Assignment-1 (BAS-103) 2024-25.pdf

Name: ………………………………… Admission No: ………………………. ABES Institute of Technology, Ghaziabad (SET-B) Subject Code: BAS103...

Name: ………………………………… Admission No: ………………………. ABES Institute of Technology, Ghaziabad (SET-B) Subject Code: BAS103 Subject Name: Engineering Mathematics-I SECTION-A Year - 1st, Branch-All Q.1 Attempt one Questions. (1×1=1) CO 1st ASSIGNMENT (ODD SEMESTER 2024-25) a. 𝑛 2 Find 𝐷 log (𝑥 + 3𝑥 + 2). 2 [Time: 1 Hours] [Total Marks: 10] Evaluate 𝑦𝑛 𝑖𝑓 𝑦 = 𝑒𝑥 𝑠𝑖𝑛2 2𝑥. b. 2 COURSE OUTCOMES CO Statements SECTION-B Remember the concept of differentiation to find successive differentiation, Leibnitz Theorem, Q.2 Attempt two Questions. (2x3=6) CO 2 and create curve tracing, and find partial and total derivatives. 𝑦 𝑥 𝑚 If cos−1 = log ( ) , then apply Leibnitz Theorem to obtain the relation a. 𝑏 𝑚 2 𝑥 2 𝑦𝑛+2 + (2𝑛 + 1)𝑥 𝑦𝑛+1 + (𝑛2 + 𝑚2 )𝑦𝑛 = 0. (SET-A) SECTION-A b. If 𝑦 = (𝑥 2 − 1)𝑛 then prove that (𝑥 2 − 1)𝑦𝑛+2 + 2𝑥 𝑦𝑛+1 − 𝑛(𝑛 + 1)𝑦𝑛 = 0. 2 Q.1 Attempt one Questions. (1×1=1) CO a. Find 𝐷𝑛 log (2𝑥 2 + 𝑥 3 ). 2 2 c. If 𝑦 = [log (𝑥 + √1 + 𝑥 2 )] 𝑡ℎ𝑒𝑛 find 𝑦𝑛 at x=0 by Leibnitz Theorem. 2 Evaluate 𝑦𝑛 𝑖𝑓 𝑦 = 𝑒 2𝑥 𝑐𝑜𝑠 2 2𝑥. b. 2 SECTION-C SECTION-B Q.3 Attempt one Questions. (1x3=3) CO Q.2 Attempt two Questions. (2x3=6) CO 2𝑥 If 𝑦 = tan−1 ( ) then show that 1−𝑥 2 If 𝑦 = sin log(𝑥 2 + 2𝑥 + 1), calculate the value of the relation a. 1 2 a. 2 𝑦𝑛 = 2 (−1)𝑛−1 (n − 1)! 𝑠𝑖𝑛𝑛 θ sin 𝑛θ where θ = tan−1 ( ). (1 + 𝑥)2 𝑦𝑛+2 + (2𝑛 + 1)(1 + 𝑥)𝑦𝑛+1 + (𝑛2 + 4)𝑦𝑛. 𝑥 If 𝑦 = 𝑎 cos(log 𝑥) + 𝑏 sin(𝑙𝑜𝑔𝑥) then b. 2 b. Obtain nth order derivative 𝑦𝑛 , for the function 𝑦 = 𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠 3 𝑥. 2 show that 𝑥 2 𝑦𝑛+2 + (2𝑛 + 1)𝑥 𝑦𝑛+1 + (𝑛2 + 1)𝑦𝑛 = 0. 𝑚 c. If 𝑦 = [𝑥 + √1 + 𝑥 2 ] 𝑡ℎ𝑒𝑛 find 𝑦𝑛 (0) by Leibnitz Theorem. 2 SECTION-C Q.3 Attempt one Questions. (1x3=3) CO 𝑥 If 𝑦 = tan−1 ( ) then show that 𝑎 a. 𝑎 2 𝑦𝑛 = (−1)𝑛−1 (n − 1)! 𝑎−𝑛 𝑠𝑖𝑛𝑛 θ sin 𝑛θ where θ = tan−1 ( ). 𝑥 b. Obtain nth order derivative 𝑦𝑛 , for the function 𝑦 = 𝑠𝑖𝑛3 𝑥 𝑐𝑜𝑠 2 𝑥. 2