Multiple choice

The quadratic equation whose roots are $l,m $ where $l=\displaystyle \lim_{\theta \rightarrow 0}\ \displaystyle \frac{3\sin\theta-4\sin^{2}\theta}{\theta}$, $m=\displaystyle \lim_{\theta \rightarrow 0}\ \frac{2\tan\theta }{\theta}$is

  1. $\mathrm{x}^{2}-5\mathrm{x}+6=0$
  2. $\mathrm{x}^{2}+5\mathrm{x}+6=0$
  3. $\mathrm{x}^{2}-\mathrm{x}+6=0$
  4. $\mathrm{x}^{2}-\mathrm{x}-6=0$
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A Correct answer
Explanation

The limit l is calculated as lim(theta->0) (3sin(theta)/theta - 4sin^2(theta)/theta) = 3 - 0 = 3. The limit m is calculated as lim(theta->0) (2tan(theta)/theta) = 2. The quadratic equation with roots 3 and 2 is (x-3)(x-2) = x^2 - 5x + 6 = 0.

AI explanation

To find the first root l, we evaluate the limit as theta approaches 0 of (3 sine theta minus 4 sine squared theta) divided by theta, which separates into 3 times the limit of (sine theta divided by theta) minus 4 times the limit of (theta times sine theta divided by theta squared). This evaluates to 3 times 1 minus 0, giving l equals 3. To find the second root m, we evaluate the limit as theta approaches 0 of (2 tangent theta divided by theta), which equals 2 times 1, giving m equals 2. The required quadratic equation is x squared minus (l plus m)x plus (l times m) equals 0, which results in x squared minus 5x plus 6 equals 0.

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