Directions: Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. What is the area of the circle ? I. An arc of length 4 cm subtends an angle of $60^{\circ}$ at the centre. II. A chord of length 5 cm subtends an angle of $90^{\circ}$ at the centre.

### Concept & Circle Geometry To find the area of a circle, we only need to determine its radius ($r$). The length of an arc is given by $l = \frac{\theta}{360^{\circ}} \times 2\pi r$. A chord subtending $90^{\circ}$ at the center forms a right-angled isosceles triangle with the two radii. ### Step-by-Step Solution * **Analyze Statement I:** Arc length = 4 cm, $\theta = 60^{\circ}$. * Using the arc length formula: $4 = \frac{60}{360} \times 2\pi r \implies 4 = \frac{1}{6} \times 2\pi r$. This allows us to solve for a unique value of $r$. Hence, I alone is sufficient. * **Analyze Statement II:** Chord length = 5 cm, $\theta = 90^{\circ}$. * The chord and the two radii form a right triangle where the chord is the hypotenuse. * Using Pythagoras theorem: $r^2 + r^2 = 5^2 \implies 2r^2 = 25 \implies r^2 = 12.5$. * Since Area = $\pi r^2$, we can directly find the area ($12.5\pi$). II alone is also sufficient. ### Exam Strategy & Shortcut If a statement provides a linear dimension (arc length, chord length) along with the corresponding central angle, the size of the circle is completely locked in. Therefore, the radius and area can always be found. ### Common Pitfall Wasting time actually calculating the final area. In Data Sufficiency, once you establish that an equation yields exactly one valid solution for the target variable, stop calculating. ### Final Answer Therefore, the correct answer is **if the data either in Statement I or in Statement II alone are sufficient to answer the question**.