Multiple choice

If $x = 2$ and $x = 3$ are roots of the equation $\displaystyle 3x^{2}-2kx+2m=0$ then $(k,m) $=

  1. $\displaystyle \left ( \frac{15}{2} ,9\right )$
  2. $\displaystyle \left ( 9,\frac{15}{2} \right )$
  3. $\displaystyle \left ( \frac{9}{2},15 \right )$

  4. ( 15 , 8)
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A Correct answer
Explanation

Roots are 2 and 3. Sum = 2+3 = 5 = 2k/3 => k = 15/2. Product = 2*3 = 6 = 2m/3 => m = 9. (k,m) = (15/2, 9).

AI explanation

Since x = 2 and x = 3 are roots of the equation 3x^2 - 2kx + 2m = 0, we can write the equation in its factored form as 3(x - 2)(x - 3) = 0. Expanding this expression gives 3x^2 - 15x + 18 = 0. Comparing the coefficients with the original equation 3x^2 - 2kx + 2m = 0, we get 2k = 15 and 2m = 18. Solving these gives k = 15/2 and m = 9, making the pair (15/2, 9).

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