CUET · MATHS · PYQ PAPER 2025

Consider the equation of the line \(\vec{r}=-\hat{i}+2 \hat{k}+\mu(4 \hat{i}-\hat{j}+2 \hat{k})\).
Match List-I with List-II
List-I List-II
(A) It passes through the point (I) \(4,-1,2\)
(B) Its direction ratios are (II) \(\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}\)
(C) Its Cartesian form is (III) \((-1,0,2)\)
(D) Its direction cosines are (IV) \(\frac{x+1}{4}=\frac{y}{-1}=\frac{z-2}{2}\)
Choose the correct answer from the options given below :

A (А) - (I), (В) - (III), (C) - (IV), (D) - (II)
B (А) - (III), (В) - (II), (C) - (IV), (D) - (I)
C (А) - (I), (B) - (II), (C) - (IV), (D) - (III)
D (А) - (III), (B) - (I), (C) - (IV), (D) - (II)

Verified Solution

Answer & Solution

Correct Answer

(D) (А) - (III), (B) - (I), (C) - (IV), (D) - (II)

Step-by-step Solution

Detailed explanation

(A) Line equation: \(\vec{r}=\vec{a}+\mu \vec{b}\) Point \(\vec{a} = -\hat{i}+0\hat{j}+2 \hat{k} \Rightarrow (-1, 0, 2)\) (A) - (III) (B) Direction vector \(\vec{b} = 4 \hat{i}-\hat{j}+2 \hat{k}\) Direction ratios: \((4, -1, 2)\) (B) - (I) (C) Cartesian form:…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match List I with List II

LIST - I	LIST - II
A. the region represented by \(x \geq 0, y \geq 0\)	i. No feasible region
B. The region represented by the inequalities \(2 x+y \geq 3, x+2 y \geq 6, x, y \geq 0\)	II. ist quadrant
C. The region represented by the inequalities \(x+2 y \leq 8,3 x+2 y \leq 12, x, y \geq 0\)	III. unbounded
D. The region represented by the inequalities \(x+y \leq 2,3 x+5 y \geq 15, x, y \geq 0\)	iv. bounded

Choose the correct answer from the options given below.

CUET 2023 Hard

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List-I	List-II
(A) It passes through the point	(I) \(4,-1,2\)
(B) Its direction ratios are	(II) \(\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}\)
(C) Its Cartesian form is	(III) \((-1,0,2)\)
(D) Its direction cosines are	(IV) \(\frac{x+1}{4}=\frac{y}{-1}=\frac{z-2}{2}\)