Multiple choice

If $\displaystyle \sin a$ and $\cos a$ are the roots of the equation $\displaystyle 4x^{2}-kx-1=0,\left ( k> 0 \right )$ then the value of $k$ is

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

Roots are sin(a) and cos(a). Sum = sin(a) + cos(a) = k/4. Product = sin(a)cos(a) = -1/4. (sin(a) + cos(a))^2 = 1 + 2sin(a)cos(a) = 1 + 2(-1/4) = 1 - 0.5 = 0.5. So (k/4)^2 = 0.5 => k^2/16 = 0.5 => k^2 = 8 => k = sqrt(8) = 2*sqrt(2).

AI explanation

By Vieta's formulas for the equation 4x^2 - kx - 1 = 0, the sum of the roots is sin a + cos a = k/4 and the product of the roots is sin a * cos a = -1/4. Using the trigonometric identity (sin a + cos a)^2 = sin^2 a + cos^2 a + 2 sin a cos a, we substitute the known sum and product to get (k/4)^2 = 1 + 2(-1/4). This simplifies to k^2/16 = 1 - 1/2, which equals 1/2. Multiplying both sides by 16 gives k^2 = 8, so k = sqrt(8) = 2*sqrt(2). Since it is given that k > 0, the value of k is 2*sqrt(2).

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