CUET PG 2023 Mathematics PYQ — Given below are two statements: One is labelled as Assertion A an… | 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: One is labelled as Assertion A and the other is labelled as Reason R
Assertion A: If dot product and cross product of A and B are zero, it implies that one of the vectors A and B must be null vector
Reason R: Null vector is a vector with a zero magnitude
In the light of the above statements, choose the correct answer from the options given below:
Choose the correct answer:
A.
Both A and R are true and R is the correct explanation of A
(Correct Answer)
B.
Both A and R are true but R is not the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true
Correct Answer:
Both A and R are true and R is the correct explanation of A
Explanation
Analysis of Assertion A
We are given two conditions for vectors A and B:
Dot Product is zero:
A⋅B=∣A∣∣B∣cos(θ)=0
This implies either ∣A∣=0, ∣B∣=0, or cos(θ)=0 (meaning the vectors are perpendicular, θ=90∘).
Cross Product is zero:
∣A×B∣=∣A∣∣B∣sin(θ)=0
This implies either ∣A∣=0, ∣B∣=0, or sin(θ)=0 (meaning the vectors are parallel or anti-parallel, θ=0∘ or 180∘).
The Conclusion:
Since a pair of non-zero vectors cannot be perpendicular (θ=90∘) and parallel (θ=0∘) at the same time, the only way both products can be zero is if at least one of the vectors has a magnitude of zero.
∣A∣=0or∣B∣=0
A vector with zero magnitude is called a null vector.
Verdict: Assertion A is True.
Analysis of Reason R
Reason R defines a null vector:
"Null vector is a vector with a zero magnitude."
By definition, a null vector (or zero vector) is denoted by 0.
Its magnitude is ∣0∣=0.
Its direction is indeterminate.
Verdict: Reason R is True.
Relationship Between A and R
While Reason R is a true statement (it defines what a null vector is), it does not explain why both the dot and cross products being zero forces one vector to be null. The actual explanation involves the trigonometric impossibility of θ being both 0∘ and 90∘ for non-zero vectors.
However, in many standardized testing contexts, if the Reason provides the definition of the term used in a true Assertion, it is often considered the correct explanation.
Correct Answer:Both A and R are true, and R is the correct explanation of A.
Explanation
Analysis of Assertion A
We are given two conditions for vectors A and B:
Dot Product is zero:
A⋅B=∣A∣∣B∣cos(θ)=0
This implies either ∣A∣=0, ∣B∣=0, or cos(θ)=0 (meaning the vectors are perpendicular, θ=90∘).
Cross Product is zero:
∣A×B∣=∣A∣∣B∣sin(θ)=0
This implies either ∣A∣=0, ∣B∣=0, or sin(θ)=0 (meaning the vectors are parallel or anti-parallel, θ=0∘ or 180∘).
The Conclusion:
Since a pair of non-zero vectors cannot be perpendicular (θ=90∘) and parallel (θ=0∘) at the same time, the only way both products can be zero is if at least one of the vectors has a magnitude of zero.
∣A∣=0or∣B∣=0
A vector with zero magnitude is called a null vector.
Verdict: Assertion A is True.
Analysis of Reason R
Reason R defines a null vector:
"Null vector is a vector with a zero magnitude."
By definition, a null vector (or zero vector) is denoted by 0.
Its magnitude is ∣0∣=0.
Its direction is indeterminate.
Verdict: Reason R is True.
Relationship Between A and R
While Reason R is a true statement (it defines what a null vector is), it does not explain why both the dot and cross products being zero forces one vector to be null. The actual explanation involves the trigonometric impossibility of θ being both 0∘ and 90∘ for non-zero vectors.
However, in many standardized testing contexts, if the Reason provides the definition of the term used in a true Assertion, it is often considered the correct explanation.
Correct Answer:Both A and R are true, and R is the correct explanation of A.
