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CUET · MATHS · PYQ PAPER 2025

Which of the following statements are true?
(A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\) is \(\vec{r}=(5 \hat{i}+2 \hat{j}-4 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}-8 \hat{k})\)
(B) Vector form of the equation of line \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) is \(\vec{r}=(5 \hat{i}-4 \hat{j}+6 \hat{k})+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})\)
(C) The direction cosines of z-axis are (1, 1, 0).
(D) If a line has direction ratios 2, -1, -2, then its direction cosines are \(-2 / 3,-1 / 3,-2 / 3\).
Choose the correct answer from the options given below:

  1. A (A), (B) and (C) only
  2. B (B), (C) and (D) only
  3. C (A) and (B) only
  4. D (C) and (D) only
Verified Solution

Answer & Solution

Correct Answer

(C) (A) and (B) only

Step-by-step Solution

Detailed explanation

Statement (A): The vector equation of a line through point \(\vec{a}\) parallel to vector \(\vec{b}\) is \(\vec{r}=\vec{a}+\lambda \vec{b}\). Given \(\vec{a}=5 \hat{i}+2 \hat{j}-4 \hat{k}\) and \(\vec{b}=3 \hat{i}+2 \hat{j}-8 \hat{k}\). This matches the given equation. (True)…