CUET · MATHS · PYQ PAPER 2023

A line makes equal angles with coordinate axis. Direction cosines of the line are:

A \(\pm(1,1,1)\)
B \(\quad \pm\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\)
C \(\pm\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)\)
D \(\quad \pm\left(\frac{-1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\quad \pm\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\)

Step-by-step Solution

Detailed explanation

Let the direction cosines be \(l, m, n\). The angles are equal: \(\alpha = \beta = \gamma\). So, \(l = m = n = \cos\alpha\). \(l^2 + m^2 + n^2 = 1\) \(3\cos^2\alpha = 1\) \(\cos\alpha = \pm\frac{1}{\sqrt{3}}\) Direction cosines are…

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 minimum value of \(f(x)=(2 x-1)^2+3\)	(I) 4
(B) The maximum value of \(f(x)=-\|x+1\|+4\)	(II) 10
(C) The minimum value of \(f(x)=\sin (2 x)+6\)	(III) 3
(D) The maximum value of \(f(x)=-(x-1)^2+10\)	(IV) 5

Choose the correct answer from the options given below:

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