NIMCET 2024 Mathematics PYQ — A man starts at the origin and walks a distance of units in the n… | Mathem Solvex | Mathem Solvex
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NIMCET 2024 — Mathematics PYQ
NIMCET | Mathematics | 2024
A man starts at the origin O and walks a distance of 3 units in the north-east direction and then walks a distance of 4 units in the north-west direction to reach the point P, then OP is equal to
Choose the correct answer:
A.
21(−i^+j^)
B.
21(i^+j^)
C.
21(i^−7j^)
D.
21(−i^+7j^)
Correct Answer:
21(−i^+7j^)
Explanation
To solve this problem, we can define the standard coordinate system for directions:
East corresponds to the positive x-axis (+i^)
West corresponds to the negative x-axis (−i^)
North corresponds to the positive y-axis (+j^)
South corresponds to the negative y-axis (−j^)
Let the first displacement be vector A and the second displacement be vector B. The final position vector OP will be the vector sum of both movements:
OP=A+B
Step 1: Determine the first displacement vector (A)
The man walks 3 units in the North-East direction.
The North-East direction forms an angle of 45∘ with both the East (positive x-axis) and North (positive y-axis) directions.
Breaking this displacement into rectangular components:
A=3cos(45∘)i^+3sin(45∘)j^
Since cos(45∘)=sin(45∘)=21:
A=23i^+23j^— (Equation 1)
Step 2: Determine the second displacement vector (B)
From that point, the man walks 4 units in the North-West direction.
The North-West direction forms an angle of 45∘ with the West (negative x-axis) and North (positive y-axis) directions.
Breaking this displacement into components (moving along negative x and positive y):
B=−4cos(45∘)i^+4sin(45∘)j^
Substituting the values of trigonometric functions:
B=−24i^+24j^— (Equation 2)
Step 3: Calculate the resultant vector OP
Now, add the two vectors A and B together component-wise:
OP=A+B
OP=(23i^+23j^)+(−24i^+24j^)
Group the i^ and j^ terms together:
OP=(23−24)i^+(23+24)j^
OP=(−21)i^+(27)j^
Taking 21 as a common factor outside the bracket:
OP=21(−i^+7j^)
Correct Answer
The correct option is (D)21(−i^+7j^).
Explanation
To solve this problem, we can define the standard coordinate system for directions:
East corresponds to the positive x-axis (+i^)
West corresponds to the negative x-axis (−i^)
North corresponds to the positive y-axis (+j^)
South corresponds to the negative y-axis (−j^)
Let the first displacement be vector A and the second displacement be vector B. The final position vector OP will be the vector sum of both movements:
OP=A+B
Step 1: Determine the first displacement vector (A)
The man walks 3 units in the North-East direction.
The North-East direction forms an angle of 45∘ with both the East (positive x-axis) and North (positive y-axis) directions.
Breaking this displacement into rectangular components:
A=3cos(45∘)i^+3sin(45∘)j^
Since cos(45∘)=sin(45∘)=21:
A=23i^+23j^— (Equation 1)
Step 2: Determine the second displacement vector (B)
From that point, the man walks 4 units in the North-West direction.
The North-West direction forms an angle of 45∘ with the West (negative x-axis) and North (positive y-axis) directions.
Breaking this displacement into components (moving along negative x and positive y):
B=−4cos(45∘)i^+4sin(45∘)j^
Substituting the values of trigonometric functions:
B=−24i^+24j^— (Equation 2)
Step 3: Calculate the resultant vector OP
Now, add the two vectors A and B together component-wise:
OP=A+B
OP=(23i^+23j^)+(−24i^+24j^)
Group the i^ and j^ terms together:
OP=(23−24)i^+(23+24)j^
OP=(−21)i^+(27)j^
Taking 21 as a common factor outside the bracket:
