Question

If \[a,b\] are roots of the equation \[2{x^2} - 35x + 2 = 0\] , then the value of \[{\left( {2a - 35} \right)^3}{\left( {2b - 35} \right)^3}\] .
  A.$ 10 $
  B.$ 100$
  C.$ 64 $
  D.$1 $

Answer 1

Hint: This question can be solved by firstly finding the product of the roots of the equation which equals \[\dfrac{c}{a}\] , and then by putting the roots \[a,b\] into the equation one by one and thereby simplifying the further equations, we can easily find the values for \[\left( {2a - 35} \right)\] and \[\left( {2b - 35} \right)\] thus, finally \[{\left( {2a - 35} \right)^3}{\left( {2b - 35} \right)^3}\] can be computed.

Complete step-by-step answer:
First of all let us write the equation given in the question,
 \[2{x^2} - 35x + 2 = 0\]
As it is given that \[a\] and \[b\] are the roots of the equation, then the product of roots is given by the equation,
 \[ab = \dfrac{c}{a} = \dfrac{2}{2} = 1\]
Now, putting \[a\] as the root in the equation,
 \[2{a^2} - 35a + 2 = 0\]
Now, simplifying this equation further
  $
  2{a^2} - 35a + 2 = 0 \\
   \Rightarrow 2{a^2} - 35a = - 2 \\
   \Rightarrow a\left( {2a - 35} \right) = - 2 \\
   \Rightarrow 2a - 35 = \dfrac{{ - 2}}{a} \;
  $
We get the value of the first factor which we needed to find.
Now putting \[b\] as the root of the equation given
  $
  2{b^2} - 35b + 2 = 0 \\
   \Rightarrow 2{b^2} - 35b = - 2 \\
   \Rightarrow b\left( {2b - 35} \right) = - 2 \\
   \Rightarrow 2b - 35 = \dfrac{{ - 2}}{b} \;
  $
Here, we got the value of the second factor.
Now writing the equation, the value of which we need to calculate.
 \[{\left( {2a - 35} \right)^3}{\left( {2b - 35} \right)^3}\]
Using the above obtained values in this equation, we get

  $
 {\left( {2a - 35} \right)^3}{\left( {2b - 35} \right)^3} \\
   \Rightarrow {\left( {\dfrac{{ - 2}}{a}} \right)^3}{\left( {\dfrac{{ - 2}}{b}} \right)^3} \\
   \Rightarrow \left( {\dfrac{{ - 8}}{{{a^3}}}} \right)\left( {\dfrac{{ - 8}}{{{b^3}}}} \right) \\
   \Rightarrow \dfrac{{64}}{{{a^3}{b^3}}} \;
  $
Now, as we estimated the value of \[ab\] above, using the value \[ab = 1 \Rightarrow {a^3}{b^3} = 1\] in the above equation, we get
 \[ \Rightarrow \dfrac{{64}}{1} = 64\]
Hence,
The factor \[{\left( {2a - 35} \right)^3}{\left( {2b - 35} \right)^3}\] equals to the value \[64\]
So, the correct answer is “Option C”.

Note: It is important to note that the step wherein we are taking the two roots \[a\] and \[b\] separately in the quadratic equation and finding the values of \[\left( {2a - 35} \right)\] and \[\left( {2b - 35} \right)\] separately first and then putting in the main equation. In this the mistake can be made if during finding the value of either \[\left( {2a - 35} \right)\] or \[\left( {2b - 35} \right)\] , but the right values give the perfect result.