Engineering Mathematics I Tutorial & Assignment 2024-2025 PDF

Sinhgad Technical Education Society's SMT. KASHIBAI NAVALE COLLEGE OF ENGINEERNG, PUNE – 411041 Department of Applied Science Academic Year: 2024 – 2025(Semester-I)...

Sinhgad Technical Education Society's SMT. KASHIBAI NAVALE COLLEGE OF ENGINEERNG, PUNE – 411041 Department of Applied Science Academic Year: 2024 – 2025(Semester-I) Tutorial-II Academic Year: 2024–25 Subject: Engineering Mathematics I [Semester: I] Date: Class: FE Batch: All 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣 If 𝑥 = 𝑢𝑡𝑎𝑛𝑣; 𝑦 = 𝑢𝑠𝑒𝑐𝑣, 𝑡𝑕𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡𝑕𝑎𝑡 = 1 𝜕𝑥 𝑦 𝜕𝑥 𝑦 𝜕𝑦 𝑥 𝜕𝑦 𝑥 𝐼𝑓 𝑥 2 = 𝑎 𝑢 + 𝑏 𝑣 𝑎𝑛𝑑 𝑦 2 = 𝑎 𝑢 − 𝑏 𝑣 where a & b are constants Prove that 𝜕𝑢 𝜕𝑥 1 𝜕𝑣 𝜕𝑦 2 = = 𝜕𝑥 𝑦 𝜕𝑢 𝑣 2 𝜕𝑦 𝑥 𝜕𝑣 𝑢 3 𝜕2𝑧 𝜕2𝑧 If 𝑧 = tan 𝑦 + 𝑎𝑥 + 𝑦 − 𝑎𝑥 2 ,then find the value of − 𝑎2 𝜕𝑦 2 3 𝜕𝑥 2 𝑥3+ 𝑦 3 𝜕2𝑢 𝜕2𝑢 𝜕2𝑢 If 𝑢 = 𝑡𝑎𝑛−1 then prove that𝑥 2 𝜕𝑥 2 + 2𝑥𝑦 + 𝑦 2 𝜕𝑦 2 = −𝑠𝑖𝑛2𝑢 𝑠𝑖𝑛2 𝑢. 4 𝑥+ 𝑦 𝜕𝑥𝜕𝑦 𝑥 +𝑦 𝜕2𝑢 𝜕2𝑢 𝜕2𝑢 −𝑠𝑖𝑛 𝑢 𝑐𝑜𝑠 2𝑢 𝐼𝑓 𝑢 = 𝑠𝑖𝑛−1 ,Prove that 𝑥 2 𝜕𝑥 2 + 2𝑥𝑦 + 𝑦 2 𝜕𝑦 2 = 𝑥+ 𝑦 𝜕𝑥𝜕𝑦 4 𝑐𝑜𝑠 3 𝑢 5 𝑥𝑦 𝑥𝑦2 If 𝑢 = cos + 𝑥2 + 𝑦2 + then find the value of 𝑥 𝑢𝑥 + 𝑦 𝑢𝑦 𝑎𝑡 3,4 𝑥2+ 𝑦2 𝑥 +𝑦 6 𝜕𝑢 𝜕2𝑢 𝑖𝑓 𝑢 = 4𝑒 −6𝑥 sin⁡( 𝑝𝑡 − 6𝑥) satisfies the partial differential equation = 𝜕𝑡 𝜕𝑥 2 7 then find value of p. 𝜕 𝜕 2 −4 If 𝑢 = 𝑙𝑜𝑔 𝑥 3 + 𝑦 3 − 𝑥 2 𝑦 − 𝑥𝑦 2 𝑝𝑟𝑜𝑣𝑒 𝑡𝑕𝑎𝑡 + 𝑢 = 8 𝜕𝑥 𝜕𝑦 𝑥+𝑦 2 If 𝑧 = 𝑓 𝑥, 𝑦 𝑤𝑕𝑒𝑟𝑒 𝑥 = 𝑒 𝑢 cos 𝑣 & 𝑦 = 𝑒 𝑢 𝑠𝑖𝑛𝑣 then 𝜕𝑧 𝜕𝑧 𝜕𝑧 9 prove that 𝑦 𝜕𝑢 + 𝑥 𝜕𝑣 = 𝑒 2𝑢. 𝜕𝑦 If x = 𝑢 + 𝑣 + 𝑤; 𝑦 = 𝑢𝑣 + 𝑣𝑤 + 𝑢𝑤, 𝑧 = 𝑢𝑣𝑤 ,∅ is a function of x,y,z then prove that 10 𝜕∅ 𝜕∅ 𝜕∅ 𝜕∅ 𝜕∅ 𝜕∅ 𝑢 +𝑣 + 𝑤 =𝑥 + 2𝑦 + 3𝑧 𝜕𝑢 𝜕𝑣 𝜕𝑤 𝜕𝑥 𝜕𝑦 𝜕𝑧  Sinhgad Technical Education Society's SMT. KASHIBAI NAVALE COLLEGE OF ENGINEERNG, PUNE – 411041 Department of Applied Science Academic Year: 2024 – 2025(Semester-I) Assignment-II Academic Year: 2024–25 Subject: Engineering Mathematics I [Semester: I] Date: Class: FE Batch: All 𝜕𝑦 𝜕𝑣 𝜕𝑢 𝜕𝑥 If 𝑢 = 2𝑥 + 3𝑦, 𝑣 = 3𝑥 − 2𝑦. Find the value of , , , 1 𝜕𝑣 𝑥 𝜕𝑦 𝑢 𝜕𝑥 𝑦 𝜕𝑢 𝑣 𝐼𝑓 𝑥 2 = 𝑎 𝑢 + 𝑏 𝑣 𝑎𝑛𝑑 𝑦 2 = 𝑎 𝑢 − 𝑏 𝑣 where a & b are constants Prove that 𝜕𝑢 𝜕𝑥 1 𝜕𝑣 𝜕𝑦 2 = = 𝜕𝑥 𝑦 𝜕𝑢 𝑣 2 𝜕𝑦 𝑥 𝜕𝑣 𝑢 𝑥 +𝑦 𝜕𝑢 𝜕𝑢 If 𝑢 = 𝑠𝑖𝑛−1 then prove that 2𝑥 + 2𝑦 = tan 𝑢 3 𝑥+ 𝑦 𝜕𝑥 𝜕𝑦 𝑥𝑦𝑧 𝑥2+ 𝑦2+ 𝑧2 𝜕𝑢 𝜕𝑢 𝜕𝑢 If 𝑢 = 2𝑥+𝑦+𝑧 + 𝑙𝑜𝑔 find 𝑥 + 𝑦 + 𝑧 𝜕𝑧 𝑥𝑦 +𝑦𝑧 𝜕𝑥 𝜕𝑦 4 𝑥3+ 𝑦 3 𝜕2𝑢 𝜕2𝑢 𝜕2𝑢 If 𝑢 = 𝑡𝑎𝑛−1 then prove that𝑥 2 𝜕𝑥 2 + 2𝑥𝑦 + 𝑦 2 𝜕𝑦 2 = −𝑠𝑖𝑛2𝑢 𝑠𝑖𝑛2 𝑢. 5 𝑥+ 𝑦 𝜕𝑥𝜕𝑦 𝑥 +𝑦 𝜕2𝑢 𝜕2𝑢 𝜕2𝑢 −𝑠𝑖𝑛𝑢 𝑐𝑜𝑠 2𝑢 𝐼𝑓 𝑢 = 𝑠𝑖𝑛−1 ,Prove that 𝑥 2 𝜕𝑥 2 + 2𝑥𝑦 + 𝑦 2 𝜕𝑦 2 = 6 𝑥+ 𝑦 𝜕𝑥𝜕𝑦 4 𝑐𝑜𝑠 3 𝑢 𝑥 𝑦 𝑧 𝜕𝑢 𝜕𝑢 𝜕𝑢 If 𝑢 = 𝑓 , , 𝑦 𝑧 𝑥 then prove that 𝑥 + 𝑦 + 𝑧 𝜕𝑧 = 0 7 𝜕𝑥 𝜕𝑦 If 𝑢 = 𝑥 2 − 𝑦 2 ,𝑉 = 2𝑥𝑦 and 𝑍 = 𝑓(𝑥, 𝑦) then show that 𝜕𝑧 𝜕𝑧 𝜕𝑧 8 𝑥 𝜕𝑥 − 𝑦 𝜕𝑦 = 2 𝑢2 + 𝑣 2 𝜕𝑢 1 If 𝑢 = 𝑓 𝑟 where 𝑟 = 𝑥 2 + 𝑦 2 then prove that 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 𝑓 ′′ 𝑟 + 𝑓 ′ (𝑟) 9 𝑟 𝜕𝑣 𝜕𝑣 𝜕𝑣 If v = f (𝑒 𝑥−𝑦 , 𝑒 𝑦 −𝑧 , 𝑒 𝑧−𝑥 ), then show that + 𝜕𝑦 + 𝜕𝑧 = 0 10 𝜕𝑥