JEE Physics Vectors and Calculus MCQs Set 06

Answer: (b) \( 10\hat i+10\sqrt2\,\hat j+10\hat k \)

Explanation:
1. Direction cosines: l = cos 60° = 1/2 and n = cos 60° = 1/2.
2. l² + m² + n² = 1 gives m² = 1/2, m = 1/√2 (horizontal direction taken northward).
3. v = 20(1/2, 1/√2, 1/2) = 10i + 10√2 j + 10k.

Teacher's Note:
1. Use l² + m² + n² = 1 for direction cosines.
2. Multiply the unit direction by the speed.

Answer: (b) \( \sqrt{45}\text{ m} \)

Explanation:
1. B = (10 - 4, 9 - 6) = (6, 3).
2. |B| = √(36 + 9) = √45 m.

Teacher's Note:
1. Subtract A from A + B component-wise.
2. Then use Pythagoras.

Answer: (d) \( \frac{\sqrt2\,ab}{a+b} \)

Explanation:
1. At time t: A (at, 0), B (0, bt), C (ct/√2, ct/√2).
2. Collinear on the line x/(at) + y/(bt) = 1, so (c/√2)(1/a + 1/b) = 1.
3. c = √2 ab/(a + b).

Teacher's Note:
1. Write the positions at time t and use the intercept form of a line.
2. C moves at 45° so its coordinates are c t/√2 each.

Answer: (b) \( \frac16\left[|\vec b\times\vec c|+|\vec c\times\vec a|+|\vec a\times\vec b|\right] \)

Explanation:
1. Because a + b + c = 0, a×b = b×c = c×a.
2. Area = (1/2)|a×b|, and (1/6) × (sum of the three equal magnitudes) = (1/6)(3|a×b|) = (1/2)|a×b|.

Teacher's Note:
1. The three cross products of the sides of a closed triangle are equal.
2. The area is half the magnitude of any one of them.

Answer: (a) \( \frac12\sqrt{107} \)

Explanation:
1. PQ = (1, -4, -1) and PR = (-2, -1, 1).
2. PQ×PR = (-5, 1, -9), magnitude √107.
3. Area = (1/2)√107.

Teacher's Note:
1. Form two edge vectors from one vertex.
2. Area is half the magnitude of their cross product.

Answer: (d) \( \frac{3}{\sqrt2} \)

Explanation:
1. (i + j + k)×3i = (0, 3, -3).
2. Its magnitude is 3√2, so the area is 3√2/2 = 3/√2.

Teacher's Note:
1. Use half the magnitude of the cross product.
2. 3√2/2 simplifies to 3/√2.

Answer: (a) iii, iv

Explanation:
1. The resultant lies closer to the larger vector, so the angle with the larger vector is smaller.
2. So α < β when A > B, and α = β when A = B.

Teacher's Note:
1. Think of the resultant as the diagonal of a parallelogram.
2. It leans toward the longer side.

Answer: (b) i, ii, iii

Explanation:
1. Three unit vectors in an equilateral arrangement give 0; two cancelling and one left gives 1; all parallel give 3.
2. The maximum possible magnitude is 3, so 3√2 is impossible.

Teacher's Note:
1. The resultant of three unit vectors lies between 0 and 3.
2. Check the extreme cases.

Answer: (d) i, ii, iii

Explanation:
1. The resultant can vanish only if the largest magnitude is at most the sum of the other two.
2. 10, 10, 10; 10, 10, 20 (collinear limit) and 10, 20, 20 satisfy this; 10, 20, 40 does not.

Teacher's Note:
1. Apply the triangle condition.
2. Largest ≤ sum of the others.

Answer: (a) i, ii, iv

Explanation:
1. Components add algebraically, so (i) is always true.
2. Rx = ΣAx ≤ Σ|A|, so it cannot exceed the sum of magnitudes.
3. It equals the sum of magnitudes when all vectors point along +x.

Teacher's Note:
1. Component of a sum is the sum of components.
2. A component never exceeds the magnitude.

Answer: (d) i, ii, iv

Explanation:
1. Dot product with 5k vanishes only for vectors with no k component.
2. 7k is parallel to 5k, not perpendicular.

Teacher's Note:
1. Perpendicular means zero dot product.
2. Vectors in the xy-plane are perpendicular to k.

Answer: (a) i, ii, iv

Explanation:
1. Addition and dot product are commutative.
2. A×B = -B×A, so (iii) is false.
3. (A - B)×(B - A) = -(A - B)×(A - B) = 0.

Teacher's Note:
1. The cross product changes sign when the order is reversed.
2. A vector crossed with itself (or its negative) gives zero.

Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

Explanation:
1. A×B is normal to the plane of A and B.
2. A + B lies in that plane, so A×B ⟂ A + B and the reason explains the assertion.

Teacher's Note:
1. Normal to a plane means perpendicular to every vector in it.
2. The reason gives exactly that geometry.

Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

Explanation:
1. B·(A - C) = 0 only says that A - C is perpendicular to B, so A need not equal C.
2. This arises because the dot product depends on cos θ, so different vectors can give the same value. The reason explains the assertion.

Teacher's Note:
1. Equal dot products do not imply equal vectors.
2. The cosine can compensate for the magnitude.

Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

Explanation:
1. Equal unit vectors make an angle of 0°.
2. A×B = AB sin 0° = 0, which is exactly what the reason states.

Teacher's Note:
1. Equal vectors are parallel.
2. The cross product of parallel vectors is zero.

Answer: (d) Assertion is false but Reason is true.

Explanation:
1. Two equal and opposite vectors add to zero, so the assertion is false.
2. The reason is true.

Teacher's Note:
1. Equal and opposite vectors cancel.
2. So zero sum is possible.

Answer: (a) 13

Explanation:
1. A + B = 3i + 4j + 12k.
2. Magnitude = √(9 + 16 + 144) = 13.

Teacher's Note:
1. Add component-wise first.
2. Then take the magnitude.

Answer: (d) zero

Explanation:
1. A×B is perpendicular to A.
2. So A·(A×B) = 0.

Teacher's Note:
1. The cross product is normal to both factors.
2. Its dot product with A vanishes.

Answer: (a) \( \frac{1}{13\sqrt5}(28\hat i-6\hat j-5\hat k) \)

Explanation:
1. A×B = (56, -12, -10) = 2(28, -6, -5).
2. |A×B| = 26√5.
3. Unit vector = (28i - 6j - 5k)/(13√5).

Teacher's Note:
1. Normalise the cross product.
2. Check 28² + 6² + 5² = 845 = 169 × 5.

Answer: (c) A-q, B-p, C-s, D-r

Explanation:
1. A·B = 8 cos θ.
2. 0 gives 90°, +8 gives 0°, 4 gives 60° and -8 gives 180°.

Teacher's Note:
1. Use A·B = AB cos θ with AB = 8.
2. Match each value of cos θ to its angle.

Answer: (d) A-s, B-r, C-p, D-q

Explanation:
1. |A×B| = 8 sin θ and |A·B| = 8|cos θ|.
2. 0 gives θ = 0°, 8 gives 90°, 4 gives 30°, and 4√2 gives 45°.

Teacher's Note:
1. Use sin θ for the cross product and cos θ for the dot product.
2. Solve each for the listed angle.