Class XI XII JEE Applications of derivatives - Part 5

Описание: For Class XI, XII, JEE, BITSAT, Engineering Entrance.
𝒈(𝒙)=∫_𝒂^𝒃▒〖𝒉(𝒙)〗 𝒅𝒙⇒𝒈^′ (𝒙)=𝒃^′ 𝒉(𝒃)−𝒂^′ 𝒉(𝒂)
(𝑵𝒆𝒘𝒕𝒐𝒏 𝑳𝒆𝒊𝒃𝒘𝒊𝒓𝒕𝒛 𝒓𝒖𝒍𝒆)

[𝐽𝐸𝐸 2025] 𝐴 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐ℎ𝑜𝑐𝑜𝑙𝑎𝑡𝑒 𝑏𝑎𝑙𝑙 ℎ𝑎𝑠 𝑎 𝑙𝑎𝑦𝑒𝑟 𝑜𝑓 𝑖𝑐𝑒−𝑐𝑟𝑒𝑎𝑚 𝑜𝑓 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑎𝑟𝑜𝑢𝑛𝑑 𝑖𝑡. 𝑊ℎ𝑒𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑐𝑒−𝑐𝑟𝑒𝑎𝑚 𝑙𝑎𝑦𝑒𝑟 𝑖𝑠 1 𝑐𝑚 , 𝑡ℎ𝑒 𝑖𝑐𝑒−𝑐𝑟𝑒𝑎𝑚 𝑚𝑒𝑙𝑡𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 81 𝑐𝑐/𝑚𝑖𝑛. 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑐𝑒−𝑐𝑟𝑒𝑎𝑚 𝑙𝑎𝑦𝑒𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 1/4𝜋.𝑐𝑚/𝑚𝑖𝑛. 𝑇ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 (𝑖𝑛 𝑐𝑚^2 ) 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑜𝑐𝑜𝑙𝑎𝑡𝑒 𝑏𝑎𝑙𝑙 (𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑡ℎ𝑒 𝑖𝑐𝑒−𝑐𝑟𝑒𝑎𝑚 𝑙𝑎𝑦𝑒𝑟) 𝑖𝑠

[𝐽𝐸𝐸 2025] 𝐿𝑒𝑡 𝑓:𝑅→𝑅 𝑏𝑒 𝑎 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 4, ℎ𝑎𝑣𝑖𝑛𝑔 𝑒𝑥𝑡𝑟𝑒𝑚𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑡 𝑥=4 𝑎𝑛𝑑 𝑥=5. 𝐼𝑓 lim┬(𝑥→0)⁡〖𝑓(𝑥)/𝑥^2 =5, 𝑡ℎ𝑒𝑛 𝑓(2)=?〗

[𝐽𝐸𝐸 2025] 𝑇ℎ𝑒 𝑠ℎ𝑜𝑟𝑡𝑒𝑠𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒𝑠 𝑦^2=8𝑥 𝑎𝑛𝑑 𝑥^2+𝑦^2+12𝑦+35=0 𝑖𝑠

[𝐽𝐸𝐸 2025] 𝐿𝑒𝑡 𝑓:𝑅→𝑅 𝑏𝑒 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓(𝑥)=|(|𝑥+2|−2|𝑥|)|. 𝐼𝑓 𝑚 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑓 𝑙𝑜𝑐𝑎𝑙 𝑚𝑖𝑛𝑖𝑚𝑎 𝑎𝑛𝑑 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑓 𝑙𝑜𝑐𝑎𝑙 𝑚𝑎𝑥𝑖𝑚𝑎 𝑜𝑓 𝑓, 𝑡ℎ𝑒𝑛 𝑚+𝑛=?

[𝐽𝐸𝐸 2025] 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 𝑅=𝑅={(𝑥,𝑦):𝑥≤ 𝑦≤ 9−(11𝑥^2)/3,𝑥≥0}. 𝑇ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑎𝑟𝑔𝑒𝑠𝑡 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑎𝑥𝑒𝑠 𝑎𝑛𝑑 𝑖𝑛𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑖𝑛 𝑅 𝑖𝑠

[𝐽𝐸𝐸 2025] 𝐿𝑒𝑡 𝑓(𝑥)= ∫24_0^(𝑥^2)▒(𝑡^2−8𝑡+15)/𝑒^𝑡 𝑑𝑡, 𝑥∈𝑅. 𝑡ℎ𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑜𝑐𝑎𝑙 𝑚𝑎𝑥𝑖𝑚𝑎 𝑎𝑛𝑑 𝑚𝑖𝑛𝑖𝑚𝑎 𝑜𝑓 𝑓 𝑎𝑟𝑒