If a b ,b c ,c a are in H.P., then - Vidyadip

MCQ

If $\frac{a}{b},\frac{b}{c},\frac{c}{a}$ are in $H.P.$, then

✓
${a^2}b,\,{c^2}a,\,{b^2}c$ are in $A.P.$
B
${a^2}b,\,{b^2}c,\,{c^2}a$ are in $H.P.$
C
${a^2}b,\,{b^2}c,\,{c^2}a$ are in $G.P.$
D
None of these

✓

Answer

Correct option: A.

${a^2}b,\,{c^2}a,\,{b^2}c$ are in $A.P.$

a
(a) $\frac{b}{a},\frac{c}{b},\frac{a}{c}$ are in $A.P.$

==> $\frac{{2c}}{b} = \frac{b}{a} + \frac{a}{c}$

$ \Rightarrow \frac{{2c}}{b} = \frac{{bc + {a^2}}}{{ac}}$

==> $2a{c^2} = {b^2}c + b{a^2}$

$\therefore \,{a^2}b,\,{c^2}a$ and ${b^2}c$ are in $A.P.$

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 "MCQ" from 8. sequence and series→8. sequence and series — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The value of $\cos 15^\circ - \sin 15^\circ $ is equal to

If $\sec\text{x}+\tan\text{x}=\text{k},\cos\text{x}=$

The line passing through the points $(3, -4)$ and $(-2, 6)$ and a line passing through $(-3,6)$ and $(9, -18)$ are

Let $f(x)=x^4+a x^3+b x^2+c$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $i \sqrt{3}$ is a root of the equation $4 x^3+3 a x^2+2 b x=0$, where $i=\sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x)=0$, then $\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2$ is equal to. . . . . .

The point $(4, 1)$ undergoes the following two successive transformation

$(i)$ Reflection about the line $y = x$

$(ii)$ Translation through a distance $2$ units along the positive $x$-axis

Then the final coordinates of the point are

If $\cot(\alpha+\beta)=0,$ then $\sin(\alpha+2\beta)$ is equal to:

If $\text{x}\sin45^\circ\cos^260^\circ=\frac{\tan^260^\circ\text{cosec }30^\circ}{\sec45^\circ\cot^230^\circ},$ then x =

Lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.

$STATEMENT -1$ : The ratio $PR$: $RQ$ equals $2 \sqrt{2}: \sqrt{5}$. because

$STATEMENT -2$ : In any triangle, bisector of an angle divides the triangle into two similar triangles.

The equation of the chord of contact, if the tangents are drawn from the point $(5, -3)$ to the circle ${x^2} + {y^2} = 10$, is

The equation of the hyperbola in the standard form (with transverse axis along the $x$ - axis) having the length of the latus rectum = $9$ units and eccentricity = $5/4$ is