Question

P and q are complex numbers such that ∣p∣ = 2/5 x + 2 , ∣q∣ = −3/4 x − 1 , and ∣p+q∣ = 2x + 4 .

Answer 1

Answer: (b)

Step-by-step explanation:

Given

$\left | p\right |=\dfrac{2}{5}x+2$

$\left | q\right |=\dfrac{-3}{4}x-1$

$\left | p+q\right |=2x+4$

for two complex number, $z_1,z_2$

$\left | z_1+z_2\right |\leq \left | z_1\right |+\left | z_2\right |$

Apply the above the property

$\Rightarrow 2x+4\leq \dfrac{2}{5}x+2-\dfrac{3}{4}x-1\\\\\Rightarrow 2x+4\leq \dfrac{8x-15x}{20}+1\\\\\Rightarrow 2x+3\leq -\dfrac{7x}{20}\\\Rightarrow x\leq -\dfrac{60}{47}$

Also, the absolute value of each complex number must be greater than or equal to zero

$\text{Case-1}\\\\\Rightarrow \dfrac{2}{5}x+2\geq 0\\\\\Rightarrow x\geq -5\\\\\text{Case-2}\\\\\Rightarrow -\dfrac{3}{4}x-1\geq 0\\\\\Rightarrow x\leq -\dfrac{4}{3}\\\\\text{Case-3}\\\\\Rightarrow 2x+4\geq 0\\\Rightarrow x\geq -2$

Taking the intersection of the above values of x, we get

$x\in [-2,-\dfrac{4}{3}]$