Engineering Mathematics - Assignment 1 PDF

B. Tech. First Year Engineering Mathematics – I(Subject Code: BAS103) Unit-1: Matrices Assignment - I UNIT - I 1. Determine eigen values & eigen vectors of the following matrices −2 2 −3...

B. Tech. First Year Engineering Mathematics – I(Subject Code: BAS103) Unit-1: Matrices Assignment - I UNIT - I 1. Determine eigen values & eigen vectors of the following matrices −2 2 −3 3 1 4 1 1 3 (𝐚) [ 2 1 −6] [2010, 2011] b) [0 2 6] c) [1 5 1] −1 −2 0 0 0 5 3 1 1 3 −2 1 1 1 3 Answers: (a) -3, -3, 5; 𝑘1 + 𝑘2 [ 1 ], 𝑘3 [ 2 ], (b) 3, 2, 5; 𝑘1 , 𝑘2 [−1], 𝑘3 1 0 −1 0 0 1 −1 1 1 (c) -2, 6, 3; 𝑘1 [ 0 ] , 𝑘2 [ 2 ], 𝑘3 [−1] 1 1 1 2 −1 1 2. Verify Cayley-Hamilton theorem for the matrix A = [−1 2 −1]. Hence compute 1 −1 2 A-1. Also evaluate 𝐴6 − 6𝐴5 + 9𝐴4 − 2𝐴3 − 12𝐴2 + 23𝐴 − 9𝐼 (A.K.T.U. 2012, 2016, 2018) 3 1 −1 9 −5 5 1 Answer: A-1 = 4 [ 1 3 1 ], 5𝐴 − 𝐼 = [−5 9 −5] −1 1 3 5 −5 9 3. Prove that the following matrices are unitary 1 1 1 1 1 1 + 𝑖 −1 + 𝑖 (i) [1 𝜔 𝜔 2 ] (ii) 2 [ ] √3 2 1+𝑖 1−𝑖 1 𝜔 𝜔 UNIT - II 4. If 𝑦 = (sin−1 𝑥)2 , Prove that (1 − 𝑥 2 )𝑦𝑛+2 − (2𝑛 + 1)𝑥𝑦𝑛+1 − 𝑛2 𝑦𝑛 = 0. Also find the value of 𝑛𝑡ℎ derivative of y for x = 0. [2014, 18] 0 ; When n is odd Ans. { 2 2 2 2. 2. 4. 6 … … (𝑛 − 2)2 ; When n is even. 𝑚 5. If 𝑦 = [𝑥 + √1 + 𝑥 2 ] , find (𝑦𝑛 )0 [2013, 18] 2 2 2 2 2 2 2 Answer. If n is even (𝑦𝑛 )0 = 𝑚 (𝑚 − 2 )(𝑚 − 4 ) + ⋯ + (𝑚 − (𝑛 − 2) ), If n is odd (𝑦𝑛 )0 = 𝑚(𝑚2 − 12 )(𝑚2 − 32 ) + ⋯ + (𝑚2 − (𝑛 − 2)2 ) 𝜕2 𝑢 𝜕2 𝑢 1 6. If 𝑢 = 𝑓(𝑟) Where 𝑟 2 = 𝑥 2 + 𝑦 2 , Prove that 𝜕𝑥 2 + 𝜕𝑦2 = 𝑓 ′′ (𝑟) + 𝑟 𝑓 ′ (𝑟) 1⁄ 1⁄ 1⁄ 2 𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢 −1 𝑥 2 +𝑦 2 𝑡𝑎𝑛𝑢 7. (i) If 𝑢 = 𝑐𝑜𝑠𝑒𝑐 ( 1⁄ 1⁄ ) , Prove that 𝑥 2 𝜕𝑥 2 + 2𝑥𝑦 𝜕𝑥𝜕𝑦 + 𝑦 2 𝜕𝑦2 = 144 (13 + 𝑡𝑎𝑛2 𝑢). 𝑥 3 +𝑦 3 −1 𝑥+𝑦 8. If 𝑢 = 𝑠𝑖𝑛 ( 𝑥+ 𝑦), Prove that √ √ 𝜕𝑢 𝜕𝑢 1 𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢 𝑠𝑖𝑛𝑢 𝑐𝑜𝑠2𝑢 (i) 𝑥 + 𝑦 = 𝑡𝑎𝑛𝑢 (ii) 𝑥 2 + 2𝑥𝑦 + 𝑦2 =−. 𝜕𝑥 𝜕𝑦 2 𝜕𝑥 2 𝜕𝑥𝜕𝑦 𝜕𝑦 2 4𝑐𝑜𝑠3 𝑢 UNIT – III 9. Expand 𝑓(𝑥, 𝑦) = 𝑒 𝑥 log(1 + 𝑦) 𝑖𝑛 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑥 𝑎𝑛𝑑 𝑦 𝑢𝑝𝑡𝑜 𝑑𝑒𝑔𝑟𝑒𝑒 3.  1 1 1 1 Ans. 𝑦 + 𝑥𝑦 − 2 𝑦 2 + 2 𝑥 2 𝑦 − 2 𝑥𝑦 2 + 3 𝑦 3 + ⋯ … … 10. Expand 𝑥 𝑦 in power of (𝑥 − 1) and (𝑦 − 1) up to the third-degree terms and hence evaluate (1.1)1.02 11. Expand sin 𝑥𝑦 in power of (𝑥 − 1) and (𝑦 − 𝜋⁄2) upto second degree 𝜋2 𝜋 𝜋 1 𝜋 2 Ans. 1 − (𝑥 − 1)2 − (𝑥 − 1) (𝑦 − ) − (𝑦 − ) − ⋯ … … … … 8 2 2 2 2 12. If 𝑥 + 𝑦 + 𝑧 = 𝑢3 + 𝑣 3 + 𝑤 3 , 𝑥 3 + 𝑦 3 + 𝑧 3 = 𝑢2 + 𝑣 2 + 𝑤 2 , 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑢 + 𝑣 + 𝑤 𝜕(𝑢,𝑣,𝑤) (𝑥−𝑦)(𝑦−𝑧)(𝑧−𝑥) then show that 𝜕(𝑥,𝑦,𝑧) = (𝑢−𝑣)(𝑣−𝑤)(𝑤−𝑢). 𝜕(𝑢,𝑣,𝑤) 13. If 𝑢, 𝑣, 𝑤 are the roots of the cubic (𝜇 − 𝑥)3 + (𝜇 − 𝑦)3 + (𝜇 − 𝑧)3 = 0 in µ, find 𝜕(𝑥,𝑦,𝑧). 2(𝑥−𝑦)(𝑦−𝑧)(𝑧−𝑥) Ans. − (𝑢−𝑣)(𝑣−𝑤)(𝑤−𝑢) 𝜕(𝑢,𝑣,𝑤) 1 14. If 𝑢 = 𝑥 + 𝑦 + 𝑧, 𝑢𝑣 = 𝑦 + 𝑧, 𝑢𝑣𝑤 = 𝑧. 𝑓𝑖𝑛𝑑 Ans. 2 𝜕(𝑥,𝑦,𝑧) 𝑢 𝑣 UNIT - IV 3 15. Evaluate ∬ 𝑥𝑦(𝑥 + 𝑦)𝑑𝑥𝑑𝑦 over the area between 𝑦 = 𝑥 2 and 𝑦 = 𝑥 [2011, 12] Ans. 56 16. Evaluate ∬𝐴 𝑥𝑦(𝑥 + 𝑦)𝑑𝑥𝑑𝑦 where A is domain bounded by x-axis and ordinate 𝑥 = 2𝑎 and the 𝑎4 curve 𝑥 2 = 4𝑎𝑦. Ans. 3 1 ∞ 𝑥𝑐 𝛤(𝑐+1) 1 𝑥 2 𝑑𝑥 1 𝑑𝑥 𝜋 17. Prove that (i) 𝛤 (2) = √𝜋. (ii) ∫0 𝑐𝑥 𝑑𝑥 = (𝑙𝑜𝑔 𝑐)𝑐+1 ; 𝑐 > 1 (iii) ∫0 √(1−𝑥 4 ) × ∫0 √(1+𝑥4 ) = 4√2. ∞ 𝑥2 5𝜋√2 2 3 16𝜋 18. Prove that (i) ∫0 (1+𝑥 4 )3 𝑑𝑥 = 128 (ii) ∫0 𝑥 √8 − 𝑥 3 𝑑𝑥 = 9 3 √ UNIT – V 19. Prove that 𝒅𝒊𝒗(𝒓𝟐 𝒓 ⃗ ) = (𝒏 + 𝟑)𝒓𝒏 , further show that 𝒓𝒏 𝒓 ⃗ is solenoidal only if 𝒏 = −𝟑. ⃗ ⃗ 20. Find the divergence and curl of the vector 𝑹 = (𝒙 + 𝒚𝒛)𝒊̂ + (𝒚𝟐 + 𝒛𝒙)𝒋̂ + (𝒛𝟐 + 𝒙𝒚)𝒌 𝟐 ̂ 21. Find the constant a, b, c so that 𝑭⃗ = (𝒙 + 𝟐𝒚 + 𝒂𝒛)𝒊̂ + (𝒃𝒙 − 𝟑𝒚 − 𝒛)𝒋̂ + (𝟒𝒙 + 𝒄𝒚 + 𝟐𝒛)𝒌 ̂ is 𝒙𝟐 𝟑𝒚𝟐 irrotational. If ⃗𝑭 = 𝒈𝒓𝒂𝒅∅, show that ∅ = − + 𝒛𝟐 + 𝟐𝒙𝒚 + 𝟒𝒙𝒛 − 𝒚𝒛. [2014, 18] 𝟐 𝟐 Ans. 𝒂 = 𝟒, 𝒃 = 𝟐, 𝒄 = −𝟏 22. Find the directional derivative of ∅ = 𝒙𝒚𝟐 + 𝒚𝒛𝟑 at the point P(2, -1, 1) in the direction of the normal to the surface 𝒙 𝐥𝐨𝐠 𝒛 − 𝒚𝟐 + 𝟒 = 𝟎 𝒂𝒕 (𝟐, −𝟏, 𝟏). Ans: -𝟑√𝟐 23. If 𝒓 ̂, show that (i) 𝒅𝒊𝒗 𝒓 ⃗ = 𝒙𝒊̂ + 𝒚𝒋̂ + 𝒛𝒌 ⃗ =𝟑 ⃗ =𝟎 (ii) curl 𝒓 24. Prove that div(grad𝒓𝒏 ) = 𝛁 𝟐 (𝒓𝒏 ) = 𝒏(𝒏 + 𝟏)𝒓𝒏−𝟐 where 𝒓 ̂. Hence show that ⃗ = 𝒙𝒊̂ + 𝒚𝒋̂ + 𝒛𝒌 𝟏 𝛁 𝟐 𝒓 = 0. [2014,2012]