CBSE Class 12 Mathematics PYQs

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From the CBSE Class 12 Mathematics Sample Paper 2027.

If vector 2î - uĵ + 3k̂ has magnitude 4, then value of u is

1. 1
2. √2
3. ±√3
4. ±2

Answer

C. ±√3

Explanation

The coefficients of i, j, k are the vector’s x, y, z components. Its magnitude is its non-negative length. The unknown component is −u, but squaring it gives u². We are finding u, not the length, so both signs may be possible.

Compute the magnitude of vector v = 2 î - uĵ + 3 k̂ using |v| = √(x² + y² + z²):
|v| = √(2² + (-u)² + 3²) = √(4 + u² + 9) = √(13 + u²).

Why this step: Length comes from the three-dimensional Pythagorean rule. (-u)²=(-u)(-u)=u² because two negative factors make a positive product.

Set the magnitude equal to 4:
√(13 + u²) = 4.

Why this step: The question states the length is 4, so equate the square-root expression to 4 before solving. Do not set the quantity under the root equal to 4.

Square both sides:
13 + u² = 16 ⇒ u² = 16 - 13 = 3.

Why this step: Squaring both sides gives 16 on the right. Subtract 13 from both sides: 16−13=3. Both original sides were non-negative, so this squaring is safe.

Solve for u:
u = ±√3. So option C is correct.

Why this step: Both √3 and −√3 square to 3 and give length 4. Unlike evaluating √3, solving u²=3 requires both signs. Thus option C is correct.

See this sample paper