha (ac - {b^2}) = 0$

==> $ac - {b^2} = 0$or $1 - 2\alpha = 0$ $ \Rightarrow $ ${b^2} = ac$ or $\alpha = \frac{1}{2}$

(As given in question)

So, ${b^2} = ac$ i.e, $a,b,c$ are in $G.P.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 3 and 4 . determinant and metrices→STD 12 - 3 and 4 . determinant and metrices — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The approximate value of $(33)^{\frac{1}{5}}$ is:

The range of values of the function $f\left( x \right) = \frac{1}{{2 - 3\sin x}}$ is

$\int_{}^{} {\frac{{dx}}{{(1 + {x^2})\sqrt {1 - {x^2}} }} = } $

The value of $\int\limits_0^1 {\left( {\prod\limits_{r = 1}^n {(x + r)} } \right)\,\,\left( {\sum\limits_{k = 1}^n {\frac{1}{{x + k}}} } \right)\,dx} $ equals

If $a, b$ be two fixed positive integers such that $f(a + x) = b + {[{b^3} + 1 - 3{b^2}f(x) + 3b{\{ f(x)\} ^2} - {\{ f(x)\} ^3}]^{\frac{1}{3}}}$ for all real $x$, then $f(x)$ is a periodic function with period

In an LPP, if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then the number of points of which Zmax occurs is:

Which of the following equation is a linear differential equation of order $3\ ?\ [$Note: The original question asks for linear equation, but it should be linear differential equation$]:$

If the direction cosine of a directed line be $a, 3a, 7a$ then $a =$

If $2f(x) - 3f\left( {\frac{1}{x}} \right) = x$, then $\int_1^2 {f(x)} \;dx$ is equal to

Let $x$ be the length of one of the equal sides of an isosceles triangle, and let $\theta$ be the angle between them. If $x$ is increasing at the rate $(1/12) \,\,m/hr$, and $\theta$ is increasing at the rate of $\pi /180 \,\,radians/hr$ then the rate in $m^2\,/hr$ at which the area of the triangle is increasing when $x = 12\, m$ and $\theta = \pi /4$