Download App

Geometric Progressions — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSGeometric Progressions3 Marks
Question
If a, b, c, d and p are different real numbers such that:
$(\text{a}^2+\text{b}^2+\text{c}^2)\text{p}^2-2(\text{ab}+\text{bc}+\text{cd})\text{p}+(\text{b}^2+\text{c}^2+\text{a}^2)\le0,$ then show that a, b, c and d are in G.P.
✓

Answer

$(\text{a}^2+\text{b}^2+\text{c}^2)\text{p}^2-2(\text{ab}+\text{bc}+\text{cd})\text{p}+(\text{b}^2+\text{c}^2+\text{a}^2)\le0,$
$\Rightarrow(\text{a}^2\text{p}^2+\text{b}^2\text{p}^2+\text{c}^2\text{p}^2)-2(\text{a}\text{b}\text{p}+\text{b}\text{c}\text{p}+\text{c}\text{d}\text{p})+(\text{b}^2+\text{c}^2+\text{d}^2)\le0$
$\Rightarrow(\text{a}^2\text{p}^2-2\text{a}\text{b}\text{p}+\text{b}^2)+(\text{b}^2\text{p}^2-2\text{b}\text{c}\text{p}+\text{c}^2)+(\text{c}^2\text{p}^2-2\text{c}\text{d}\text{p}+\text{d}^2)\le0$
$\Rightarrow(\text{a}\text{p}-\text{b})^2+(\text{b}\text{p}-\text{c})^2+(\text{c}\text{p}-\text{d})^2\le0$
$\Rightarrow(\text{a}\text{p}-\text{b})^2+(\text{b}\text{p}-\text{c})^2+(\text{c}\text{p}-\text{d})^2=0$
$\Rightarrow(\text{a}\text{p}-\text{b})^2=0$
$\Rightarrow\text{p}=\frac{\text{b}}{\text{a}}$
Also, $(\text{bp}-\text{c})^2=0$
$\Rightarrow\text{p}=\frac{\text{c}}{\text{b}}$
Similiarly, $\Rightarrow(\text{cp}-\text{d})^2=0$
$\Rightarrow\text{p}=\frac{\text{d}}{\text{c}}$
$\therefore\frac{\text{b}}{\text{a}}=\frac{\text{c}}{\text{b}}=\frac{\text{d}}{\text{c}}$
Thus, a, b, c and d 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 "3 Marks Question" from Geometric Progressions→Geometric Progressions — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
$\text{A}\times(\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
→
Find the equation of the straight line which joints the point $(3,5)$ to the point of intersection of lines $4 x+y=1$ and $7 x-3 y-35=0$. Prove that this line is at equal distance from origin and point $(8,34)$.
→
Show that the three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which C divides AB.
→
The means and standard deviations of heights ans weights of 50 students of a class are as follows:
  Weights Heights
Mean 63.2kg 63.2 inch
Standard deviation 5.6kg 11.5 inch
Which shows more variability, heights or weights?
→
Find the coefficient of $\frac{1}{\text{x}^{17}}$ in the expansion of $\Big(\text{x}^4-\frac{1}{\text{x}^3}\Big)^{15}.$
→
Find the equation of a line drawn perpendicular to the line $\frac{\text{x}}{4}+\frac{\text{y}}{6}=1$ through the point where it meets the y-axis.
→
Evaluate $\lim\limits_{\text{x}\rightarrow0}\text{f(x)},$ where $\text{f(x)}=\begin{cases}\frac{|\text{x}|}{\text{x}}, & \text{x} \ne0\\0, &\text{x} = 0\end{cases}.$
→

Using binomial theorem, write down the expansions of the following:

(2x + 3y)5

→
What does the equation $(\text{a} − \text{b}) (\text{x}^2 + \text{y}^2) −2\text{abx} = 0$ become if the origin is shifted to the point $\Bigg(\frac{\text{a}\text{b}}{\text{a}-\text{b}},0\Bigg)$ without rotation?
→
If $\text{f(x)}=\begin{cases}\text{x}^2,&\text{when }\text{ x}<0\\\text{x},&\text{when }\ 0\leq\text{x}<1\\\frac{1}{\text{x}},&\text{when }\text{ x}>0\end{cases}$
Find:
  1. $\text{f}\Big(\frac{1}{2}\Big)$
  2. $\text{f}(-2)$
  3. $\text{f}(1)$
  4. $\text{f}(\sqrt{3})$
  5. $\text{f}(\sqrt{-3})$
→