NIMCET 2022 Mathematics PYQ — There are two circles in xy-plane whose equations are and . A poi… | Mathem Solvex | Mathem Solvex
Tip:A–D to answerE for explanationV for videoS to reveal answer
NIMCET 2022 — Mathematics PYQ
NIMCET | Mathematics | 2022
There are two circles in xy-plane whose equations are x2+y2−2y=0 and x2+y2−2y−3=0. A point (x, y) is chosen at random inside the larger circle. then, the probability that the point has been taken from smaller circle is
Choose the correct answer:
A.
1/3
B.
2/3
C.
1/2
D.
1/4
(Correct Answer)
Correct Answer:
1/4
Explanation
1. The equation of the first circle is given as x2+y2−2y=0. This equation can be rewritten by completing the square for the y terms: x2+(y2−2y+1)=1, which simplifies to x2+(y−1)2=12. 2. The center of the first circle is (0,1) and its radius, r1, is 1. 3. The equation of the second circle is given as x2+y2−2y−3=0. This equation can be rewritten by completing the square for the y terms: x2+(y2−2y+1)=3+1, which simplifies to x2+(y−1)2=4. 4. The center of the second circle is (0,1) and its radius, r2, is 4=2. 5. The larger circle is the second circle with radius r2=2, and the smaller circle is the first circle with radius r1=1.
Areas of the Circles
1. The area of the smaller circle, A1, is calculated using the formula A=πr2: A1=π(1)2=π. 2. The area of the larger circle, A2, is calculated using the formula A=πr2: A2=π(2)2=4π.
Probability Calculation
1. The probability that a point chosen at random inside the larger circle is also inside the smaller circle is given by the ratio of the area of the smaller circle to the area of the larger circle. 2. The probability, P, is calculated as P=A2A1=4ππ. 3. Simplifying the expression, the probability is found to be 41.
Final Answer The probability that the point has been taken from the smaller circle is 41.
Explanation
1. The equation of the first circle is given as x2+y2−2y=0. This equation can be rewritten by completing the square for the y terms: x2+(y2−2y+1)=1, which simplifies to x2+(y−1)2=12. 2. The center of the first circle is (0,1) and its radius, r1, is 1. 3. The equation of the second circle is given as x2+y2−2y−3=0. This equation can be rewritten by completing the square for the y terms: x2+(y2−2y+1)=3+1, which simplifies to x2+(y−1)2=4. 4. The center of the second circle is (0,1) and its radius, r2, is 4=2. 5. The larger circle is the second circle with radius r2=2, and the smaller circle is the first circle with radius r1=1.
Areas of the Circles
1. The area of the smaller circle, A1, is calculated using the formula A=πr2: A1=π(1)2=π. 2. The area of the larger circle, A2, is calculated using the formula A=πr2: A2=π(2)2=4π.
Probability Calculation
1. The probability that a point chosen at random inside the larger circle is also inside the smaller circle is given by the ratio of the area of the smaller circle to the area of the larger circle. 2. The probability, P, is calculated as P=A2A1=4ππ. 3. Simplifying the expression, the probability is found to be 41.
Final Answer The probability that the point has been taken from the smaller circle is 41.
