Consider a line $\vec r= (\hat i + 4\hat j) +λ(2\hat i − 2\hat j + 3\hat k)$, then which of the following statements are correct?

(A) it passes through point (9, -4, 12)
(B) it passes through point (1, 4, -1)
(C) its direction cosine's are $\frac{2}{\sqrt{17}},\frac{-2}{\sqrt{17}},\frac{3}{\sqrt{17}}$
(D) its Cartesian equation is $\frac{x-1}{2}=\frac{y-4}{-2}=\frac{z}{3}$

Choose the correct answer from the options given below:

(A), (C) and (D) only

The correct answer is Option (1) → (A), (C) and (D) only

$\vec r=(1\,\hat i+4\,\hat j)+\lambda(2\,\hat i-2\,\hat j+3\,\hat k)$

Point on line (put $\lambda=0$): $(1,4,0)$

Direction ratios: $(2,-2,3)$

Direction cosines:

$\ell=\frac{2}{\sqrt{17}},\; m=\frac{-2}{\sqrt{17}},\; n=\frac{3}{\sqrt{17}}$

Cartesian equation:

$\frac{x-1}{2}=\frac{y-4}{-2}=\frac{z-0}{3}$

Checking options:

(A) $(9,-4,12)$ on line → true

(B) $(1,4,-1)$ not on line → false

(C) direction cosines correct → true

(D) Cartesian form correct → true

Correct statements: (A), (C) and (D)