In the following question, two statements (i) and (ii) are given. Choose the valid statement. i. The roots of equation 6x2 − 5x + 1= 0 are $\frac{1}{2}$ and $\frac{1}{3}$. ii. The sum of two numbers is 7 and the sum of their reciprocals is $\frac{7}{12}$. Then smaller number is 3.

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Both (i) and (ii) Explanation: Statement (i): Given equation: 6x2 − 5x + 1 = 0 6x2 − 5x + 1 = (3x − 1) (2x − 1) = 0 x = $\frac{1}{3}$, $\frac{1}{2}$ Statement (i) is true. Statement (ii): Let the two numbers be aaa and b. Given: a + b = 7 1/a + 1/b = $\frac{7}{12}$ (a+b)/(ab) = $\frac{7}{12}$ => 7/(ab) = $\frac{7}{12}$ $\Rightarrow$ ab = 12 So the numbers satisfy: x2 − 7x + 12 = 0 $\Rightarrow$ (x − 3) (x − 4) = 0 Numbers are 3 and 4. Smaller number is 3. Statement (ii) is true.

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