Question

The smallest positive integral value of $a$, for which all the roots of $x^4-ax^2+9=0$ are real and distinct, is equal to

• A. 3
• B. 9
• C. 7
• D. 4

Hint:

For biquadratic equations, reduce degree first and apply conditions for both the substituted variable and original variable.

Answer 1

The Correct Option is C

Step 1: Substitute $x^2=t$. 
\[ t^2-at+9=0 \] Step 2: Condition for real and distinct roots. 
Discriminant must be positive: \[ a^2-36>0 \Rightarrow a>6 \] Step 3: Roots of $t$ must be positive. 
Product of roots $=9>0$ and sum $=a>0$, so both roots are positive. 
Step 4: Smallest integer satisfying the condition. 
\[ a>6 \Rightarrow a_{\min}=7 \]