Using integration, find the area of the ellipse:
\[
\frac{x^2}{16} + \frac{y^2}{4} = 1,
\]
included between the lines \(x = -2\) and \(x = 2\).
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Solution and Explanation
The equation of the ellipse is: \[ \frac{x^2}{16} + \frac{y^2}{4} = 1. \]
Rearrange to solve for \(y^2\): \[ \frac{y^2}{4} = 1 - \frac{x^2}{16}. \] \[ y^2 = 4\left(1 - \frac{x^2}{16}\right) = 4 - \frac{x^2}{4}. \] \[ y = \pm \sqrt{4 - \frac{x^2}{4}}. \]
Step 1: Use symmetry to simplify the calculation. The ellipse is symmetric about the \(x\)-axis. The area between \(x = -2\) and \(x = 2\) can be calculated as twice the area above the \(x\)-axis: \[ \text{Area} = 2 \int_{-2}^{2} \sqrt{4 - \frac{x^2}{4}} \, dx. \]
Step 2: Change the limits and integrate. Since the integrand is even (symmetric about the \(y\)-axis), we can further simplify: \[ \text{Area} = 4 \int_{0}^{2} \sqrt{4 - \frac{x^2}{4}} \, dx. \]
Step 3: Substitution for simplification.
Let: \[ u = 4 - \frac{x^2}{4}, \quad \text{so} \quad du = -\frac{x}{2} \, dx \quad \text{and} \quad x \, dx = -2 \, du. \] When \(x = 0\), \(u = 4\), and when \(x = 2\), \(u = 4 - \frac{2^2}{4} = 3\).
The integral becomes: \[ \int_{0}^{2} \sqrt{4 - \frac{x^2}{4}} \, dx = \int_{4}^{3} \sqrt{u} \cdot (-2) \, du. \] Simplify: \[ \int_{0}^{2} \sqrt{4 - \frac{x^2}{4}} \, dx = 2 \int_{3}^{4} \sqrt{u} \, du. \]
Step 4: Evaluate the integral. The integral of \(\sqrt{u}\) is: \[ \int \sqrt{u} \, du = \frac{2}{3} u^{3/2}. \] Evaluate from \(u = 3\) to \(u = 4\): \[ \int_{3}^{4} \sqrt{u} \, du = \frac{2}{3} \left[4^{3/2} - 3^{3/2}\right]. \]
Simplify: \[ 4^{3/2} = (2^2)^{3/2} = 2^3 = 8, \quad 3^{3/2} = \sqrt{3^3} = \sqrt{27}. \]
Thus: \[ \int_{3}^{4} \sqrt{u} \, du = \frac{2}{3} \left[8 - \sqrt{27}\right]. \]
Step 5: Final area. Substitute back into the expression for the area: \[ \text{Area} = 4 \cdot 2 \cdot \frac{2}{3} \left[8 - \sqrt{27}\right] = \frac{16}{3} \left[8 - \sqrt{27}\right]. \]
Final Answer: \[ \text{Area} = \frac{16}{3} \left[8 - \sqrt{27}\right]. \]
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- Letter Writing
Read the following passage carefully:
(1) Gossip. All humans partake in some form, despite the age-old adage, "If you have nothing nice to say, don't say anything at all." Whether it's workplace chatter, the sharing of family news or group texts between friends, it's inevitable that anyone who participates in the above, talks about other people.
(2) People tend to think of gossip as synonymous with malicious rumours, put-downs, or the breathless propagation of a tabloid scoop. But researchers often define it more broadly as "talking about people who aren't present," says an assistant professor of psychology. "It's something that comes very naturally to us" - an integral part of conversation, information sharing, and even community building.
(3) "It's not necessarily negative," adds a professor of psychology. "It can be positive or neutral." Some researchers argue that gossip helped our ancestors survive. Evolutionary psychologists first pioneered this idea, comparing gossip to the grooming primates engage in as a means of bonding.
(4) People are usually resistant to thinking about gossip as anything but bad. It is true that there are some types of gossip that should be avoided, such as gossip that is purely harmful and serves no purpose like mean comments on someone's way of dressing. In such a scenario no one benefits. Gossip can also be both active and passive. Active gossiping refers to directly sharing negative information about someone else. An active gossiper initiates the conversation and actively spreads it. Passive gossiping on the other hand involves more subtle behaviour like hinting, dropping suggestive comments or participating in gossip by listening without actively contributing, but still allowing the gossip to spread through their presence.
(5) It's interesting to note that gossiping also says something about relationships people have with one another. The act of gossiping often implies a level of trust and closeness with someone, which are key aspects of a good friendship.
(6) Research has found that gossip can stave off loneliness, while other studies have found it can facilitate bonding and closeness and serve as a form of entertainment. So, keep on talking. And when your conversation turns to gossip, as it inevitably will, remember that some good can come of it - with the right intentions, of course.
Answer the following questions, based on the above passage:- CBSE CLASS XII - 2025
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Read the following passage carefully:
(1) Introduction: Humans are social creatures with a deep desire to interact with one another. Recent developments in technology have made it possible for billions of people worldwide to use cell phones to satisfy this need. The current study aims to analyse the impact of smartphone use among various categories of people in Kerala, India. The present study will provide an insight into the extent and magnitude of people's reliance on smartphones to communicate in this modern age and the dwindling nature of social relationships.
(2) Methodology: A google form created questionnaire was used to gather information on the subject from a sample of participants who were from various districts in Kerala, India.
(3) Survey Examination: Among the 131 study participants, 66 were men and the remaining 65 were women. For convenience, the study considered only the main purpose for which a smartphone is used by any given individual.
(4) Results:
Individuals differed in their interests and preferences regarding their choice of communication medium. Some preferred direct conversation, while others conveyed their messages through virtual mediums of communication, that did not involve any face-to-face interaction.
(5) Implications for Interventions: It was observed that individuals were interested in using their smartphones even when they were dining. This constant use can have several disadvantages such as reckless use of mobile phones, including prolonged hours of radiation exposure, over-calorie intake due to binge eating, leading to obesity, and other health problems. It is also a vital tool to achieve social progress.
The use of phones while travelling can sometimes detract from an individual's enjoyment of the serenity of the environment, the scenic beauty of picnic spots, and affect the way an individual behaves and interacts with his or her co-travellers.
(6) Conclusion: It is true that most of the population under different age and gender categories use smartphones every day, but there is no significant fall in the duration spent in direct face-to-face conversation with fellow beings, except in very few individuals. The majority of people agree that traditional face-to-face communication is more comfortable, reliable, and expressive than text messages, phone calls, or electronic mail.
Answer the following questions, based on the above passage:- CBSE CLASS XII - 2025
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Read the following extracts and answer the questions
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Find even the ivory needle hard to pull.
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