CUET · MATHS · PYQ PAPER 2025

The shortest distance between the lines \(\vec{r}_1=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})\) and \(\vec{r}_2=2 \hat{i}+\hat{j}-\hat{k}+\mu(4 \hat{i}-2 \hat{j}+2 \hat{k})\) is :

A \(\frac{\sqrt{66}}{6}\)
B \(6 \sqrt{66}\)
C \(\sqrt{66}\)
D 6

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{\sqrt{66}}{6}\)

Step-by-step Solution

Detailed explanation

\(\vec{b}_1 = 2 \hat{i}-\hat{j}+\hat{k}\), \(\vec{b}_2 = 4 \hat{i}-2 \hat{j}+2 \hat{k} = 2(2 \hat{i}-\hat{j}+\hat{k}) = 2\vec{b}_1\). Lines are parallel. Let \(\vec{b} = 2 \hat{i}-\hat{j}+\hat{k}\).…

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.

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Match List - I With List - II

List - I	List - II
(A) The area of parallelogram determined by vectors \(2 \hat{i}\) and \(3 \hat{j}\)	(I) 2
(B) The value of \((\hat{i} \times \hat{j}) \cdot \hat{k}+(\hat{j} \times \hat{k}) \cdot \hat{i}\)	(II) 4
(C) The value of a for which the vectors \(2 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(a \hat{i}-6 \hat{j}+8 \hat{k}\) are collinear.	(III) 0
D. The value of \(\lambda\) for which the vectors \(2 \hat{i}+\hat{j}+\hat{k}\) and \(2 \hat{i}-4 \hat{j}+\lambda \hat{k}\) are perpendicular.	(IV) 6

Choose the correct answer from the options given below.

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Match List - I with List - II

List - I (Differential Equations)	List - II (Order and degree)
(A) \(ydx + xlog(y/x)dy - 2xdy = 0\)	(I) Order: 2, degree:1
(B) \(\left(\frac{d^3 y}{d x^3}\right)^2+3 \frac{d^2 y}{d x^2}+2\left(\frac{d y}{d x}\right)^4=y^2\)	(II) Order :1, degree:1
(C) \(\frac{d y}{d x}+\log \left(\frac{d y}{d x}\right)+x=y\)	(III) Order: 3, degree:2
(D) \(\left(\frac{d s}{d t}\right)^4+2 s \frac{d^2 s}{d t^2}=0\)	(IV) Order: 1, degree: Not defined

Choose the correct answer from the options given below:

CUET 2025 Medium

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List I – Name of the part	List II – Characteristics
A. Non-albuminous seed	III. Residual endosperm absent
B. Albuminous seed	I. Residual endosperm present
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D. Pericarp	II. Fruit wall

Choose the correct answer from the options given below:

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List-I	List-II
(A) adj(A)	(I) \(B^{-1} A^{-1}\)
(B) \((A B)^{-1}\)	(II) \(\|A\|^{-1}\)
(C) \(\left\|A^{-1}\right\|\)	(III) \(\|A\|^2\)
(D) \(\|\operatorname{adj} A\|\)	(IV) \(\|A\| A^{-1}\)