Applied Mathematics I Question Bank 2024-25 (Annasaheb Dange College of Engineering and Technology) PDF

Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 CH.1 - RANK OF MATRIX and SYSTEM OF LINEAR EQUATION 1....

Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 CH.1 - RANK OF MATRIX and SYSTEM OF LINEAR EQUATION 1. NORMAL FORM Que. Reduce the following matrix to normal form & Find its rank 1 −1 2 −1 2 1 −1 3 8 1 1 1 −1 −2 1) A = [4 2 −1 2] 2) A=[ ] 3 2 −1 0 6 2 2 −2 0 0 4 3 2 −8 2 −4 3 1 0 0 1 −3 −1 1 −2 1 −4 2 1 0 1 1 3) A =[ ] 4) 𝐴= [ ] 0 1 −1 3 1 3 1 0 2 4 −7 4 −4 5 1 1 −2 0 1 2 3 1 2 3 −1 −2 2 4 3 2 1 −1 −2 −4 5) 𝐴=[ ] 6) 𝐴=[ ] 3 2 1 3 3 1 3 −2 6 8 7 6 6 3 0 −8 2 3 −1 −1 3 5 −2 4 −1 6 1 −1 −2 −4 −6 −2 1 1 −2 −2 7) 𝐴=[ ] 8) 𝐴=[ ] 3 1 3 −2 5 4 1 0 5 10 6 3 0 −7 2 −1 −2 5 −8 −6 6 1 3 8 1 2 1 0 4 2 6 −1 9) 𝐴=[ ] 10) 𝐴 = [−2 4 3 0] 10 3 9 7 16 4 12 15 1 0 −1 −8 1 1 1 1 1 2 3 11) 𝐴 = [1 −1 −1] 12) A = [1 2 3 2] 3 1 1 0 −1 −1 4 Ans. 1. 𝑟 = 3 2. 𝑟 = 4 3. 𝑟 = 3 4. 𝑟 = 2 5. 𝑟 = 3 6. 𝑟 = 3 7. 𝑟 = 3 8. 𝑟 = 3 9. 𝑟 = 2 10. 𝑟 = 3 11. 𝑟 = 2 12. 𝑟 = 3 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 1  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 2. ECHELON FORM: Que. Reduce the following matrix to Echelon form and hence determine the rank 0 1 −3 −1 8 1 3 6 1 0 1 1 1) 𝐴=[ ] 2) 𝐴=[ 0 3 2 2] 3 1 0 2 1 1 −2 0 −8 1 −3 4 2 3 −1 −1 1 −1 −2 −4 Ans. 1. 𝑟 = 2 2. 𝑟 = 3 3) 𝐴=[ ] 3 1 3 −2 3. 𝑟 = 4 6 3 0 −7 3. SIMULTANEOUS LINEAR EQUATIONS: a) Non Homogeneous Linear Equations Que. Test for consistency and if possible solve 1) 𝑥 + 𝑦 + 𝑧 = 3, 𝑥 + 2 𝑦 + 3𝑧 = 4, 𝑥 + 4𝑦 + 9𝑧 = 6 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 3 = 𝑛 unique solution 𝑥 = 2, 𝑦 = 1, 𝑧 = 0 2) 𝑥 + 𝑦 + 2𝑧 + 𝑤 = 5 ; 2𝑥 + 3𝑦 – 𝑧 − 2𝑤 = 2; 4𝑥 + 5𝑦 + 3𝑧 = 7 Ans. 𝜌(𝐴) = 2 ≠ 𝜌(𝐴: 𝐵) = 3 = 𝑛 inconsistent system 3) 𝑥1 + 2𝑥2 − 𝑥3 = 1, 3𝑥1 − 2𝑥2 + 2𝑥3 = 2, 7𝑥1 − 2𝑥2 + 3𝑥3 = 5 3−𝑡 1+5𝑡 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 3 = 𝑟 < 𝑛 = 3 infinite solution 𝑥 = , 𝑦= , 𝑧=𝑡 4 8 4) x + y + z = 3, 2 x - y + 3z = 1, 4x + y +5 z = 2 3x -2 y + z = 4 Ans. 𝜌(𝐴) = 2 ≠ 𝜌(𝐴: 𝐵) = 3 = 𝑛 inconsistent system 5) 𝑥1 + 2𝑥2 – 𝑥3 = 3, 3𝑥1 − 𝑥2 + 2𝑥3 = 1, 2𝑥1 − 2𝑥2 + 3𝑥3 = 2, 𝑥1 − 𝑥2 + 𝑥3 = −1 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 3 = 𝑛 unique solution 𝑥 − 1, 𝑦 = 4, 𝑧 = 4 6) 3𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 1, 4𝑥 − 3𝑦 + 2𝑧 = 4, 3𝑥 + 2𝑦 + 𝑧 = 10 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 3 = 𝑛 unique solution 𝑥 = 1, 𝑦 = 2, 𝑧 = 3 7) 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 − 𝑦 + 2𝑧 = 5, 3𝑥 + 𝑦 + 𝑧 = 8 2𝑥 − 2 𝑦 + 3𝑧 = 7 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 3 = 𝑛 unique solution 𝑥 = 1, 𝑦 = 2, 𝑧 = 3 8) 5𝑥 + 7𝑦 + 2𝑧 = 5; 3𝑥 + 𝑦 + 5𝑧 = 4; 2𝑥 + 4𝑦 − 15𝑧 + 7 = 0, 4𝑥 − 4𝑦 + 13 = 7 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 2  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 1 2 3 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 3 = 𝑛 unique solution 𝑥 = 5 , 𝑦 = 5 , 𝑧 = 5 9) 2𝑥 + 𝑧 = 4, 𝑥 − 2𝑦 + 2𝑧 = 7, 3 𝑥 + 2𝑦 = 1 4−𝑡 3𝑡−10 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 2 = 𝑟 < 𝑛 infinite solution 𝑥 = , 𝑦= , 𝑧=𝑡 2 4 10) 5 𝑥 + 3𝑦 + 7𝑧 = 4, 3𝑥 + 26𝑦 + 2𝑧 = 9, 7𝑥 + 2𝑦 + 10𝑧 = 5 7−16𝑡 3+𝑡 Ans. 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 2 = 𝑟 < 𝑛 infinite solution 𝑥 = , 𝑦= , 𝑧=𝑡 11 11 b) Non Homogeneous Equation with parameters 1) Find λ and μ if the following equations has an infinite no. Of solutions. 2𝑥 − 5𝑦 + 2𝑧 = 8; 2𝑥 + 4𝑦 + 6𝑧 = 5; 𝑥 + 2𝑦 + 𝜆 𝑧 = 𝜇 Ans. 5 𝜆 = 3, 𝜇 = 2 2) For what value of k the equations 𝑥 + 𝑦 + 𝑧 = 1, 𝑥 + 2𝑦 + 4𝑧 = 𝑘, 𝑥 + 4𝑦 + 10𝑧 = 𝑘 2 Have a solution and solve them completely for one of the value of k Ans. 1) 𝑘 = 1, 𝑥 = 1 + 2𝑡, 𝑦 = −3𝑡, 𝑧 = 𝑡, 2) 𝑘 = 2 𝑥 = 2𝑡, 𝑦 = 1 − 3𝑡, 𝑧 = 𝑡 3) For what values of λ & μ do the system of equations x + y + z = 6, x + 2 y + 3 z = 10, x + 2y + λ z = μ have i) no solution ii) unique solution and iii) infinite solutions. Also obtain unique by selecting proper values of λ & μ Ans. i) 𝜆 = 3, 𝜇 ≠ 10 ii) 𝜆 ≠ 3, iii) 𝜆 = 3, 𝜇 = 10 4) Find the value of k for which the equations 3𝑥 − 𝑦 + 4𝑧 = 3, 𝑥 + 2𝑦 − 3𝑧 + 2 = 0, 6𝑥 + 5𝑦 + 𝑘𝑧 + 3 = 0 have infinite solution and hence find the solution. Ans. 4−5𝑡 13𝑡−9 𝑘 = −5, 𝑥 = , 𝑦= , 𝑧=𝑡 7 7 5) Use matrix method to determine the value of k for which the equation 𝑥 + 2𝑦 + 𝑧 = 3, 𝑥 + 𝑦 + 𝑧 = 𝑘, 3𝑥 + 𝑦 + 3𝑧 = 𝑘 2 Are consistent and solve completely Ans. 1) 𝑘 = 2, 𝑥 = 1 − 𝑡, 𝑦 = 1, 𝑧 = 𝑡, 2) 𝑘 = 3 𝑥 = 3 − 𝑡, 𝑦 = 0, 𝑧 = 𝑡 c) Homogeneous linear Equations 1) Use matrix method to solve the equations 𝑥 + 3𝑦 − 2𝑧 = 0, 2𝑥 − 𝑦 + 4𝑧 = 0, 𝑥 − 11𝑦 + 14𝑧 = 0. Ans. −10𝑡 8𝑡 𝜌(𝐴) = 2 = 𝑟 < 𝑛 nontrival solution 𝑥 = , 𝑦= , 𝑧=𝑡 7 7 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 3  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 2) Solve 2𝑥– 𝑦 + 3𝑧 = 0, 3𝑥 + 2𝑦 + 𝑧 = 0, 𝑥– 4𝑦 + 5𝑧 = 0. Ans. 𝜌(𝐴) = 2 = 𝑟 < 𝑛 non-trivial solution 𝑥 = −𝑡, 𝑦 = 𝑡, 𝑧 = 𝑡 3) Determine the rank of the coefficient matrix for the system of equations 𝑥 + 2𝑦 + 3𝑧 = 0; 𝑥 + 4𝑦 + 2𝑧 = 0; 2𝑥 + 6𝑦 + 5𝑧 = 0 and hence find non-trivial solution −𝑡 Ans. 𝜌(𝐴) = 2 = 𝑟 < 𝑛 non-trivial solution 𝑥 = −4𝑡, 𝑦 = , 𝑧=𝑡 2 4) Use matrix method to solve following set of equations. 𝑥 + 3𝑦 + 2𝑧 = 0; 2𝑥 − 𝑦 + 3𝑧 = 0; 3𝑥– 5𝑦 + 4𝑧 = 0; 𝑥 + 17𝑦 + 4𝑧 = 0 Ans. −11𝑡 −𝑡 𝜌(𝐴) = 2 = 𝑟 < 𝑛 non-trivial solution 𝑥 = , 𝑦= , 𝑧=𝑡 7 7 5) Solve 3𝑥 + 𝑦 − 5𝑧 = 0, 5𝑥 − 3𝑦 − 6𝑧 = 0, 𝑥 + 𝑦 − 2𝑧 = 0, 𝑥– 5𝑦 + 𝑧 = 0 Ans. 3𝑡 𝑡 𝜌(𝐴) = 2 = 𝑟 < 𝑛 non-trivial solution 𝑥 = , 𝑦 = 2, 𝑧 = 𝑡 2 6) Solve the equation 𝑥 + 2𝑦 + 3𝑧 = 0; 2𝑥 + 3𝑦 + 𝑧 = 0; 4𝑥 + 5𝑦 + 4𝑧 = 0; 𝑥 + 𝑦 − 2𝑧 = 0 Ans. 𝜌(𝐴) = 3 = 𝑟 = 𝑛 trivial solution 𝑥 = 0, 𝑦 = 0, 𝑧 = 0 7) Solve by matrix method: 𝑥 + 𝑦 + 2𝑧 = 0; 𝑥 + 2𝑦 + 3𝑧 = 0; 𝑥 + 3𝑦 + 4𝑧 = 0, 3𝑥 + 4𝑦 + 7𝑧 = 0. Ans. 𝜌(𝐴) = 2 = 𝑟 < 𝑛 non-trivial solution 𝑥 = −𝑡, 𝑦 = −𝑡, 𝑧 = 𝑡 d) Homogeneous Equation with parameters 1) Find the value of 𝜆 for which the following system of equations has non-trivial solution and find the non-trivial solution 3𝑥 + 𝑦 − 𝜆 𝑧 = 0; 2𝑥 + 4𝑦 + 𝜆 𝑧 = 0; 8𝑥 − 4𝑦 − 6𝑧 = 0 𝑡, 𝑡, Ans. 𝜆 = 1, 𝑥 = 𝑦 = −2, 𝑧 = 𝑡 2 2) Determine the values of k for which following system has non-zero solution & find the solution for each value of k, 3𝑥 + 𝑦– 𝑘𝑧 = 0; 4𝑥 − 2𝑦 − 3𝑧 = 0; 2𝑘𝑥 + 4𝑦 + 𝑘𝑧 = 0 Ans. 𝑡 𝑡 3𝑡 9𝑡, 1) 𝑘 = 1, 𝑥 = 2 𝑦 = − 2 , 𝑧 = 𝑡, 2) 𝑘 = 2 𝑥 = − 𝑦=− , 𝑧=𝑡 2 2 3) Find the value of K for which the equations 2𝑥 + 3𝑦 − 2𝑧 = 0, 3𝑥 − 𝑦 + 3𝑧 = 0, 7𝑥 + 𝑘𝑦 − 𝑧 = 0 has non-trivial solution and hence find the solution. Ans. −7𝑡, 12𝑡, 𝑘 = 5, 𝑥 = 𝑦= , 𝑧=𝑡 11 11 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 4  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 CH.2 - EIGEN VALUES AND EIGEN VECTORS 1. LINEAR DEPENDENT AND LINEAR INDEPENDENT: Que. Examine for L.D. or L. I. Of vectors. If dependent find the relation between them 1) [3 1 1]; [2 0 − 1]; [4 2 1]. Ans. Vectors are linearly independent 2) [ 1 3 4 2]; [3 − 5 2 6]; [2 − 1 3 4] Ans. Vectors are linearly dependent 𝑋1 + 𝑋2 = 2𝑋3 3) (1 2 1); (2 1 4); (4 5 6) Ans. Vectors are linearly dependent 2𝑋1 + 𝑋2 = 𝑋3 4) [1 1 0 1]; [1 1 1 1]; [4 4 1 1]; [1 0 0 1]; Ans. Vectors are linearly independent 5) [3 1 − 4]; [2 2 − 3]; [0 − 4 1] Ans. Vectors are linearly dependent 2𝑋1 = 3𝑋2 + 𝑋3 7 −1 3 6 1 4 6) Determine the linear dependence of row vectors of [ ] 2 4 8 −2 −1 2 Ans. Vectors are linearly dependent 3𝑋2 + 𝑋4 = 2𝑋1 + 𝑋3 Define linear dependence of vectors & show that vectors [1 2 3] T; [3 -2 1]T 7) [1 -6 -5] T are L. D. If dependent find the relation between them Ans. Vectors are linearly dependent 2𝑋1 + 𝑋3 = 𝑋2 8) (1 1 3); (1 3 − 3); (−2 − 4 − 4); (− 9 − 25 9) Ans. Vectors are linearly dependent 5𝑋2 + 𝑋4 = 2𝑋1 + 3𝑋3 9) [ 1 1 1 3]; [1 2 3 4]; [2 3 4 7]. Ans. Vectors are linearly dependent 𝑋1 + 𝑋2 = 𝑋3 10) [1 1 3 1]; [2 2 7 -1]; [3 -1 2 4] Ans. Vectors are linearly independent 11) [2 3 - 1 - 1]; [1 – 1 - 2 - 4]; [3 1 3 - 2]; [6 3 0 - 7] Ans. Vectors are linearly dependent 𝑋1 + 𝑋2 + 𝑋3 = 𝑋4 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 5  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 2. EIGEN VALUES AND EIGEN VECTOR 1) 2 1 1 Find Eigen values of the matrix A= [1 2 1 ] And hence find the Eigen value of 0 0 1 𝐴𝑇 ; (𝐴𝑑𝑗 − 𝐴) And 𝐴−1 Cha eqn. 𝜆3 − 5𝜆2 + 7𝜆 − 3 = 0 ; Eigen value of A = 1, 1, 3 = Eigen value of 𝐴𝑇 ; 1 1 −1 1 Ans. Eigen value of (𝐴𝑑𝑗 𝐴) = 3 , 3 , 1 and Eigen value of 𝐴 = 1, 1, 3 Eigen vector for 𝜆 = 1, [−1, 1, 0], [−1 0 1];Eigen vector for 𝜆 = 3, [1 1 0] 2) 3 10 5 Find the Eigen vector for least Eigen value of the matrix [−2 −3 −4] 3 5 7 n 3 2 Cha eq. 𝜆 − 7𝜆 + 16𝜆 − 12 = 0 ; Eigen value of A = 2, 2, 3 Ans. Eigen vector for 𝜆 = 2, [−5 − 2 5]; Eigen vector for 𝜆 = 3, [−1 − 1 2] 3) −2 5 4 Find Eigen values of the matrix A= [ 5 7 5 ] And Eigen vector corresponding to 4 5 −2 the smallest Eigen value. Cha eqn. 𝜆3 − 3𝜆2 − 90𝜆 − 216 = 0 ; Eigen value of A = -6, -3, 12 Ans. Eigen vectors for 𝜆 = −6, [−1 0 1]; for 𝜆 = −3, [1 − 1 1]; for 𝜆 = 12, [1 2 1] 4) 6 −2 2 Find the Eigen value of the matrix [−2 3 −1 ]And hence Eigen values of 2 −1 3 (𝐴𝑑𝑗 𝐴) And 𝐴−1 Cha eqn. 𝜆3 − 12𝜆2 + 36𝜆 − 32 = 0 ; Eigen value of A = 2, 2, 8 1 1 1 1 1 1 Ans. Eigen values of (𝐴𝑑𝑗 𝐴) = , , 4 And Eigen values of 𝐴−1 = 2 , , 16 16 2 8 Eigen vectors for 𝜆 = 2, [1 2 0], [−1 0 2]; for 𝜆 = 8, [2 − 1 1] 5) 2 1 1 Find Eigen values of the matrix A= [2 3 2 ] And hence find the Eigen value of 𝐴−1 3 3 4 ,𝐴2 And 3𝐴 Cha eqn. 𝜆3 − 9𝜆2 + 15𝜆 − 7 = 0 ; Eigen value of A = 1, 1, 7 Eigen values of 3𝐴 = 3, −1 1 2 Ans. 3, 21 And Eigen values of 𝐴 = 1, 1, 7 Eigen value of 𝐴 = 1, 1, 49 Eigen vectors for 𝜆 = 1, [−1 0 1], [−1 1 0]; for 𝜆 = 7 [1 2 3] Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 6  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 6) 1 1 1 Find the Eigen values of the matrix 𝐴 = [1 2 1] And Eigen vector corresponding to 3 2 3 the smallest Eigen value. Cha eqn. 𝜆3 − 6𝜆2 + 5𝜆 = 0 ; Eigen value of A = 0, 1, 5 Ans. Eigen vectors for 𝜆 = 0, [−1 0 1], for 𝜆 = 1, [0 − 1 1]; for 𝜆 = 5, [4 5 11] 7) 8 −6 2 Find Eigen values of the matrix A= [−6 7 −4 ] And Eigen vector corresponding to 2 −4 3 the greatest Eigen value of matrix A. Cha eqn. 𝜆3 − 18𝜆2 + 45𝜆 = 0 ; Eigen value of A = 0, 3, 15 Ans. Eigen vectors for 𝜆 = 0, [1 2 2], for 𝜆 = 3, [−2 − 1 2]; for 𝜆 = 15, [2 − 2 1] 8) 4 2 −2 i) Find Eigen values of the matrix A= [−5 3 2 ] And Eigen vector corresponding to −2 4 1 the greatest Eigen value of matrix A. ii) Find 𝐴−1 By using characteristic equation of matrix A Cha eqn. 𝜆3 − 8𝜆2 + 17𝜆 − 10 = 0 ; Eigen value of A = 1, 2 5 Ans. Eigen vectors for 𝜆 = 1, [2 1 4], for 𝜆 = 2, [1 1 2]; for 𝜆 = 5, [0 1 1] 9) 1 1 3 Find Eigen values and Eigen vector of the matrix A = [1 5 1 ] 3 1 1 Cha eqn. 𝜆3 − 7𝜆2 + 36 = 0 ; Eigen value of A = -2, 3, 6 Ans. Eigen vectors for 𝜆 = −2, [−1 0 1], for 𝜆 = 3, [1 − 1 1]; for 𝜆 = 6, [1 2 1] 10) Obtain the Eigen values of adjoint of A and also find Eigen vector corresponding to 3 −1 1 largest Eigen value of A where 𝐴 = [−1 5 −1] 1 −1 3 Cha eqn. 𝜆3 − 11𝜆2 + 36𝜆 − 36 = 0 ; Eigen value of A = 2, 3, 6 Ans. Eigen vectors for 𝜆 = 2, [−1 0 1], for 𝜆 = 3, [1 1 1]; for 𝜆 = 6, [1 − 2 1] 11) 6 −2 2 Find Eigen values and Eigen vector of the matrix A = [ −2 3 −1] 2 −1 3 Cha eqn. 𝜆3 − 12𝜆2 + 36𝜆 − 32 = 0 ; Eigen value of A = 2, 2, 8 Ans. Eigen vectors for 𝜆 = 2, [−1 0 2], [1 2 0]; for 𝜆 = 8, [2 − 1 1] 12) −2 2 −3 Find Eigen values of the matrix A= [ 2 1 −6 ] And Eigen vector corresponding to −1 −2 0 the greatest Eigen value of matrix A. Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 7  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 Cha eqn. 𝜆3 + 𝜆2 − 21𝜆 − 45 = 0 ; Eigen value of A = -3, -3, 5 Ans. Eigen vectors for 𝜆 = −3, [−2 1 0], [3 0 1]; for 𝜆 = 5, [−1 − 2 1] 13) Obtain the Eigen values of A and A2 and also find Eigen vectors of A 3 10 5 Where A = [−2 −3 −4] 3 5 7 Cha eqn. 𝜆3 − 7𝜆2 + 16𝜆 − 12 = 0 ; Eigen value of A = 2, 2, 3 Eigen vectors for 𝜆 = 2, Ans. [−5 , −2, 5], for 𝜆 = 3, [−1 − 1 2] Eigen value of A2 = 4, 4,9 14) 4 0 1 Find the Eigen value of the matrix A = [−2 1 0] Hence determine Eigen value of 𝐴−1 −2 0 1 Cha eqn. 𝜆3 − 6𝜆2 + 11𝜆 − 6 = 0 ; 1 1 Ans. Eigen value of A = 1, 2, 3 and Eigen value of 𝐴−1 = 1, 2 , 3 Eigen vectors for 𝜆 = 1, [0 1 0], for 𝜆 = 2, [−1 2 2]; for 𝜆 = 3, [−1 1 1] 3. CAYLEY HAMILTON’S THEOREM Que. Verify Cayley Hamilton’s theorem 1 2 −2 1) A = [−1 3 0 ] Use if to find𝐴−1 And 𝐴4 0 −2 1 Cha eqn. 𝜆3 − 5𝜆2 + 8𝜆 − 1 = 0 −1 12 −4 −13 42 −2 −55 104 24 A2 = [−4 7 2 ] , A3 = [−11 9 10 ] , A4 = [−20 −15 32 ], Ans. 2 −8 1 10 −22 −3 32 −40 −23 3 2 6 A−1 = [1 1 2] 2 2 5 1 2 3 2) A = [2 4 5 ] 3 5 6 Cha eqn. 𝜆3 − 11𝜆2 − 4𝜆 + 1 = 0 Ans. 14 25 31 157 283 353 A2 = [25 45 56 ] , A3 = [283 510 636 ] 31 56 70 353 636 793 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 8  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 1 1 3 3) A=[ 1 3 −3 ] Also use it to find 𝐴−1, find 𝐴4 −2 −4 −4 Cha eqn. 𝜆3 − 20𝜆 + 8 = 0 −4 −8 −12 12 20 60 −88 −168 −264 A2 = [ 10 22 6 ] , A3 [ 20 52 −60 ] , A4 = [ 192 416 144 ], Ans. 2 2 22 −40 −80 −88 56 72 472 3 1 3/2 −1 −5/4 −1/4 −3/4 A =[ ] −1/4 −1/4 −1/4 4 3 1 4) A = [2 1 −2 ] Hence obtain 𝐴4 1 2 1 Cha eqn. 𝜆3 − 6𝜆2 + 6𝜆 − 11 = 0 Ans. 23 17 −1 125 84 −12 656 435 −55 A2 = [ 8 3 −2 ] , A3 = [ 36 23 0 ] , A4 = [190 131 −10 ], 9 7 −2 48 30 −7 245 160 −19 1 2 3 5) 𝐴 = [2 −1 4 ] And find 𝐴−1 3 1 −1 Cha eq. 𝜆 + 𝜆2 − 18𝜆 − 40 = 0 n 3 Ans. 14 3 8 44 33 46 −3/40 1/8 11/40 2 3 −1 A = [12 9 −2 ] , A = [24 13 74 ] , A = [ 7/20 −1/4 1/20 ] 2 4 14 52 14 8 1/8 1/8 −1/8 2 −1 1 6) A = [−1 2 −1 ] 1 −1 2 n 3 2 Cha eq. 𝜆 − 6𝜆 + 9𝜆 − 4 = 0 Ans. 6 −5 5 22 −21 21 2 3 A = [−5 6 −5 ] , A = [−21 22 −21 ] 5 −5 5 21 −21 22 1 1 2 7) A = [3 1 1 ] 2 3 1 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 9  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 Cha eqn. 𝜆3 − 3𝜆2 − 7𝜆 − 11 = 0 Ans. 8 8 5 42 31 29 A2 = [ 8 7 8 ] , A3 = [45 39 31 ], 13 8 8 53 45 42 8 −8 −2 8) Show that the following matrix satisfies its characteristic equation [4 −3 −2 ] 3 −4 1 Cha eqn. 𝜆3 − 6𝜆2 + 11𝜆 − 6 = 0 Ans. 26 −32 −2 74 −104 −10 A2 = [14 −15 −4 ] , A3 = [ 40 −51 −2 ] 11 −16 3 33 −52 13 2 1 1 Find the characteristic equation of the matrix 𝐴 = [0 1 0 ] And hence use it to find 9) 1 1 2 inverse of matrix A Cha eqn. 𝜆3 − 5𝜆2 + 7𝜆 − 3 = 0 Ans. 5 4 4 14 13 13 2/3 −1/3 −1/3 2 A = [0 1 0 ] , A3 = [ 0 1 −1 0 ],A = [ 0 1 0 ], 4 4 5 13 13 14 −1/3 −1/3 2/3 1 3 7 Find the characteristic equation of the matrix [4 2 3 ] And Show that the 10) 1 2 1 characteristic equation is satisfied by A Cha eqn. 𝜆3 − 5𝜆2 − 20𝜆 − 35 = 0 Ans. 20 23 23 135 152 232 2 A = [15 22 37 ] , A3 = [140 163 208 ] 10 9 14 60 76 111 1 4 Find Characteristics equation for A = [ ] And use it to find 11) 2 3 𝐴5 + 5𝐴4 − 6𝐴3 + 2𝐴2 − 4𝐴 − 7𝐼 In terms of A. Cha eqn. 𝜆2 − 4𝜆 − 5 = 0 Ans. 9 16 1861 3676 A2 = [ ] , 𝐴5 + 5𝐴4 − 6𝐴3 + 2𝐴2 − 4𝐴 − 7𝐼 = 919𝐴 + 942𝐼 = [ ] 8 17 1838 3699 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 10  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 7 −1 3 12) Use Cayley Hamilton Thm & find A-1 where A = [6 1 4] 2 4 8 Cha eqn. 𝜆3 − 16𝜆2 + 55𝜆 − 50 = 0 Ans. 49 4 41 449 119 491 −4/25 2/5 −7/50 2 3 −1 A = [56 11 54 ] , A = [566 171 644 ] , A = [ −4/5 1 −1/5 ], 54 34 86 754 324 986 11/25 −3/5 13/50 CH.3 – Numerical solution of Simultaneous system of linear equations Solving the following equations using the methods 1) Gauss elimination method, 2) Gauss -Jordan method, 3) Jacobi’s iteration method and 4) Gauss Seidel method 1. Solve 2𝑥 + 𝑦 + 𝑧 = 10, 3𝑥 + 2𝑦 + 3𝑧 = 18, 𝑥 + 4𝑦 + 9𝑧 = 16 2. Solve 𝑥 + 2𝑦 + 𝑧 = 8, 2𝑥 + 3𝑦 + 4𝑧 = 20,4𝑥 + 3𝑦 + 2𝑧 = 16 3. Solve 20𝑥 + 𝑦 − 2𝑧 = 17, 3𝑥 + 20𝑦 − 𝑧 = −18, 2𝑥 − 3𝑦 + 20𝑧 = 25 4. Solve 10𝑥 + 𝑦 − 𝑧 = 11, 𝑥 + 10𝑦 + 𝑧 = 28, − 𝑥 + 𝑦 + 10𝑧 = 35.61 correct to 2 decimal 5. Solve 2𝑥 + 20𝑦 − 3𝑧 = 19, 3𝑥 − 3𝑦 + 25𝑧 = 22, 15𝑥 + 2𝑦 − 𝑧 = 18 up to 3rd iteration 6. Solve 10𝑥 + 𝑦 + 𝑧 = 12, 2𝑥 + 10𝑦 + 𝑧 = 13, 𝑥 + 𝑦 + 5𝑧 = 7 up to third iteration M-2014 7. Solve 20𝑥 + 𝑦 − 2𝑧 = 17, 3𝑥 + 20𝑦 − 𝑧 = −18,2𝑥 − 3𝑦 + 20𝑧 = 25 8. Solve 𝑥 − 𝑦 + 5𝑧 = 34, 𝑥 − 3𝑦 + 22𝑧 = 12, 𝑥 − 4𝑦 + 8𝑧 = 8 9. Solve 2𝑥 − 𝑦 + 𝑧 = 20, 𝑥 − 2𝑦 + 4𝑧 = 6, 𝑥 + 3𝑦 − 2𝑧 = 10 10. Solve 2𝑥 + 𝑦 + 6𝑧 = 17, 3𝑥 + 12𝑦 − 𝑧 = −18, 12𝑥 − 𝑦 + 6𝑧 = 25 11. Solve 𝑥 + 𝑦 + 2𝑧 + 𝑤 = 5 ; 2𝑥 + 3𝑦 – 𝑧 − 2𝑤 = 2; 4𝑥 + 5𝑦 + 3𝑧 = 7 12. Solve 𝑥1 + 2𝑥2 − 𝑥3 = 1, 3𝑥1 − 2𝑥2 + 2𝑥3 = 2, 7𝑥1 − 2𝑥2 + 3𝑥3 = 5 13. Solve x + y + z = 3, 2 x - y + 3z = 1, 4x + y +5 z = 2 3x -2 y + z = 4 14. Solve 𝑥1 + 2𝑥2 – 𝑥3 = 3, 3𝑥1 − 𝑥2 + 2𝑥3 = 1, 2𝑥1 − 2𝑥2 + 3𝑥3 = 2, 𝑥1 − 𝑥2 + 𝑥3 = −1 15. Solve 3𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 1, 4𝑥 − 3𝑦 + 2𝑧 = 4, 3𝑥 + 2𝑦 + 𝑧 = 10 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 11  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 16. Solve 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 − 𝑦 + 2𝑧 = 5, 3𝑥 + 𝑦 + 𝑧 = 8 2𝑥 − 2 𝑦 + 3𝑧 = 7 17. Solve 5𝑥 + 7𝑦 + 2𝑧 = 5; 3𝑥 + 𝑦 + 5𝑧 = 4; 2𝑥 + 4𝑦 − 15𝑧 + 7 = 0, 4𝑥 − 4𝑦 + 13 = 7 18. Solve 2𝑥 + 𝑧 = 4, 𝑥 − 2𝑦 + 2𝑧 = 7, 3 𝑥 + 2𝑦 = 1 19. Solve 5 𝑥 + 3𝑦 + 7𝑧 = 4, 3𝑥 + 26𝑦 + 2𝑧 = 9, 7𝑥 + 2𝑦 + 10𝑧 = 5 20. Solve 6 𝑥 + 3𝑦 − 3𝑧 = 14, 4𝑥 + 6𝑦 − 5𝑧 = 19, 2𝑥 + 3𝑦 + 8𝑧 = 15 21. Solve 2𝑥 + 𝑦 + 𝑧 = 10, 3𝑥 + 2𝑦 + 3𝑧 = 18, 𝑥 + 4𝑦 + 9𝑧 = 16 22. Solve 𝑥 + 2𝑦 + 𝑧 = 8, 2𝑥 + 3𝑦 + 4𝑧 = 20,4𝑥 + 3𝑦 + 2𝑧 = 16 CH.4 – CURVE FITTING 1. The following figures relate to advertising expenditure and sales. Advertising expenditure X (Rs. Lakhs): 60 62 65 70 75 71 73 Sales Y (Rs. Crores) 10 11 13 15 16 19 14 Estimate the sales for advertising expenditure of Rs. 90 lakhs. (Ans. Y =23.4286) 2. A panels of two judges A and B graded seven debaters independently awarded the following marks. Marks by A (X): 40 34 28 30 44 38 31 Marks by B (Y): 32 39 26 30 38 34 28 Find the regression equation of Y on X and Also find the value of Y when X = 36 (Ans. Y = 0.5874 X +11.8696, when X =36, Y = 33) 3. Obtain the equation of two lines of regression for the following data. Also obtain the estimate of X for Y = 70 X: 65 66 67 67 68 69 70 72 Y: 67 68 65 68 72 72 69 71 (Ans. Y = 0.665 X + 23.78, X = 0.54Y +30.74, Y = 70 ⟹ X = 68.54) 4. for the following data obtain Find the regression equation of Y on X and hence estimate the value of when X = 50 X: 78 36 98 25 75 82 90 62 65 39 Y: 84 51 91 60 68 62 86 58 53 47 (Ans. Y = 0.5009 X +44.4396 and when X = 50, Y = 58.486) Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 12  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 5. From following data find the equation of the line of regression of Y on X and estimate the most probable value of Y when X=80 X: 89 86 74 65 64 63 66 67 72 79 Y: 92 91 84 75 73 72 71 75 78 84 (Ans. Y = 0.808 X +20.94 and Y = 85.56) 6. Find the regression line of Y on X for the following data and estimate Y when X = 9 X: 18 26 28 31 25 19 35 Y: 11 16 19 17 14 11 24 (Ans. Y = 0.71 X – 2.5714 when X = 9, Y = 3.857) 7. From the following data find the equations of the line of regression and estimate the value of Y when X = 80 and of X when X = 85 X: 89 86 74 55 64 63 67 67 72 79 Y: 92 91 84 75 73 72 21 75 78 84 (Ans. i) Y= 0.8076 X + 20.95, X = 1.177 Y – 21.08, X = 80 ⟹ Y= 85.55, Y = 85 ⟹ X = 78.97) 8. Following data gives the height in inches (X) and the weight in lbs (Y) of random sample of 10 students from large group of student of age 17years. X: 61 68 68 64 65 70 63 62 64 67 Y: 112 123 130 115 110 125 100 113 116 126 Estimate the weight of the student of height 59 inches (Ans-126.4) 9. The table below gives the respective heights X and Y of a sample of 10 fathers and their sons: i) Find the regression line of Y on X. ii) Find the regression line of X on Y iii) Estimate son’s height if father’s height is 65 inches. iv)Estimate father’s height if son’s height is 60 inches. v) compute correlation coefficient between X and Y. Height of father X 65 63 67 64 68 62 70 66 68 67 (inches) Height of son Y (inches) 68 66 68 65 69 66 68 65 71 67 (Ans. i) Y = 0.4821 X + 35.4821, ii) X = 0.8411Y + 9.3940 iii) Y = 66.8179 iv) X = 59.86 v) r = 0.6368 10. Following data are related to marks in Accountancy (X) and marks in statistics (Y) of 10 students. X 66 65 68 68 67 66 70 64 69 67 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 13  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 Y 68 67 67 70 65 68 70 66 68 66 Calculate regression coefficients, hence correlation coefficient between X and Y. ii) Estimate marks in statistics of a student who has scored 76 marks in Accountancy. iii)Estimate marks in Accountancy of a student who has obtained 60 marks in statistics. (Ans. i) bYX = 0.5, bXY = 0.61, r = 0.55, ii) 77, iii) 52.) 11. Following are the marks in Statistics (X) and Mathematics (Y) of ten students X: 56 55 58 57 56 60 54 59 57 54 Y: 68 67 67 70 65 68 70 66 68 66 Calculate the coefficient of correlation and estimate marks in mathematics of a students who score 62 marks in statistics 12. Obtain the two line of regression equation and estimate the yield of crop when the rainfall is 29 cm and rainfall when yield is 600 kg Yield in kg(X) rainfall in cm(Y) Mean 508.4 26.7 S. D. 36.8 4.6 and r = 0.52 (Ans. Y = 0.0650X – 6.3460, Y = 32.6540, X = 4.16Y + 397.3280 and X = 517.9680) 13. Given the data for two tests: Marks in Hindi (X) Marks in Marathi (Y) Mean 75 70 S. D. 6 8 and r = 0.72 Obtain the two lines of regression equations and estimate the marks in Hindi when a student gets 5 marks in Marathi. (Ans. i) Y = 0.96X – 1.9999, ii) X = 0.54Y + 37.2, when Y = 5, X = 39.9) 14. Given the following values, estimate the yield of wheat, when the rainfall is 15.5cms Yield of wheat (kgs. Per unit area) 10.7 8.1 Annual rainfall (cms.) 20.5 5.0 Correlation coefficient = 0.52 (Ans. X = 0.3210Y + 4.1198, when Y =15.5, X = 9.0951) 15. Given the following information about marks of 60 student’s data for two tests: Accountancy (X) Economics (Y) Mean 80 50 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 14  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 S. D. 15 10 and r = 0.4 Estimate the marks of student in accountancy who scored 60 marks in Economics (Ans. i) X = 0.6Y + 50 when Y = 60 , X = 86) 16. Given the following information: Mean height (𝑋̅) = 120.5cm, Mean age (𝑌̅) = 10.37 years, S.D. of X = 12.7cm, S.D. of Y = 2.39 years correlation coefficient between and Y = 0.93 i) Fit the two regression lines. Ii) Estimate the height of a boy of 12 years. (Ans. i)Y = 0.175X − 10.7194 ii) X = 4.9418 Y + 69.2533 iii) 128.56) 17. Following is the information about the bivariate frequency distribution: 𝑛 = 20 ∑ 𝑋 = 80 , ∑ 𝑌 = 40 , ∑ 𝑋 2 = 1680 , ∑ 𝑌 2 = 320 , ∑ 𝑋𝑌 = 480 i) Obtain the regression lines. ii) Estimate Y for X = 3and estimate X for Y = 3 (Ans. i) 3X = 4Y + 4, 17Y =4X + 18 ii) X =5.3333, Y = 1.7647) 18. You are given the following information about two variables X and Y: 𝑛 = 10 ∑ 𝑋 2 = 385 , ∑ 𝑌 2 = 192 , ∑ 𝑋𝑌 = 185 , 𝑋̅ = 5.5, 𝑌̅ = 4 Find i) Regression lines of Y on X ii) regression line of X on Y (Ans. Y= - 0.4242X+6.3331 ii) X = -1.09375Y+9.875) 19. Following is the information about the bivariate frequency distribution: 𝑛 = 8 ∑ 𝑋 = 59 , ∑ 𝑌 = 40 , ∑ 𝑋 2 = 524 , ∑ 𝑌 2 = 256 , ∑ 𝑋𝑌 = 364 Find the regression equation of X on Y. (Ans. i) X = 1.2312Y + 1.2143) 20. You are given the following information about two variables X and Y: 𝑛 = 7 ∑ 𝑋 = 56 , ∑ 𝑌 = 56 ∑ 𝑋 2 = 476 , ∑ 𝑋𝑌 = 469. Find i) Regression lines of Y on X ii) Estimate the income of a person who has completed 13 years of services (Ans. i) Y= 0.75X + 2 ii) when X = 13, Y =11.75(in Rs. 100) = 1175Rs.) 21. The equation 40X − 18Y − 214 = 0 and 8X − 10Y + 66 = 0 are the lines of regression of X on Y and on Y respectively. Find i) Mean values of X and Y. ii) correlation coefficient between X and Y. (Ans. Mean of X = 13 and Mean of Y = 17, ii) r = 0.6) 22. For a certain data of two variables, marks in Maths and marks in Accountancy, two regression lines are ̅ and ̅Y ii) r given as, 3X + 2Y − 26 = 0, 6X + Y − 31 = 0 Find i) X Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 15  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 (Ans. Mean of X = 4 and Mean of Y = 7, ii) r = −0.5) 23. For two variables X and Y, the regression equation of X on Y is 𝑋 = 4𝑌 − 3 regression equation of Y on X is 9Y = X + 13 Find i) ̅ X and ̅Y ii) r (Ans. Mean of X = 5 and Mean of Y = 2, ii) r = 0.6667) 4 3 24. For two variables X and Y, the regression equation of X on Y is 𝑋 = 5 𝑌 − 5 regression equation of Y on X is 9Y = 5X + 13 Find i) ̅ X and ̅Y ii) r (Ans. Mean of X = 1 and Mean of Y = 2, ii) r = 0.6667) 25. For two variables X and Y, the regression equations are given by 10𝑋 + 3𝑌 − 62 = 0, 6𝑋 + 5Y − 50 = 0 Find i) ̅ X and ̅Y ii) r (Ans. Mean of X = 5 and Mean of Y = 4, ii) r = −0.6) Fit The Curve: 1. Fit a first degree curve to the following data and estimate the value of y when x = 73 x: 10 20 30 40 50 60 70 80 y: 1 3 5 10 6 4 2 1 (Ans. y = 4.64 - 0.0143 x, y = 3.5691) 2. Fit a straight line to the given data regarding x as the independent variable: X 1 2 3 4 6 8 Y 2.4 3.1 3.5 4.2 5.0 6.0 (Ans. y = 2.0253 + 0.502x) 3. Fit a straight line y = a + b x to the following data by the method of least square: X 0 1 3 6 8 Y 1 3 2 5 4 (Ans. y = 1.6 + 0.38x) 4. Obtain the relation of the form 𝑦 = 𝑎 + 𝑏𝑥 from the following data x: 2 3 4 5 6 y: 8.3 15.3 33.1 65.1 127.4 (Ans. 𝑦 = − 0.6536 + 28.8 𝑥) 5. Obtain the relation of the form 𝑦 = 𝑐 + 𝑚 𝑝 from the following data p: 100 120 140 160 180 200 y: 0.45 0.55 0.60 0.70 0.80 0.8 (𝐀𝐧𝐬. y = 0.0476 + 0.0041x) 6. Obtain the relation of the form 𝑦 = 𝑎 + 𝑏𝑥 from the following data Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 16  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 x: 0 1 2 3 4 y: 1 1.8 1.3 2.5 6.3 (Ans. 𝑦 = − 065.36 + 28.8 𝑥) 7. Find the least square approximation of the form 𝑦 = 𝑎 + 𝑏𝑥 2 for the following data: X 0 0.1 0.2 0.3 0.4 0.5 Y 1 1.01 0.99 0.85 0.81 0.75 (Ans. y = 1.0032 + 1.1081 𝑥 2 ) 8. Fit a second degree parabola to the following data by the least square method: x: 1 2 3 4 5 y: 1090 1220 1390 1625 1915 (Ans. y = 27.5x 2 + 40.5x + 1024) 9. Fit a second-degree parabola to the following data by least squares method. X 1929 1930 1931 1932 1933 1934 1935 1936 1937 y 352 356 357 358 360 361 361 360 359 ( Ans. y = − 1010135.08 + 1044.69 x − 0.27x2) 10. Fit a second degree parabola to the following data: x: 0 1 2 y: 1 6 17 (Ans. y =1 + 2 x + 3 𝑥 2 ) 11. Fit a second degree parabola to the following data: x: 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y: 1.1 1.3 1.6 2.0 2.7 3.4 4.1 (Ans. y = 1.04 + 1.93x + 0.243𝑥 2 ) 12. Fit a second degree parabola to the following data by the least square method: x: 1 2 3 4 5 y: 1090 1220 1390 1625 1915 (Ans. y = 27.5x 2 + 40.5x + 1024) 13. Fit a second degree parabola to the following data by the least square method: x: 1 2 3 4 5 6 7 8 9 y: 2 6 7 8 10 11 11 10 9 (Ans. y = - 0.27x 2 + 3.55x - 1) 14. Fit a parabola y = a + b x + c𝑥 2 to the following data: x: 2 4 6 8 10 y: 3.07 12.85 31.47 57.38 91.29 (Ans. y = 0.34 + 0.78 x + 0.99𝑥 2 ) 15. Fit a second degree parabola to the following data : Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 17  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 x: 1 2 3 4 5 y: 1090 1220 1390 1625 1915 (Ans. y = 0.55x 2 + 1.07x + 1.24) 16. Determine the constants a and b by the method of least squares such that 𝑦 = 𝑎𝑒 𝑏𝑥 fits the following data: x: 2 4 6 8 10 y: 4.077 11.084 30.128 81.897 222.62 (Ans. 𝑦 = 1.49989𝑒 0.50001𝑥 ) 17. Fit a second degree parabola in the following data: x: 0 1 2 3 4 y: 1 4 10 17 30 (Ans. y = 1 + 2x + 3𝑥 2 ) 18. Fit an exponential curve of the form 𝑦 = 𝑎𝑏 𝑥 to the following data: x: 1 2 3 4 5 6 7 8 y: 1 1.2 1.8 2.5 3.6 4.7 6.6 9.1 CH.5 - CALCULUS 1. MACLAURIN’S THEOREM: 1) Expand log(1 + sin 𝑥) By Maclaurin’s theorem 𝜋 2) Expand tan( 4 + 𝑥) Using Maclaurin’s expansion. Also find value of tan(46.50 ) 3) Expand log(1 + 𝑒 𝑥 ) By Maclaurin’s theorem 4) 𝜋 𝑥3 𝑥5 Using Maclaurin’s series prove that log [tan (4 + 𝑥)] = 2𝑥 + 4 +6 + … 3 5 5) Using Maclaurin’s series prove that 𝑒 𝑥 Sec 𝑥 = 1 + 𝑥 + 𝑥 2 + ⋯ … … … 6) 𝑥4 Using Maclaurin’s series prove that 𝑒 𝑥 𝑠𝑖𝑛2 𝑥 = 𝑥 2 + 𝑥 3 + +⋯ 6 2.STANDARD EXPANSIONS: 𝑥2 𝑥3 11𝑥 4 Show that 𝑒 𝑥 cos 𝑥 = 1 + 𝑥 + − − ⋯⋯⋯⋯⋯ 1) 2 3 24 2) 𝑥2 1 𝑥4 Show that log(1 + sin 𝑥) = 𝑥 − + 6 𝑥 3 − 12 … 2 3) 2𝑥 4 Prove that sec 2 𝑥 = 1 + 𝑥 2 + +⋯ 3 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 18  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 4) Obtain the expansion of sec 2 𝑥 and log sec 𝑥 5) Find expansion of (1 + 𝑥)𝑥 Up to 𝑥 5. 6) 1⁄ Obtain expansion for log(log(1 + 𝑥) 𝑥) up to 𝑥 3. 3. TAYLOR’S THEOREM Expand 2𝑥 3 + 7𝑥 2 + 𝑥 − 6 in powers of (𝑥 − 2) 1) 2) Using Taylor’s series express (𝑥 − 2)4 − 3(𝑥 − 2)3 + 4(𝑥 − 2)2 + 5 in powers of x. 3) Expand 𝑥 4 − 3𝑥 3 + 2𝑥 2 − 𝑥 + 1 in powers of (𝑥 − 3) 4) 𝜋 Using Taylor’s theorem find expansion of tan (𝑥 + 4 ) In ascending power of 𝑥 upto 𝑥 4 And find approximately value of Tan(430 ) 5) Using Taylor’s theorem find approximate value of Sin(300 30′ ) 6) Arrange in powers of x , by Taylor’s theorem 17 + 6(𝑥 + 2) + 3(𝑥 + 2)3 + (𝑥 + 2)4 − (𝑥 + 2)5 7) Expand the function√𝑥 + ℎ Using Taylor’s series. Hence find the value of √25.15 8) Using Taylor’s series express (𝑥 + 2)4 − 3(𝑥 + 2)3 + 4(𝑥 + 2)2 + 5 in powers of x 9) 1 Given Sin 300 = 2 Use Taylor’s theorem to evaluate Sin 310 Up to 4 decimal 4. INDETERMINATE FORM 𝑥 1 Evaluate lim [𝑥−1 − log 𝑥] (∞ − ∞)form 1) 𝑥→1 2) sin 2𝑥+𝑝𝑠𝑖𝑛𝑥 0 If lim is finite, find the value of p and find limit (0) form 𝑥→0 𝑥3 3) log tan 𝑥 ∞ Evaluate lim (∞) form 𝑥→0 log 𝑥 4) 𝑒 𝑎𝑥 −𝑒 −𝑎𝑥 0 Evaluate lim log(1+𝑏𝑥) (0) form 𝑥→0 5) 𝑥−sin 𝑥 0 Evaluate lim (0) form 𝑥→0 tan3 𝑥 1 6) 𝑥 𝑦 −𝑦 𝑥 0 Evaluate i) lim(cos 𝑥)𝑥 (1∞ ) form ii) lim 𝑥 𝑥 −𝑦 𝑦 (0) form 𝑥→0 𝑥→𝑦 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 19  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 iii) lim tan 𝑥 log 𝑥 (0. ∞) form 𝑥→0 7) Evaluate lim(1 + cot 𝑥)tan 𝑥 (∞0 ) form 𝑥→0 8) Evaluate lim[log(2 − 𝑥)] cot(𝑥 − 1) (0. ∞) form 𝑥→1 1 9) 𝑒 2𝑥 −(1+𝑥)2 0 Evaluate i) lim (0) form ii) lim (1 − 𝑥 2 )log(1−𝑥) (00 ) form 𝑥→0 x log(1+𝑥) 𝑥→1 10) sin−1 √ 𝑎−𝑥 𝑎+𝑥 0 𝑥 Evaluate i) lim (0) form ii) lim log(2 − 𝑎)cot(𝑥−𝑎) (1∞ ) form 𝑥→𝑎 sin √𝑎2 −𝑥 2 𝑥→𝑎 𝜋𝑥 11) 𝑥 tan( ) (1∞ ) form Evaluate lim log(2 − 𝑎) 2𝑎 𝑥→𝑎 12) tan 𝑥 12 Evaluate lim lim( )𝑥 (1∞ ) form 𝑥→𝑎 𝑥→0 𝑥 1 1 1 13) 2 ⁄𝑥 +3 ⁄𝑥 +5 ⁄𝑥 𝑥 𝑥𝑒 𝑥 −log(1+𝑥) 0 Evaluate lim [ ] AND Evaluate lim (0) form 𝑥→∞ 3 𝑥→0 𝑥2 14) Evaluate lim 𝑒 𝑥 sin 𝑥−𝑥−𝑥 2 (1∞ ) form 𝑥 2 +𝑥𝑙𝑜𝑔(1−𝑥) 𝑥→0 15) Evaluate lim (sinh 𝑥)tanh 𝑥 (00 ) form 𝑥→0 CH. 6 COMPLEX NUMBER 1. DE-MOIVRE’S THEOREM 1) 6 6 Evaluate (−1 + 𝑖 √3) + (−1 – 𝑖 √3) 2) (cos 3𝜃+𝑖 Sin 3𝜃)5 (Cos 2𝜃−𝑖 Sin 2𝜃)3 Simplify (cos 4𝜃−𝑖 Sin 4𝜃)−9 (Cos 5𝜃+𝑖 Sin 5𝜃)9 3) (cos 3𝜃+𝑖 Sin 3𝜃)5 (Cos 4𝜃−𝑖 Sin 4𝜃)4 Simplify (cos 5𝜃−𝑖 Sin 5𝜃)3 (Cos 6𝜃+𝑖 Sin 6𝜃)2 4) (cos 2𝜃−𝑖 Sin 2𝜃)7 (Cos 3𝜃+𝑖 Sin 3𝜃)−5 Simplify (cos 4𝜃+𝑖 Sin 4𝜃)12 (Cos 5𝜃−𝑖 Sin 5𝜃)−6 5) If 𝛼 and 𝛽are the roots of 𝑧 2 sin2 𝜃 − 𝑧 sin 2𝜃 + 1 = 0 then prove that 𝛼 𝑛 + 𝛽 𝑛 = 2 cos 𝑛𝜃 csc n 𝜃 ) n= (𝛼 - 𝛽 ) sin (n𝜃) cosec n𝜃 where n is an integer 6) If 𝑧 = −1 + 𝑖√3 and n is an integer prove that 22𝑛 + 2𝑛 𝑧 𝑛 𝑧 2𝑛 is zero if n is not a multiple of 3 7) If α = (1 + i); β = (1 - i) and 𝑐𝑜𝑡 𝜃 = (𝑥 + 𝑖) then prove that (𝑥 + 𝛼 )𝑛 – (𝑥 + 𝛽 ) 𝑛 = (𝛼 − 𝛽 ) 𝑠𝑖𝑛 (𝑛𝜃) 𝐶𝑜𝑠𝑒𝑐 𝑛𝜃 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 20  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 8) Prove that (1 + 𝑖 √3)8 + (1 − 𝑖 √3)8 = - 28 9) 𝜋 𝜋 8 1 + sin( )+ 𝑖 cos( ) 8 8 Simplify [ 𝜋 𝜋 ] 1 + sin( )− 𝑖 cos( ) 8 8 𝑛 10) 1+𝑠𝑖𝑛 𝛼+𝑖 𝑐𝑜𝑠 𝛼 Prove that ( 1+𝑠𝑖𝑛 𝛼−𝑖𝑐𝑜𝑠𝛼 ) = cos 𝑛( 𝜋⁄2 − 𝛼) + 𝑖 sin n( 𝜋⁄2 − 𝛼) 11) cos 𝜃+𝑖 sin 𝜃 4 Simplify ( ) sin 𝜃+𝑖 cos 𝜃 2. ROOTS OF COMPLEX NUMBER 1) Find all the roots of equation x10 + 11x5 +10 = 0 2) 1 Find all the roots of (1 + 𝑖) ⁄5 3) Use De-Moivre’s Theorem to solve𝑥 8 + 𝑥 5 + 𝑥 3 + 1 = 0 4) 1⁄ Find all the values of (1 + 𝑖√3) 5 5) 1 Find all values of (−1) ⁄3 6) Solve 𝑥 4 + 𝑥 3 + 𝑥 2 + 𝑥 + 1 = 0 3 7) 1 𝑖√3 4 Find all the values of (2 + ) & show that their continued product is 1 D-2008 2 8) Solve the equation 𝑥 9 − 𝑥 5 + 𝑥 4 − 1 = 0 9) Find the roots of x6 – i = 0 10) Find all values of (−1)1⁄5 11) Find nth root of unity & prove that their sum is 0 & their product is (-1)n-1 12) Solve the equation 𝑥 4 − 𝑥 3 + 𝑥 2 − 𝑥 + 1 = 0 13) Solve the equation 𝑥 7 + 𝑥 4 + 𝑥 3 + 1 = 0 M-2008 14) Find the roots of the equation 𝑥 7 + 𝑥 4 + 𝑖𝑥 3 + 𝑖 = 0 15) Find all values of (−1 − 𝑖)1⁄5 2𝑛𝜋 16) 5 5 − 3+4𝑖 Sin 5 Show that the roots of (𝑥– 1) = 32(𝑥 + 1) Are x = 2𝑛𝜋 , n = 0, 1, 2, 3, 4 5−4 Cos 5 17) Solve the equation x5+1=0 18) Find the roots common to equation x6 –i = 0 & x4 + 1 = 0 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 21  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 3. EXPANSION 1) 1 Prove that sin7 θ = [35 sin θ −21sin 3θ +7sin 5θ − sin 7θ] 26 2) Prove that cos 5θ = 5 cos θ − 20 cos3 θ + 16 cos 5 θ 3) Prove that sin 5θ = 5 sin θ − 20 sin3 θ + 16 sin5 θ 4) Express tan5θ in terms of powers of tan θ 5) sin 7θ Prove that = 7 − 56 sin2 θ + 112 sin4 θ − 64 sin6 θ sin θ 6) sin 6θ Prove that = 6 cos θ − 32 cos3 θ + 32 cos5 θ sin θ 7) Express sin 8θ in term of powers of sin θ , cos θ 4. HYPERBOLIC FUNCTION 1) If sin(x + iy) = [cos α + i sin α]then prove that sin2 α = cos 4 x 2) iπ If α + i β = tanh (x + 4 ) then prove that α2 + β2 = 1 3) 1+√1−x2 Prove that sech−1 x = log ( ) x 4) Define cosh x & sinh x also prove that cosh2 x − sinh2 x = 1 5) x2 x2 If sin(α + iβ) = x + iy, prove that cosh2 β + sinh2 β = 1 6) π 2x If tan ( 6 + iα) = x + iy then prove that x 2 + y 2 + =1 √3 7) sin(x + iy) Prove that log [sin(x − iy)] = 2i tan−1(cot x tanh y) 8) Solve the equation 5 sinh x – cosh x = 5 and hence find tanh x 9) 1 sin(θ−α) If cos ((θ + iø) = r (cos α + i sin α) prove that ø= log ( ) 2 sin(θ+α) 10) If tan(x + i y) =i where x & y are real, p.t. x is indeterminate & y is infinite m-2008 11) If tan(α + i β) = eiθ then prove that α = nπ + π and β = 1 log tan (π + θ) 2 4 2 4 2 12) Prove that sinh u = tan x and cosh u sec x if tan (x/2) = tanh (u/2) 13) Express sinh7x in terms of hyperbolic sines of multiples of x 14) If sin(θ + i∅) = tan α + i sec α show that cos 2θ cosh 2∅ = 3 15) Solve the equation for real value of x for 7cos h x + 8 sin h x = 1 16) If tan (α + i β) = sin (x + iy) then prove that tan x sin(2α) = tan h y sin h (2β) 17) Expand cosh7 x in terms of hyperbolic cosines of multiples of x 18) If cosh x = sec θ prove that i) x = log(sec θ + tan θ) and ii)θ = π − 2 tan−1 e−x 2 Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 22  Annasaheb Dange College of Engineering and Technology APPLIED MATHEMATICS - I Question Bank 2024-25 19) If cosec (π + ix) = u + i v then prove that (u2 + v2) 2 = 2 (u2 – v2) 4 5. INVERSE HYPERBOLIC FUNCTION 1) Show that sinh−1 z = log( z + √z 2 + 1) 2) x Prove that tanh−1 x = sinh−1 (√1−x2) 3) π x θ If cos h x = sec θ then show that θ = 2 − 2 tan−1 e−x and tanh 2 = ± tan 2 4) Prove that tanh−1 (sin θ) = cosh−1 (sec θ) 5) If cosh−1 (x + iy) + cosh−1 (x − iy) = cosh−1 α then prove that 2(a − 1)x 2 + 2a(a + 1)y 2 = a2 − 1 6) 1 x+a Prove that coth-1 (x/a) = 2 log (x−a) 7) Prove that tanh−1 (sin θ) = cosh−1 (sec θ) x 8) Prove that sinh−1 x = tanh−1 √(1+x2 ) 9) Prove that cos h−1 x = log(x + √x 2 − 1) 10) Prove that sech −1 (sinθ) = log [cot(θ/2)] Prepared by Savita Mohite Assistant Professor ADCET Ashta Page 23