Download App

STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
MCQ
For any three positive real numbers $a,b,c$ ; $9\left( {25{a^2} + {b^2}} \right) + 25\left( {{c^2} - 3ac} \right) = 15b\left( {3a + c} \right)$ then
  • A
    $a,b,c$ are in $G.P.$
  • B
    $b,c,a$ are in $G.P.$
  • ✓
    $b,c,a$ are in $A.P.$
  • D
    $a,b,c$ are in $A.P.$
✓

Answer

Correct option: C.
$b,c,a$ are in $A.P.$
c
We have 

$9\left( {25{a^2} + {b^2}} \right) + 25\left( {{c^2} - 3ac} \right) = 15b\left( {3a + c} \right)$

$ \Rightarrow 225{a^2} + 9{b^2} + 25{c^2} - 75ac = 45ab + 15bc$

$ \Rightarrow {\left( {15a} \right)^2} + {\left( {3b} \right)^2} + 5{c^2} - 75ac - 45ab - 15bc = 0$

$\frac{1}{2}\left[ {{{\left( {15a - 3b} \right)}^2} + {{\left( {3b - 5c} \right)}^2} + {{\left( {5c - 15a} \right)}^2}} \right] = 0$

it is possible when $15a - 3b = 0,3b - 5c = 0$ and $5c - 15a = 0$

$ \Rightarrow 15a = 3b = 5c$

$ \Rightarrow b\frac{{5c}}{3},a = \frac{c}{3}$

$ \Rightarrow a + b = \frac{c}{3} + \frac{{5c}}{3} = \frac{{6c}}{3}$

$ \Rightarrow a + b = 2c$

$ \Rightarrow b,c,a$ 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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If  $E_1$ is the event that die $A$ shows up four, $E_2 $ is the event that die $B$ shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true $?$
→
The number of straight lines joining $8$ points on a circle is
→
Which of the following functions differentiable at $x = 0$ ?
→
Let $f(x) =  - 1 + \left| {x - 2} \right|,$ and $g(x) = 1 - \left| x \right|;$ then the set of all points where $\text{fog}$ is discontinuous is
→
Let ${a_1},{a_2},{a_3}$ be any positive real numbers, then which of the following statement is not true
→
If both $\left( {A - \frac{I}{2}} \right)$ and ${A + \frac{I}{2}}$ are orthogonal matrices, then  
→
Let $O$ be the origin. Let $\overline{ OP }= x \hat{ i }+ y \hat{ j }-\hat{ k }$ and $\overline{ OQ }=-\hat{ i }+2 \hat{ j }+3 x \hat{ k }, x , y \in R , x >0,$ be such that $|\overline{ PQ }|=\sqrt{20}$ and the vector $\overline{ OP }$ is perpendicular to $\overline{ OQ }$. If $\overline{ OR }=3 \hat{ i }+ z \hat{ j }-7 \hat{ k }, z \in R ,$ is coplanar with $\overline{ OP }$ and $\overline{ OQ },$ then the value of $x ^{2}+ y ^{2}+ z ^{2}$ is equal to ...... .
→
Let $a, b, c, d, e$ be real numbers such that $a + b < c + d$, $b + c < d + e, c + d < e + a, d + e < a + b$. Then,
→
The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is
→
A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta(h)=$ area of the triangle $P Q R, \Delta_1=\max _{1 / 2 \leq h \leq 1} \Delta(h)$ and $\Delta_2=\min _{1 / 2 \leq h \leq 1} \Delta(h)$, then $\frac{8}{\sqrt{5}} \Delta_1-8 \Delta_2=$
→