Question

Let \(S\) be the set of all rational numbers \(r\) such that if \[ r=\frac{p}{q} \] with \(p,q\in\mathbb{Z}\), \(q\neq0\), then the two roots of the quadratic equation \[ x^{2}+2px+q^{2}=0 \] are equal. Then the number of elements in \(S\) is:

• A. 2
• B. 4
• C. 1
• D. Infinite

Hint:

Whenever a question contains the phrase "roots are equal", immediately think of the discriminant condition \(D=0\). This is usually the fastest route to the answer.

Answer 1

The Correct Option is A

Concept: A quadratic equation has equal roots when its discriminant is zero. For \[ ax^2+bx+c=0 \] equal roots occur when \[ b^2-4ac=0 \]

Step 1: Apply the discriminant condition. Given: \[ x^2+2px+q^2=0 \] Thus, \[ a=1,\quad b=2p,\quad c=q^2 \] Equal roots require: \[ (2p)^2-4(1)(q^2)=0 \] \[ 4p^2-4q^2=0 \] \[ p^2=q^2 \] \[ p=\pm q \]

Step 2: Determine possible values of \(r\). Since \[ r=\frac{p}{q} \] If \[ p=q \] then \[ r=1 \] If \[ p=-q \] then \[ r=-1 \]

Step 3: Count the elements. Therefore, \[ S=\{1,-1\} \] Number of elements: \[ n(S)=2 \] \[ \boxed{2} \] Hence option (A) is correct.