MCQ
If $\alpha ,\;\beta ,\;\gamma $ are the geometric means between $ca,\;ab;\;ab,\;bc;\;bc,\;ca$ respectively where $a,\;b,\;c$ are in A.P., then ${\alpha ^2},\;{\beta ^2},\;{\gamma ^2}$ are in
- ✓$A.P.$
- B$H.P.$
- C$G.P.$
- DNone of the above
✓
Answer
Correct option: A.
$A.P.$
a
(a) By hypothesis, ${\alpha ^2} = {a^2}bc,\;{\beta ^2} = {b^2}ca,\;{\gamma ^2} = {c^2}ab$ and $2b = a + c$.
(a) By hypothesis, ${\alpha ^2} = {a^2}bc,\;{\beta ^2} = {b^2}ca,\;{\gamma ^2} = {c^2}ab$ and $2b = a + c$.
.Hence ${2^{n - 1}} > 100$ 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.
Explore more
Similar questions
If $f(x) = x^3-x^2+100\,x \, +1001\,;$ then
→Let â be a unit vector perpendicular to the vectors $\overrightarrow{ b }=\hat{ i }-2 \hat{ j }+3 \hat{ k }$ and $\overrightarrow{ c }=2 \hat{ i }+3 \hat{ j }-\hat{ k }$, and makes an angle of $\cos ^{-1}\left(-\frac{1}{3}\right)$ with the vector $\hat{ i }+\hat{ j }+\hat{ k }$. If $\hat{ a }$ makes an angle of $\frac{\pi}{3}$ with the vector $\hat{ i }+\alpha \hat{ j }+\hat{ k }$, then the value of $\alpha$ is :
→There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i$, $i =1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats.
→($1$) The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and $NONE$ of the remaining students gets the seat previously allotted to him/her is
$(A)$ $\frac{3}{40}$ $(B)$ $\frac{1}{8}$ $(C)$ $\frac{7}{40}$ $(D)$ $\frac{1}{5}$
($2$) For $i =1,2,3,4$, let $T _{ i }$ denote the event that the students $S _{ i }$ and $S _{ i +1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _1 \cap T _2 \cap T _3 \cap T _4$ is
$(A)$ $\frac{1}{15}$ $(B)$ $\frac{1}{10}$ $(C)$ $\frac{7}{60}$ $(D)$ $\frac{1}{5}$
Give the answer or quetion ($1$) and ($2$)
If a rectangle is inscribed in an equilateral triangle of side length $2 \sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is $....$
→The value of the integral $\int {\frac{{\sin \,\left( {\ln \,(2 + 2x)} \right)}}{{x + 1}}dx} $ is
→An ordinary cube has four blank faces, one face marked $2$ another marked $3$. Then the probability of obtaining a total of exactly $12$ in $5$ throws, is
→If $f(x)=\int_{0}^{x}(5+|1-t|) d t, \quad x>2$
→$\quad \quad \quad \quad \quad 5 x+1,\quad \quad \quad \quad \quad x \leq 2$, then
If $I = {\kern 1pt} \int\limits_0^{\frac{\pi }{6}} {\frac{{\cos x}}{x}dx,J = \int\limits_{\frac{\pi }{3}}^{\frac{\pi }{2}} {\frac{{\cos x}}{x}dx.} } $ Which of the following is CORRECT ?
→If both the roots of the quadratic equation ${x^2} - 2kx + {k^2} + k - 5 = 0$ are less than $5$, then $k$ lies in the interval
→Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is....................
→