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NVAMediumJEE 2025Centre of Mass

JEE Physics 2025 Question with Solution

Two particles are located at equal distance from origin. The position vectors of those are represented by A⃗=2i^+3j^+2k^\vec{A} = 2\hat{i} + 3\hat{j} + 2\hat{k} and B⃗=2i^−2j^+4k^\vec{B} = 2\hat{i} - 2\hat{j} + 4\hat{k}, respectively. If both the vectors are at right angle to each other, the value of n−1n^{-1} is:

Answer

Correct answer:3

Step-by-step solution

Standard Method

Given: The vectors are perpendicular and their magnitudes are equal.

Find: The value of n−1n^{-1}.

From the solution, the intended variable form used in the working is

A⃗=2i^+3nj^+2k^,B⃗=2i^−2j^+4pk^\vec{A} = 2\hat{i} + 3n\hat{j} + 2\hat{k}, \qquad \vec{B} = 2\hat{i} - 2\hat{j} + 4p\hat{k}

Using perpendicularity,

A⃗⋅B⃗=0\vec{A} \cdot \vec{B} = 0

so

4−6n+8p=0(1)4 - 6n + 8p = 0 \tag{1}

Using equality of magnitudes,

∣A⃗∣=∣B⃗∣|\vec{A}| = |\vec{B}|

therefore

4+9n2+4=4+4+16p2\sqrt{4 + 9n^2 + 4} = \sqrt{4 + 4 + 16p^2}

Squaring both sides,

4+9n2+4=4+4+16p24 + 9n^2 + 4 = 4 + 4 + 16p^2

which gives

9n2=16p29n^2 = 16p^2

Hence,

p=±34np = \pm \frac{3}{4}n

Now substitute into equation (1)(1).

For

p=34np = \frac{3}{4}n

we get

4−6n+8(34n)=04 - 6n + 8\left(\frac{3}{4}n\right) = 0 4−6n+6n=04 - 6n + 6n = 0 4=04 = 0

which is impossible.

So take

p=−34np = -\frac{3}{4}n

Then

4−6n+8(−34n)=04 - 6n + 8\left(-\frac{3}{4}n\right) = 0 4−6n−6n=04 - 6n - 6n = 0 4−12n=04 - 12n = 0 n=13n = \frac{1}{3}

Therefore,

n−1=3n^{-1} = 3

So the final answer is 33.

Note: The given question text and the solution notation are inconsistent; the answer has been derived from the solution, which is treated.

Common mistakes

  • Using only the perpendicularity condition and ignoring the equal-magnitude condition is wrong because one equation is not enough to determine the unknowns. Use both A⃗⋅B⃗=0\vec{A} \cdot \vec{B} = 0 and ∣A⃗∣=∣B⃗∣|\vec{A}| = |\vec{B}|.

  • Taking only p=34np = \frac{3}{4}n from 9n2=16p29n^2 = 16p^2 is incorrect because squaring introduces both signs. You must test both p=±34np = \pm \frac{3}{4}n.

  • Reporting n=13n = \frac{1}{3} as the final answer is incomplete because the question asks for n−1n^{-1}. After finding nn, take the reciprocal to get 33.

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