CUET PG 2023 Mathematics PYQ — Given below are two statements: Statement I : The angle between t… | Mathem Solvex | Mathem Solvex
Tip:A–D to answerE for explanationV for videoS to reveal answer
CUET PG 2023 — Mathematics PYQ
CUET PG | Mathematics | 2023
Given below are two statements: Statement I : The angle between the vectors 2i^+3j^+k^ and 2i^−j^−k^ is π/2 Statement II :The vector a×(b×c) is coplanar with a and b In the light of the above statement, choose the correct answer from the options given below.
Choose the correct answer:
A.
Both Statement I and Statement II are true
B.
Both Statement I and Statement II are False
C.
Statement I is true but Statement II is False
(Correct Answer)
D.
Statement I is false but Statement II is true
Correct Answer:
Statement I is true but Statement II is False
Explanation
Analysis of Statement I
To find the angle between two vectors, we check their dot product. If the dot product is zero, the angle is π/2 (90∘).
Let u=2i^+3j^+k^ and v=2i^−j^−k^.
u⋅v=(2)(2)+(3)(−1)+(1)(−1)
u⋅v=4−3−1
u⋅v=0
Since the dot product is 0, the vectors are indeed perpendicular.
Verdict: Statement I is True.
Analysis of Statement II
Statement II discusses the Vector Triple Product, a×(b×c).
The standard expansion formula for a vector triple product is:
a×(b×c)=(a⋅c)b−(a⋅b)c
This formula shows that the resulting vector is a linear combination of b and c.
Therefore, the vector a×(b×c) lies in the plane containing b and c.
Statement II claims it is coplanar with a and b, which is generally False (unless a also happens to be in the same plane).
Verdict: Statement II is False.
Final Conclusion
Statement I is true.
Statement II is false.
Explanation
Analysis of Statement I
To find the angle between two vectors, we check their dot product. If the dot product is zero, the angle is π/2 (90∘).
Let u=2i^+3j^+k^ and v=2i^−j^−k^.
u⋅v=(2)(2)+(3)(−1)+(1)(−1)
u⋅v=4−3−1
u⋅v=0
Since the dot product is 0, the vectors are indeed perpendicular.
Verdict: Statement I is True.
Analysis of Statement II
Statement II discusses the Vector Triple Product, a×(b×c).
The standard expansion formula for a vector triple product is:
a×(b×c)=(a⋅c)b−(a⋅b)c
This formula shows that the resulting vector is a linear combination of b and c.
Therefore, the vector a×(b×c) lies in the plane containing b and c.
Statement II claims it is coplanar with a and b, which is generally False (unless a also happens to be in the same plane).
