Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The number of pairs of real numbers, such that whenever is a root of the equation <im

The number of pairs (a, b) of real numbers, such that whenever \alpha is a root of the equation x^{2}+a x+b=0, \alpha^{2}-2 is also a root of this equation, is
Option: 1 6
Option: 2 8
Option: 3 4
Option: 4 2

Answers (1)

best_answer

x^{2}+ax+b= 0

Let\, \alpha & \beta be its roots
 

Case 1: when \alpha = \beta
It means \alpha =\alpha ^{2}-2
\Rightarrow \alpha ^{2}-\alpha -2= 0
\Rightarrow \left ( \alpha -2 \right )\left ( \alpha +1 \right )= 0
\Rightarrow \alpha = 2\, or\: \alpha = -1
So either (2,2) are roots or (-1,-1) are roots.

For (2,2) being roots,
a= -\left ( \alpha +\beta \right )= -4,b= \alpha \beta = 4
\left ( a,b \right )is\left ( -4,4 \right )

Similarly when \left ( -1,1 \right ) are roots \left ( a,b \right )is\: \left ( 2,1 \right )
 

Case 2:  when \alpha \neq \beta

Subcase 1: \alpha = \alpha ^{2}-2\; \; and\; \; \beta = \beta ^{2} -2
                     \Rightarrow \left ( \alpha ,\beta \right ) can be \left ( 2,-1 \right )or\left ( -1,2 \right )
For these cases \left ( a,b \right )is\left ( -1,-2 \right )
 

Subcase 2 : \alpha = \beta ^{2}-2\; \; and\beta = \alpha ^{2}-2
Subtract these :

\alpha - \beta = \left ( \beta -\alpha \right )\left ( \beta +\alpha \right )
\Rightarrow \alpha +\beta = -1\Rightarrow a= 1

Adding these , \alpha +\beta = \alpha ^{2}+\beta ^{2}-4
\Rightarrow -1= \left ( 1 \right )-2\alpha \beta -4
\Rightarrow \alpha \beta = -1\Rightarrow b= -1
\therefore \left ( a,b \right )\: is\: \left ( 1,-1 \right )


Subcase 3 :  \alpha = \alpha ^{2}-2= \beta ^{2}-2
\Rightarrow \alpha ^{2}-2= \beta ^{2}-2\; \; and\left ( \alpha = -1\, or \; 2\right )
\Rightarrow \alpha = -\beta \; \; and\left ( \alpha = -1\, or\: 2 \right )
\Rightarrow \left ( \alpha ,\beta \right ) can be \left ( -1,1 \right )\: \: or\: \left ( 2,-2 \right )
\Rightarrow \left ( a,b \right ) can be \left ( 0,-1 \right )\: or\left ( 0,-4 \right )
 

Subcase 4 :   \beta = \beta ^{2}-2= \alpha ^{2}-2
This can be solved similar to subcase 3 and (a,b) values are (0,-4) or (0,-1)
So total 6 (a,b) pairs are there

Posted by

Kuldeep Maurya

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JAC Chandigarh Counselling 2025: Round 1 seat allotment result out for BTech admissions
anam-khan

JoSAA round 5 seat allotment result 2025 out on josaa.nic.in
anam-khan

BTech aspirant from north-east or UT? Apply for CSAB NEUT counselling 2025 today

Latest Question