JEE Main 2021 · 27 July · morning · Single correct · +4 / −1
Assertion A : If A, B, C, D are four points on a semi-circular are with centre at 'O' such that |AB⃗|=|BC⃗|=|CD⃗|, then AB⃗+AC⃗+AD⃗=4AO⃗+OB⃗+OC⃗
Reason R : Polygon law of vector addition yields AB⃗+BC⃗+CD⃗+AD⃗=2AO⃗
In the light of the above statements, choose the most appropriate answer from the options given below :
- (a) A is correct but R is not correct.
- (b) A is not correct but R is correct.
- (c) Both A and R are correct and R is the correct explanation of A.
- (d) Both A and R are correct but R is not the correct explanation of A.
Answer and solution
Answer (a) A is correct but R is not correct.
Solution
Physics sir · 4 steps
Route every vector through the centre O, and remember that AD is a diameter, so AD = 2AO.
- Use the diameter. A and D are the two ends of the semicircle and O is its centre, so O is the midpoint of AD. That gives AD⃗ = 2AO⃗.
- Check Assertion A. Break each vector at O using the triangle law: AB⃗ = AO⃗ + OB⃗ and AC⃗ = AO⃗ + OC⃗. Add AD⃗ = 2AO⃗ to these.
AB⃗+AC⃗+AD⃗ = 4AO⃗+OB⃗+OC⃗
- Check Reason R. By the polygon law, AB⃗+BC⃗+CD⃗ = AD⃗. Adding one more AD⃗ gives 2AD⃗ = 4AO⃗, not 2AO⃗ as R claims.
- So. Assertion A is correct and Reason R is wrong, which is option (a). The equal chord lengths are not even needed for A.
The idea to keep. Break each vector at a convenient point with the triangle law. A chain of head-to-tail vectors adds up to one arrow from the first tail to the last head. Then check every coefficient.
Answer from the NTA final answer key. Solution by PrepWiser.