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Mathematics

Assertion (A) : If x = 1121\dfrac{1}{2} is a solution of the equation 2x2 + px - 6 = 0, then value of p is 1.

Reason (R) : If α is a root of quadratic equation ax2 + bx + c = 0, where a, b and c ∈ R and a ≠ 0, then aα2 + bα + c = 0.

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true.

  4. Both A and R are false.

Quadratic Equations

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Answer

Given,

x = 112=321\dfrac{1}{2} = \dfrac{3}{2} is a solution of the equation 2x2 + px - 6 = 0.

Substituting value of x in above equation, we get :

⇒2×(32)2+p×32−6=0⇒2×94+3p2−6=0⇒92+3p2−6=0⇒9+3p−122=0⇒3p−32=0⇒3p−3=0⇒3(p−1)=0⇒p−1=0⇒p=1.\Rightarrow 2 \times \Big(\dfrac{3}{2}\Big)^2 + p \times \dfrac{3}{2} - 6 = 0 \\[1em] \Rightarrow 2 \times \dfrac{9}{4} + \dfrac{3p}{2} - 6 = 0\\[1em] \Rightarrow \dfrac{9}{2} + \dfrac{3p}{2} - 6 = 0 \\[1em] \Rightarrow \dfrac{9 + 3p - 12}{2} = 0 \\[1em] \Rightarrow \dfrac{3p - 3}{2} = 0 \\[1em] \Rightarrow 3p - 3 = 0 \\[1em] \Rightarrow 3(p - 1) = 0 \\[1em] \Rightarrow p - 1 = 0 \\[1em] \Rightarrow p = 1.

∴ Assertion (A) is true.

Given,

α is a root of quadratic equation ax2 + bx + c = 0. Substituting values we get :

⇒ aα2 + bα + c = 0.

∴ Reason (R) is true.

Hence, Option 3 is the correct option.

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