Multiple choice

If the roots of the quadratic equation $\displaystyle 4x^{2}-kx-1=0$ are $\displaystyle \sin \theta $ and $\cos \theta $ , then $\displaystyle \left | k \right | $ has the value equal to

  1. $2$
  2. $8$
  3. $\displaystyle 4\sqrt{2}$
  4. $\displaystyle 2\sqrt{2}$
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D Correct answer
Explanation

For 4x^2 - kx - 1 = 0, the sum of roots sin(theta) + cos(theta) = k/4 and the product sin(theta)cos(theta) = -1/4. Squaring the sum: sin^2(theta) + cos^2(theta) + 2sin(theta)cos(theta) = k^2/16. Substituting 1 + 2(-1/4) = k^2/16 gives 1 - 1/2 = k^2/16, so 1/2 = k^2/16, k^2 = 8. Thus |k| = sqrt(8) = 2*sqrt(2).

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