ExitExam+ESSLCE Support

📌Call for Bussiness Incubation ✅Anyone can apply to this competition

📝Solution1⃣2️⃣ To determine when the particle changes direction, we need to find the values of t where the velocity function changes sign. Velocity is the derivative of position, so we need to find the zeros of the derivative of x(t). The velocity function is given by v(t) = x'(t) = 6t^2 - 6t - 12. To find the zeros of v(t), we set v(t) = 0 and solve for t: 6t^2 - 6t - 12 = 0. Dividing through by 6, we get: t^2 - t - 2 = 0. Factoring the quadratic equation, we have: (t - 2)(t + 1) = 0. Setting each factor equal to zero, we find t = 2 and t = -1 as the zeros of v(t). Since we are interested in the time when the particle changes direction, we discard the negative value (-1) as it is not within the given time frame. Therefore, the particle changes direction at t = 2. The correct answer is c) t = 2. 🙏Join, Share👇👇 https://t.me/tutorialpointeth

✅️Question👆👆 (#Application of #Derivative) A rectangular box with a square base is to be constructed with a fixed volume of 64 cubic units. What dimensions should be used to minimize the surface area? 📝Solution: Let the length, width, and height of the box be x, x, and h, respectively(bc the base is square). The volume of the box is given by V = x^2 * h = 64. We need to minimize the surface area, which is given by S = 2x^2 + 4xh. From the volume equation, we can express h in terms of x as: h = 64 / x^2. Substituting this expression for h into the surface area equation, we get: S = 2x^2 + 4x(64 / x^2) = 2x^2 + 256 / x. To minimize the surface area, we take the derivative of S with respect to x and set it equal to zero: dS/dx = 4x - 256 / x^2 = 0. Simplifying the equation, we have: 4x^3 - 256 = 0, x^3 - 64 = 0, (x - 4)(x^2 + 4x + 16) = 0. The only real solution is x = 4. Substituting this value back into the volume equation, we find h = 4. Therefore, the correct answer is b) 4 units by 4 units by 4 units. 🙏Join, Share👇👇 https://t.me/tutorialpointeth

✅️Solution 8️⃣ To find the derivative of sqrt(4x + 3), we use the chain rule. The chain rule states that the derivative of 📚 f(g(x)) is f'(g(x)) * g'(x)📚 Applying the chain rule to our function, we have: f(x) = sqrt(4x+3), g(x) = 4x + 3. Differentiating f(x) and g(x), we get: f'(x) = 1 / (2sqrt(4x+3)), g'(x) = 4. Substituting these values into the chain rule formula, we obtain: (1 / (2sqrt(4x + 3))) * 4, which simplifies to: 2 / sqrt(4x + 3). Therefore, the correct answer is b) 2 / sqrt(4x + 3).

✅QUESTION 6 What is the limit of the function f(x) = (2x^3 - 5x^2 + x + 3) / (x^3 + 4x^2 + 1) as x approaches -1? 📝Solution To find the limit of the function f(x) as x approaches -1, we can substitute -1 for x in the function and simplify: f(x) = (2x^3 - 5x^2 + x + 3) / (x^3 + 4x^2 + 1) f(-1) = (2(-1)^3 - 5(-1)^2 + (-1) + 3) / ((-1)^3 + 4(-1)^2 + 1) f(-1) = (-2 - 5 - 1 + 3) / (-1 + 4 + 1) f(-1) = -5 / 4 Therefore, the limit of the function f(x) as x approaches -1 is -5/4.

Answer: d) Birds Explanation: Birds are known for exhibiting various forms of parental care, including building nests, incubating eggs, and feeding and protecting their offspring. 1️⃣2️⃣ Which of the following is a characteristic of cartilaginous fish? a) Bony skeleton b) External fertilization c) Lungs for respiration d) Placental nourishment Answer: b) External fertilization Explanation: Cartilaginous fish, such as sharks and rays, practice external fertilization, where the female releases eggs into the water and the male fertilizes them externally. 1️⃣3️⃣ The group of vertebrates that includes animals capable of echolocation is: a) Fish b) Birds c) Reptiles d) Mammals Answer: d) Mammals Explanation: Certain mammals, such as bats and dolphins, have the ability to navigate and locate objects using echolocation, emitting sounds and interpreting the echoes that bounce back. 1️⃣4️⃣Which of the following groups of vertebrates exhibits a diaphragm for respiration? a) Fish b) Birds c) Reptiles d) Mammals Answer: d) Mammals Explanation: Mammals possess a diaphragm, a muscular sheet below the lungs, which aids in respiration by facilitating inhalation and exhalation. 1️⃣5️⃣ Which of the following groups of vertebrates exhibits viviparous reproduction? a) Fish b) Amphibians c) Reptiles d) Mammals Answer: d) Mammals Explanation: Viviparous reproduction refers to giving birth to live young. Mammals, such as humans and dogs, exhibit viviparous reproduction. 1️⃣6️⃣ Which of the following is an example of a monotreme? a) Platypus b) Kangaroo c) Tiger d) Gorilla Answer: a) Platypus Explanation: Monotremes are a unique group of egg-laying mammals. The platypus is an example of a monotreme. 1️⃣7️⃣Which of the following vertebrate groups exhibits the highest diversity of species? a) Birds b) Mammals c) Fish d) Reptiles Answer: c) Fish Explanation: Fish, including both cartilaginous and bony fish, exhibit the highest diversity of species among vertebrates. 1⃣8️⃣Which of the following is a characteristic of endothermic vertebrates? a) Ectothermic metabolism b) Cold-bloodedness c) Limited activity in cold environments d) Maintenance of constant internal body temperature Answer: d) Maintenance of constant internal body temperature Explanation: Endothermic vertebrates, such as birds and mammals, can regulate their body temperature internally, regardless of the external environment. 1️⃣9️⃣Which of the following vertebrates exhibits the highest brain-to-body size ratio? a) Fish b) Reptiles c) Birds d) Primates Answer: d) Primates Explanation: Primates, including humans, have the highest brain-to-body size ratio among vertebrates, indicating a relatively larger brain in proportion to their body size. =========================== 🙏Join, Share👇👇 https://t.me/tutorialpointeth