If the projection of the vector i ^ + 2 j ^ + k ^ on the sum of the two vectors 2 i ^ + 4 j ^ - 5 k ^ and -…

Sum of two vectors $=2\hat{i}+4\hat{j}-5\hat{k}+(-\lambda \hat{i}+2\hat{j}+3\hat{k})=(2-\lambda )\hat{i}+6\hat{j}-2\hat{k}$
Projection $=1=\frac{1(2-\lambda )+2\left(6\right)+1\times (-2)}{\sqrt{(2-\lambda {)}^2+6^2+2^2}}$

$\Rightarrow \frac{12-\lambda }{\sqrt{{\lambda }^2-4\lambda +44}}=1$
$\Rightarrow (12-\lambda {)}^2={\lambda }^2-4\lambda +44$
$\Rightarrow 144+{\lambda }^2-24\lambda ={\lambda }^2-4\lambda +44$
$\Rightarrow 100=20\lambda$
$\Rightarrow \lambda =5$