Item contains a Question followed by two Statements. Answer each item using the following instructions : 

Question: What are the unique values of a, b and c if 2 is a root of the equation ax2 + bx + c = 0 ? 

Statement I: Ratio of c to a is 1. 

Statement II: Ratio of b to a is (-5/2).  

This question was previously asked in
CDS Elementary Mathematics 16 April 2023 Official Paper
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  1. Choose this option if the Question can be answered by one of the Statements alone but not by the other.  
  2. Choose this option if the Question can be answered by either Statement alone.
  3. Choose this option if the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.  
  4. Choose this option if the Question cannot be answered even by using both Statements together.

Answer (Detailed Solution Below)

Option 4 : Choose this option if the Question cannot be answered even by using both Statements together.
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Detailed Solution

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Concept:

For the quadratic equation ax2 + bx + c = 0

Sum of root (α + β) = -b/a

Produt of root (αβ)   = c/a

The quadratic equation can also be written as 

k[x2 - x (α + β) + α β] = 0

Where k is any integer.   

Calculation:

We have 

ax2 + bx + c = 0      ----(1)

One of the roots is 2 (let α)

Statement: 1 Ratio of c to a is 1.

Product of root 

αβ = c/a = 1

2 × β = 1 

⇒ β = 1/2

The quadratic equation will be 

k[x2 - x (2 + 1/2) + 2 × (1/2)] = 0

k[x2 - 5x/2 + 1] = 0

k[2x2 - 5x + 2] = 0

On comparing it to equation (1)

a =  2k, b = -5k, c = 2k

So there is no unique value of a, b & c.

Statement:2  Ratio of b to a is (-5/2)

2 + β = 5/2

β = 1/2 

So again there is no unique value of a, b & c.

∴ Option 4 will be correct.

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