Question

Assertion A : If A, B, C, D are four points on a semi-circular arc with centre at 'O' such that $|\vec{AB}| = |\vec{BC}| = |\vec{CD}|$, then $\vec{AB} + \vec{AC} + \vec{AD} = 4\vec{AO} + \vec{OB} + \vec{OC}$ 
Reason R : Polygon law of vector addition yields $\vec{AB} + \vec{BC} + \vec{CD} = \vec{AD} = 2\vec{AO}$ 

• A. Both A and R are correct and R is the correct explanation of A.
• B. Both A and R are correct but R is not the correct explanation of A.
• C. A is correct but R is not correct.
• D. A is not correct but R is correct.

Hint:

In assertion-reason questions, first check if each statement is independently true. Then, check if the reason logically and completely explains the assertion. A correct reason must use the premises given in the assertion to derive the conclusion.

Answer 1

The Correct Option is B

Analysis of Reason (R):
By the polygon law of vector addition, $\vec{AB} + \vec{BC} + \vec{CD} = \vec{AD}$. This part is correct.
The statement $\vec{AD} = 2\vec{AO}$ implies $\vec{OD} - \vec{OA} = -2\vec{OA}$, which simplifies to $\vec{OD} = -\vec{OA}$. This is true if and only if AD is a diameter of the circle with center O. So, R is a correct statement provided AD is a diameter.
Analysis of Assertion (A):
The condition $|\vec{AB}| = |\vec{BC}| = |\vec{CD}|$ means the lengths of the chords AB, BC, and CD are equal. Equal chords in a circle subtend equal angles at the center.
Since A, B, C, D are on a semi-circular arc, the total angle $\angle AOD = 180^\circ$.
Let $\angle AOB = \angle BOC = \angle COD = \theta$. Then $3\theta = 180^\circ$, which means $\theta = 60^\circ$.
This confirms that the points A and D are at the ends of a diameter, so $\vec{OD} = -\vec{OA}$.
Now, let's evaluate the LHS of the assertion's equation, taking O as the origin:
LHS = $\vec{AB} + \vec{AC} + \vec{AD} = (\vec{OB}-\vec{OA}) + (\vec{OC}-\vec{OA}) + (\vec{OD}-\vec{OA})$
LHS = $\vec{OB} + \vec{OC} + \vec{OD} - 3\vec{OA}$
Substitute $\vec{OD} = -\vec{OA}$:
LHS = $\vec{OB} + \vec{OC} - \vec{OA} - 3\vec{OA} = \vec{OB} + \vec{OC} - 4\vec{OA}$
Now evaluate the RHS:
RHS = $4\vec{AO} + \vec{OB} + \vec{OC} = -4\vec{OA} + \vec{OB} + \vec{OC}$
Since LHS = RHS, the Assertion (A) is correct.
Conclusion:
Both Assertion (A) and Reason (R) are correct statements. However, Reason (R) does not explain Assertion (A). The reason simply states the polygon law and a geometric fact ($\vec{AD} = 2\vec{AO}$) without explaining why this fact is true based on the given condition $|\vec{AB}| = |\vec{BC}| = |\vec{CD}|$. The assertion's validity depends critically on this condition, which is not used in the reasoning. Therefore, R is not the correct explanation of A.