MCQ
If $a,\;b,\;c$ are in $A.P.$ and $a,\;b,\;d$ in $G.P.$, then $a,\;a - b,\;d - c$ will be in
- A$A.P.$
- ✓$G.P.$
- C$H.P.$
- DNone of these
$ \Rightarrow b = \frac{{a + c}}{2}$…..$(i)$
and ${b^2} = ad$….. $(ii)$
Hence $a,\;a - b,\;d - c$ are in $G.P. $ because
${(a - b)^2} = {a^2} - 2ab + {b^2} = a(a - 2b) + ad$
$ = a(a - a - c) + ad = ad - ac$.
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$(A)$ There exist $r , s \in R$, where $r < s$, such that $f$ is one-one on the open interval $( r , s )$
$(B)$ There exists $x 0 \in(-4,0)$ such that $\left| f ^{\prime}\left( x _0\right)\right| \leq 1$
$(C)$ $\lim _{x \rightarrow \infty} f(x)=1$
$(D)$ There exists a $\in(-4,4)$ such that $f(a)+f^{\prime \prime}(a)=0$ and $f^{\prime}(a) \neq 0$