Engineering Mathematics 1 19MA1ICMAT Module 1- Linear Algebra PDF

Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 1 – LINEAR ALGEBRA Q.No Question 1. 0 1 −3 −1...

Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 1 – LINEAR ALGEBRA Q.No Question 1. 0 1 −3 −1 1 0 1 1 Find the Rank of a Matrix A= [ ] 3 1 0 2 1 1 −2 0 2. 2 1 3 4 0 3 4 1 Find the rank of the matrix A= [ ] 2 3 7 5 2 5 11 6 3. −2 −1 −3 −1 Find the Rank of the Matrix [1 2 3 −1 ] 1 0 1 1 0 1 −1 −1 4. Test the following system of equations are consistent 𝑥 + 2𝑦 + 3𝑧 = 1, 2𝑥 + 3𝑦 + 8𝑧 = 2, 𝑥 + 𝑦 + 𝑧 = 3 5. Find the value of 𝜇, the system possesses a solution. Solve completely in each case 𝑥 + 𝑦 + 𝑧 = 1, 𝑥 + 2𝑦 + 4𝑧 = 𝜇, 𝑥 + 4𝑦 + 10𝑧 = 𝜇 2 6. Find the values of 𝛽 and 𝜇 for which the system 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 = 10, 𝑥 + 2𝑦 + 𝛽𝑧 = 𝜇, may have a) a unique solution b) infinitely many solution c) no solution. 7. Test for consistency and solve the following system of equations, 𝑥 + 3𝑦 − 2𝑧 = 0, 2𝑥 − 𝑦 + 4𝑧 = 0, 𝑥 − 11𝑦 + 14𝑧 = 0 8. Find the value of 𝜇 for the following system of equation to have a non-trivial solution and also find the solution. 𝑥 + 𝑦 + 3𝑧 = 0, 4𝑥 + 3𝑦 + 𝜇𝑧 = 0, 2𝑥 + 𝑦 + 2𝑧 = 0 9. Find the value of 𝜇 for the following system of equation to have a non-trivial solution and also find the solution. 4𝑥 + 9𝑦 + 𝑧 = 0, 𝜇𝑥 + 3𝑦 + 𝜇𝑧 = 0, 𝑥 + 4𝑦 + 2𝑧 = 0 10. Solve the following system of equation by Gauss elimination Method 𝑥 + 𝑦 + 𝑧 = 9, 𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 3 11. Solve the following system of equation by Gauss elimination Method 2𝑥1 + 𝑥2 + 3𝑥3 = 1, 4𝑥1 + 4𝑥2 + 7𝑥3 = 1, 2𝑥1 + 5𝑥2 + 9𝑥3 = 3 12. Solve the following system of equation by Gauss elimination Method 𝑥 + 5𝑦 + 6𝑧 = 8 , 4𝑥 − 5𝑦 + 𝑧 = 5, 5𝑥 + 𝑦 + 6𝑧 = 11 13. Solve the following system of equation by Gauss elimination Method 𝑥 − 𝑦 + 𝑧 = 1 , −3𝑥 + 2𝑦 − 3𝑧 = −6 , 2𝑥 − 5𝑦 + 3𝑧 = 5 14. Solve by Gauss elimination Method, balance the chemical reaction equation 𝐶5 𝐻8 + 𝑂2 → 𝐶𝑂2 + 𝐻2 𝑂 15. Solve by Gauss elimination Method, balance the chemical reaction equation 𝐻3 𝐵𝑂3 → 𝐻4 𝐵6 𝑂11 + 𝐻2 𝑂 16. Using power method find the dominant Eigen value and the corresponding Eigen 4 1 −1 vector of the matrix A= [ 2 3 −1]initial vector [1 0.8 −0.8]𝑇 up to 6th −2 1 5 iterations 17. Using power method find the dominant Eigen value and the corresponding Eigen 2 −1 0 vector of the matrix A= [−1 2 −1] initial vector [1 0 0]𝑇 (Carryout 6 0 −1 2 iterations) 18. Using Rayleigh’s power method find the largest Eigen value and Eigen vector, 10 2 1 given[ 2 10 1 ] with initial Eigen vector [0 0 1]𝑇 up to 6th iterations 2 1 10 19. −1 3 Diagonalize the following matrix, if possible A = [ ] −2 4 20. 7 2 Diagonalize to the matrix A = [ ] −4 1  Engineering Mathematics –I Subject Code: 19MA1ICMAT Module – 2 - DIFFERENTIAL CALCULUS Q.No Question 1. Find the angle between the radius vector and the tangent. 2. Find the angle of intersection of the curves 𝑟 = 𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃 and 𝑟 = 2𝑠𝑖𝑛𝜃. Are they orthogonal? 𝑎 3. Find the angle between the curves 𝑟 = 𝑎𝑐𝑜𝑠𝜃, 𝑟 = 2. 4. Find the angle between the curves 𝑟 𝑛 𝑐𝑜𝑠𝑛𝜃 = 𝑎𝑛 and 𝑟 𝑛 𝑠𝑖𝑛𝑛𝜃 = 𝑏 𝑛. 5. Find the angle between the curves 𝑟 = 𝑎𝜃 and 𝑟 = 𝑎 1+𝜃 1+𝜃2 6. Show that the tangents to the cardioid 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃) at the Points 𝜃 = 3 and 𝜃 = 𝜋 2𝜋 are 3 respectively parallel and perpendicular to the initial line. 7. Derive radius of curvature in Cartesian form 8. Find the radius of curvature for the curve 𝑥 3 + 𝑦 3 = 3𝑎𝑥𝑦 at the point (3𝑎⁄2 , 3𝑎⁄2). 9. Find the radius of curvature for the curve 𝑦 2 = 𝑎2 (𝑎−𝑥) , where the curve meets the x-axis. 𝑥 10. Derive radius of curvature in Polar form. 11. Show that for the curve 𝑟 2 𝑠𝑒𝑐2𝜃 = 𝑎2 is 𝜌 = 3𝑟. 𝑎2 12. Find the radius of curvature 2𝑎 = 1 + 𝑐𝑜𝑠𝜃. 𝑟 𝜋 13. Obtain Taylor’s series expansion of 𝑙𝑜𝑔𝑐𝑜𝑠𝑥 about the point 𝑥 = 3 upto the 4th degree terms. 𝜋 14. Obtain Taylor’s series expansion of tan 𝑥 at 𝑥 = 4 upto the 4th degree terms 15. Obtain the Maclaurin’s expansion of 𝑦 = log(1 + 𝑠𝑖𝑛𝑥) upto the terms of third degree. 16. Expand 𝑒 𝑠𝑖𝑛𝑥 using Maclaurin’s series upto the term containing 𝑥 4. 17. Find the coordinate of the Centre of curvature at any point of the parabola 𝑥 2 = 4𝑎𝑦. 18. Find the Centre of curvature of the curve 𝑥𝑦 2 + 𝑥 2 𝑦 = 2 at the point (1,1) 19. Show that the Circle of curvature at the origin for the curve 𝑥 + 𝑦 = 𝑎𝑥 2 + 𝑏𝑦 2 + 𝑒𝑥 3 is (𝑎 + 𝑏)(𝑥 2 + 𝑦 2 ) = 2(𝑥 + 𝑦). 20. Show that the Evolute of the parabola 𝑦 2 = 4𝑎𝑥 is 27𝑎𝑌 2 = 4(𝑋 − 2𝑎)3.  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 3 (Partial Differentiation) Q.No Question 1. If 𝑢 = 𝑒 𝑎𝑥+𝑏𝑦 𝑓(𝑎𝑥 − 𝑏𝑦) prove that 𝑏 𝜕𝑥 + 𝑎 𝜕𝑦 = 2𝑎𝑏𝑢. 𝜕𝑢 𝜕𝑢 2. 𝜕𝑧 𝜕𝑧 2 𝜕𝑧 𝜕𝑧 If 𝑧(𝑥 + 𝑦) = 𝑥 2 + 𝑦 2 show that [𝜕𝑥 − 𝜕𝑦] = 4 [1 − 𝜕𝑥 − 𝜕𝑦] = 3. If 𝑥 𝑥 𝑦 𝑦 𝑧 𝑧 = 𝑐, prove that at 𝑥 = 𝑦 = 𝑧 , 𝜕2 𝑧 = −1 = −1 𝜕𝑥𝜕𝑦 𝑥(1+𝑙𝑜𝑔𝑥) 𝑥𝑙𝑜𝑔(𝑒𝑥) 4. If 𝑢 = 𝑓(𝑟) 𝑤ℎ𝑒𝑟𝑒 𝑟 2 = 𝑥 2 + 𝑦 2 prove that 𝜕2 𝑢 + 𝜕2 𝑢 1 = 𝑓 ′′ (𝑟) + 𝑓′(𝑟). 𝜕𝑥 2 𝜕𝑦 2 𝑟 5. If 𝑢 = 𝑓(𝑦 − 𝑧, 𝑧 − 𝑥, 𝑥 − 𝑦) find the value of 𝜕𝑢 + 𝜕𝑢 𝜕𝑢 + 𝜕𝑧 𝜕𝑥 𝜕𝑦 6. If 𝑢 = 𝑡𝑎𝑛−1 𝑥 𝑦 𝑤ℎ𝑒𝑟𝑒 𝑥 = 𝑒 𝑡 − 𝑒 −𝑡 , 𝑦 = 𝑒 𝑡 + 𝑒 −𝑡 , find 𝑑𝑢 𝑑𝑡 7. 2 3 𝐼𝑓 𝑢 = 𝑥 + 3𝑦 − 𝑧 , 𝑣 = 4𝑥 𝑦𝑧, 𝑤 = 2𝑧 − 𝑥𝑦 2 2 evaluate 𝜕(𝑢,𝑣,𝑤) 𝑎𝑡 𝑡ℎ𝑒 point (1, −1,0). 𝜕(𝑥,𝑦,𝑧) 8. If 𝑥 = 𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠∅, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛∅, 𝑧 = 𝑟 𝑐𝑜𝑠𝜃 prove that 𝐽 = 𝑟 2 𝑠𝑖𝑛𝜃. 9. If u = 𝑦𝑧 , v= 𝑧𝑥 , w= 𝑥𝑦 𝜕(𝑢,𝑣,𝑤) Prove that 𝜕(𝑥,𝑦,𝑧) = 4. 𝑥 𝑦 𝑧 𝜋 10. Expand sin(𝑥𝑦) in powers of (𝑥 − 1)& (𝑦 − 2 ) upto second degree term by Taylor’s series. 11. 𝜋 Expand by Taylor’s series of 𝑒 𝑥 𝑐𝑜𝑠𝑦 at (0, 4 ) upto second degree terms 12. Expand 𝑓(𝑥) = 𝑒 𝑥𝑦 in Taylor’s series at (1,1) upto second degree terms. 13. 𝑦 Expand tan−1 (𝑥 ) about the point (1, 1) upto second degree term by Taylor’s series. 14. Find the extreme value of the function 𝑓(𝑥) = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦, 𝑎 > 0. 15. Find the extreme values of 𝑥𝑦(𝑎 − 𝑥 − 𝑦); 𝑎 > 0. 16. Find the extreme values of 𝑓(𝑥, 𝑦) = 𝑥 3 𝑦 2 (1 − 𝑥 − 𝑦) 17. 1 𝑥 ∝ −1 Evaluate∫0 𝑑𝑥, ∝ ≥ 0, by applying differentiation under integral sign. 𝑙𝑜𝑔𝑥 18. Show that ∞ 𝑡𝑎𝑛−1 𝑎𝑥 ∫0 𝑥(1+𝑥 2 ) 𝑑𝑥 𝜋 = 2 log(1 + 𝑎) , 𝑎 ≥ 0 by applying differentiation under integral sign. 19. ∞ Evaluate∫0 𝑒 −𝛼𝑥 𝑠𝑖𝑛𝑥 𝑑𝑥 , by applying differentiation under integral sign and hence show 𝑥 ∞ 𝑠𝑖𝑛𝑥 𝜋 that ∫0 𝑥 𝑑𝑥 = 2. 20. ∞ 𝑒 −𝑥 (1−𝑒 −𝑎𝑥 ) Show that ∫0 𝑑𝑥 = log(1 + 𝑎) , 𝑎 ≥ −1. by applying differentiation under 𝑥 integral sign  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 4 (Integral Calculus) Q.No Question 1. 1 √𝑥 Evaluate: ∫0 ∫𝑥 (𝑥 2 + 𝑦 2 )𝑑𝑦𝑑𝑥 2. Evaluate ∫0 𝜋⁄ 4 √𝑐𝑜𝑠2𝜃 𝑟 𝑑𝑟 𝑑𝜃 ∫0 (1+𝑟 2 )2 3. 𝑐 𝑏 𝑎 Evaluate: ∫−𝑐 ∫−𝑏 ∫−𝑎(𝑥 2 + 𝑦 2 + 𝑧 2 ) 𝑑𝑥𝑑𝑦𝑑𝑧 4. 1 √1−𝑥 2 √1−𝑥 2 −𝑦 2 Evaluate: ∫0 ∫0 ∫0 𝑥𝑦𝑧 𝑑𝑥𝑑𝑦𝑑𝑧. 5. Evaluate∬ 𝑥𝑦𝑑𝑥𝑑𝑦 over the positive quadrant of the circle 𝑥2 + 𝑦2 = 𝑎2 6. Evaluate ∫ ∫ 𝑥𝑦(𝑥 + 𝑦)𝑑𝑦 𝑑𝑥 taken over the area between 𝑦 = 𝑥 2 and y = x. 7. 3 Evaluate ∫0 ∫1 √4−𝑦 (𝑥 + 𝑦)𝑑𝑥 𝑑𝑦 by changing the order of integration. 8. 1 √𝑥 Evaluate ∫0 ∫𝑥 𝑥𝑦 𝑑𝑦 𝑑𝑥 , by changing order of integration ∞ ∞ 9. Evaluate∫0 ∫0 𝑒 −(𝑥 2 +𝑦 2 ) 𝑑𝑥𝑑𝑦 by changing to polar coordinates. Hence show that ∞ 2 √𝜋 ∫0 𝑒 −𝑥 𝑑𝑥 =. 2 10. 𝑎 √𝑎2 −𝑦 2 Evaluate ∫0 ∫0 𝑦√𝑥 2 + 𝑦 2 𝑑𝑦𝑑𝑥 by transforming to polar co-ordinates. 11. Evaluate the area enclosed between the parabola 𝑦 = 𝑥 2 and the straight line y = x. 12. Find the area enclosed by the ellipse 𝑎2 + 𝑏2 = 1 𝑥2 𝑦2 13. Calculate the volume of the solid bounded by the planes x =0 ,y =0, x + y + z =1 and z =0. 14. 1 1 4 Evaluate ∫0 [log(𝑥)] dx 15. Prove that 𝛽(𝑚, 𝑛) = Γ(𝑚)Γ(𝑛) Γ(𝑚+𝑛) 16. Express ∞ 𝑑𝑥 ∫0 1+𝑥 4 in terms of Beta function and hence evaluate. 𝜋 𝜋 17. 1 Prove that ∫02 √sin 𝑥 𝑑𝑥 × ∫02 𝑑𝑥 = 𝜋 √sin 𝑥 18. 1 Show that ∫−1(1 + 𝑥)𝑝−1 (1 − 𝑥)𝑞−1 𝑑𝑥 = 2𝑝+𝑞−1 𝛽(𝑝 , 𝑞) 19. Prove that 𝛽(𝑚, 𝑛) = 𝛽(𝑛, 𝑚) ∞ ∞ 𝜋 20. 8 Prove that ∫0 𝑥 𝑒 −𝑥 𝑑𝑥 𝑋 ∫0 𝑥 2 𝑒 −𝑥 𝑑𝑥 = 16√2 4  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 5 (Ordinary Differential Equations) Q.No Question 1. Solve 𝑑𝑦 1 1 − 2 (1 + 𝑥) 𝑦 + 3𝑦 3 =0 𝑑𝑥 𝑥 2. 𝑑𝑦 Solve 𝑑𝑥 − 𝑦𝑡𝑎𝑛𝑥 = 𝑦2 𝑠𝑒𝑐𝑥 3. 𝑑𝑟 Solve 𝑟 sin 𝜃 − 𝑐𝑜𝑠𝜃 𝑑𝜃 = 𝑟 2 4. Solve(5𝑥4 + 3𝑥2 𝑦2 − 2𝑥𝑦3 ) 𝑑𝑥 + (2𝑥3 𝑦 − 3𝑥2 𝑦2 − 5𝑦4 ) 𝑑𝑦 = 0 5. Solve 𝑠𝑒𝑐 2 𝑥 𝑡𝑎𝑛𝑦 𝑑𝑥 + 𝑠𝑒𝑐 2 𝑦 𝑡𝑎𝑛𝑥 𝑑𝑦 = 0 6. Solve (𝑥2 𝑦 − 2𝑥𝑦2 )𝑑𝑥 − (𝑥3 − 3𝑥2 𝑦)𝑑𝑦 = 0 7. Solve (𝑥 2 𝑦)𝑑𝑥 − (𝑥 3 +𝑦 3 )𝑑𝑦 = 0 8. Solve (𝑥 2 + 𝑦 2 + 𝑥)𝑑𝑥 + 𝑥𝑦𝑑𝑦 = 0 9. Solve (𝑥𝑦 + 𝑦 2 )𝑑𝑥 + (𝑥 + 2𝑦 − 1)𝑑𝑦 = 0 10. Solve (𝑥𝑦 𝑠𝑖𝑛 𝑥𝑦 + cos 𝑥𝑦)𝑦 𝑑𝑥 + (𝑥𝑦 sin 𝑥𝑦 − 𝑐𝑜𝑠𝑥𝑦)𝑥𝑑𝑦 = 0 11. Solve 𝑦 𝑑𝑥 − 𝑥𝑑𝑦 + 3𝑥 2 𝑦 2 𝑒 𝑥 𝑑𝑥 = 0 3 12. Find the orthogonal trajectories of 𝑦 2 = 4 𝑎 (𝑥 + 𝑎) and show that it is self-orthogonal where a is the parameter. 13. Find the orthogonal trajectory of the family of curves 𝑟 𝑛 = 𝑎 sin 𝑛𝜃 14. A body kept in air with temperature 25𝑜 C cools from 140𝑜 C to 80𝑜 C in 20 mins. Find the temperature when the body cools down to 35𝑜 C. 15. If a substance cools from 370 k to 330 k in 10 mins, when the temperature of the surrounding air is 290 k , find the temperature of the substance after 40 mins. 16. The L-R series circuit the differential equation for the current i in an electrical circuit containing an inductance L and a resistance R in series and acted on by an electromagnetic 𝑑𝑖 force E sin 𝜔t satisfies the equation L𝑑𝑡 + 𝑅𝑖 = 𝐸 sin 𝜔𝑡. Find the value of the current at any time t, if initially there is no current in the circuit. 17. Solve 𝑝2 + 2𝑝𝑦 𝑐𝑜𝑡𝑥 = 𝑦2 18. Solve 𝑝3 + 2𝑥𝑝2 − 𝑦 2 𝑝2 − 2𝑥𝑦 2 𝑝 = 0 19. Solve 𝑝 = sin(𝑦 − 𝑥𝑝).Also find its singular solution 20. Solve 𝑝 = log(𝑝𝑥 − 𝑦). Also find its singular solution  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 1 – LINEAR ALGEBRA Q.No. Questions 1. 3 4 5 The matrix 𝐴 = (0 6 9) is in _______ form. 0 0 0 a) Echelon b) Normal c) Matrix d) None of these 2. The rank of the matrix A in the echelon form is equal to _______. a) Number of non-zero columns c) Number of zero rows b) Number of non-zero rows d) Number of zero columns 3. 2 3 Find the Rank of the matrix A= [ ] 4 6 a) 0 b) 2 c) 1 d) None of these 4. 1 2 5 ∶ 3 Find the value of 𝐾, if (𝐴: 𝐵) = ( ) and 𝜌(𝐴) = 𝜌(𝐴: 𝐵) 0 0 0 ∶𝐾−1 a) – 1 b) – 2 c) 2 d) 1 5. If 𝜌(𝐴) = 𝜌(𝐴: 𝐵) = 𝑛(number of unknowns) then system of non-homogeneous equations are_______ a) Consistent and gives Unique solution c) Inconsistent and gives Unique solution b) Consistent and gives Trivial solution d) Inconsistent and gives Trivial solution 6. The system of linear equations are inconsistent if _____ a) 𝜌(𝐴) = 𝜌(𝐴: 𝐵) b) 𝜌(𝐴) ≠ 𝜌(𝐴: 𝐵) c) 𝜌(𝐴) < 𝜌(𝐴: 𝐵) d) 𝜌(𝐴) > 𝜌(𝐴: 𝐵) 7. If 𝜌(𝐴) = 𝜌(𝐴: 𝐵) < 𝑛(number of unknowns) then system of linear homogeneous equations are_______ a) Consistent and gives infinite number of solution b) Inconsistent and gives Trivial solution c) Consistent and gives Trivial solution d) Inconsistent and gives infinite number of solution 8. In Gauss elimination method, coefficient matrix is reduced to ________matrix a) Unit b) Upper triangular c) Lower triangular d) Diagonal 9. In Diagonalization 𝑃 −1 𝐴 𝑃 = ___________ a) P b) A c) D d) 𝑃−1 10. Which method is used to calculate the Dominant Eigen value and the corresponding Eigen vector of the square matrix A a) Gauss elimination method c) Rayleigh Power method b) Diagonalization method d) None of these  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 2 - DIFFERENTIAL CALCULUS Q.No. Questions 1. The angle between the radius vector and the tangent is______. 1 𝑑𝑟 1 𝑑𝑟 𝑑𝑟 𝑑𝑟 a) 𝑐𝑜𝑡 ∅ = 𝑟 𝑑𝜃 b) 𝑡𝑎𝑛 ∅ = 𝑟 𝑑𝜃 c) 𝑐𝑜𝑡 ∅ = 𝑟 𝑑𝜃 d) 𝑡𝑎𝑛 ∅ = 𝑟 𝑑𝜃 2 If the two curves intersect orthogonally, then the angle between these two curves is______. 𝜋 𝜋 𝜋 𝜋 a) 3 b) 2 c) 6 d) 4 3. The rate at which the curve is bending, is called ______. a) Radius of curvature b) radius c) curvature d) arc length 4. The radius of curvature for curve 𝑦 = 𝑓(𝑥) is given by is ______. 3 3 3 3 (1+𝑦1 )2 (1+𝑦22 )2 (1+𝑦12 )2 (1+𝑦2 )2 a) 𝜌 = b) 𝜌 = c) 𝜌 = d) 𝜌 = 𝑦2 𝑦1 𝑦2 𝑦1 5. The radius of curvature of the circle with the radius 𝑟 is ______. a) r = constant b) r ≠ 0 c) r = 0 d) r ≠ constant 6. The radius of curvature for polar curve 𝑟 = 𝑓(𝜃) is given by_______. 3 3 3 3 (𝑟 2 −𝑟12 )2 (𝑟 2 +𝑟12 )2 (𝑟 2 −𝑟12 )2 (𝑟 2 +𝑟12 )2 a) 𝜌 = b) 𝜌 = c) 𝜌 = d) 𝜌 = 𝑟 2 −2𝑟12 −𝑟𝑟2 𝑟 2 +2𝑟12 −𝑟𝑟2 𝑟 2 −2𝑟12 +𝑟𝑟2 𝑟 2 +2𝑟12 +𝑟𝑟2 7. Write Maclaurin’s series expansion of 𝑒 𝑥. 𝑥2 𝑥2 𝑥2 𝑥2 a) 1 + 𝑥 + 2 +⋯ b) 1 − 𝑥 + 2 +⋯ c) 1 + 𝑥 − 2 +⋯ d) 1 − 𝑥 − 2 +⋯ 8. If (𝑎, 𝑏) are the coordinates of the centre of curvature whose curvature is 𝑘, then the equation of the circle of curvature is________. 1 a) (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑘 2 c) (𝑥 + 𝑎)2 + (𝑦 + 𝑏)2 = 𝑘 2 1 b) (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 2 d) (𝑥 + 𝑎)2 + (𝑦 + 𝑏)2 = 𝑘 2 𝑘 9. The locus of the centre of curvature for a curve is called ________ a) Involute b) Evolute c) circle d) None of these 10. Centre of curvature at any point 𝑃(𝑥, 𝑦) on the curve 𝑦 = 𝑓(𝑥) then 𝑦̅ = ________ 1+𝑦12 1+𝑦12 1−𝑦12 1−𝑦12 a) 𝑦 + b) 𝑦 − c) 𝑦 + d) 𝑦 − 𝑦2 𝑦2 𝑦2 𝑦2  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 3 (Partial Differentiation) Q.No. Questions 1. 𝑦 𝜕𝑢 If 𝑢 = 𝑥 , then find. 𝜕𝑦 a) 𝑦 𝑥 𝑙𝑜𝑔𝑥 b) 𝑥 𝑦 𝑙𝑜𝑔𝑦 c) 𝑥 𝑦 𝑙𝑜𝑔𝑥 d) 𝑦 𝑥 𝑙𝑜𝑔𝑦 2. 𝜕𝑢 If 𝑢 = log(𝑥 2 + 𝑦 2 + 𝑧 2 ) , 𝑡ℎ𝑒𝑛 is -------- 𝜕𝑧 2𝑥 2𝑦 2𝑧 2𝑧 a) 𝑥 2 +𝑦 2+𝑧 2 b) 𝑥 2 +𝑦 2+𝑧 2 c) 𝑥 2 +𝑦 2 +𝑧 2 d) 𝑥 2 +𝑦 2 −𝑧 2 3. If 𝑓(𝑥, 𝑦)and y is a function of x, then 𝑑𝑢 𝜕𝑢 𝜕𝑢 𝑑𝑦 𝜕𝑢 𝑑𝑢 𝜕𝑢 𝑑𝑦 𝑑𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑦 𝜕𝑢 𝑑𝑢 𝜕𝑢 𝜕𝑦 a) 𝑑𝑥 = 𝜕𝑥 + 𝜕𝑦 𝑑𝑥 b) 𝜕𝑥 = 𝑑𝑥 + 𝜕𝑦 𝑑𝑥 c) 𝑑𝑥 = 𝜕𝑥 + 𝜕𝑦 𝜕𝑥 d) 𝜕𝑥 = 𝑑𝑥 + 𝜕𝑦 𝜕𝑥 4. 𝑢 𝜕(𝑥,𝑦) If 𝑥 = 𝑢𝑣 , 𝑦 = 𝑣 , then is equal to --------- 𝜕(𝑢,𝑣) −2𝑢 −2𝑣 a) 1 b) c) d) Zero 𝑣 𝑢 5. 𝑢,𝑣,𝑧 If 𝑢 = 𝑥 + 𝑦 + 𝑧, 𝑣 = 𝑦 + 𝑧, 𝑧 = 𝑧 𝑡ℎ𝑒𝑛 𝐽 (𝑥,𝑦,𝑧)is equal to a) 1 b) -1 c) 0 d) None of these 6. The Jacobian of transformation from the Cartesian to polar coordinates system is a) 𝑟 3 b) 𝑟 c) 𝑟 2 d) 𝑆𝑖𝑛𝑟 7. If 𝑢 = 𝑓(𝑥, 𝑦) and let 𝑦 be a function of 𝑥 and also 𝑓(𝑥, 𝑦) = 𝑐, c being a constant 𝑑𝑦 then 𝑑𝑥 = ________. 𝑢𝑥 𝑢 𝑢𝑦 𝑢𝑦 a) b) − 𝑢𝑥 c) d) − 𝑢 𝑢𝑦 𝑦 𝑢𝑥 𝑥 8. The set of necessary conditions for f(x , y) to have a maximum or minimum at (a, b) is that --------- a) 𝑓𝑥 (𝑎, 𝑏) = 0 = 𝑓𝑦 (𝑎, 𝑏) c) 𝑓𝑥 (𝑎, 𝑏) ≠ 0 ≠ 𝑓𝑦 (𝑎, 𝑏) b) 𝑓𝑥 (𝑎, 𝑏) = 1 = 𝑓𝑦 (𝑎, 𝑏) d) 𝑓𝑥 (𝑎, 𝑏) ≠ 1 ≠ 𝑓𝑦 (𝑎, 𝑏) 9. The stationary points of the function 𝑓(𝑥, 𝑦) = 𝑥 2 + 𝑥𝑦 2 + 𝑦 4 are ------------ a) (1,-1) b) (0,0) c) (2,3) d) None of these 10. If the sum of three numbers is constant, then their product is maximum when the numbers are a) equal b) not equal c) (1,2,3) d) None of these  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 4 (Integral Calculus) Q.No. Questions 1. Evaluate 𝑎 𝑏 ∫0 ∫0 𝑑𝑥𝑑𝑦 ab (a) 0 (b) (c) 2ab (d) ab 2 2. For evaluation of double integral by changing in to polar form, we use 𝑑𝑥 𝑑𝑦 = _____. (a) 𝑟 𝑑𝑟 𝑑𝜃 (b) 𝑟 2 𝑑𝑟 𝑑𝜃 (c) −𝑟 𝑑𝑟 𝑑𝜃 (d) 2𝑟 𝑑𝑟 𝑑𝜃 ∞ ∞ 3. For∫0 ∫𝑥 𝑓(𝑥, 𝑦) 𝑑𝑥 𝑑𝑦, the change of order is _____. ∞ ∞ ∞ ∞ (a) ∫𝑥 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 (b) ∫0 ∫𝑦 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 ∞ 𝑦 ∞ 𝑥 (c) ∫0 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 (d) ∫0 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 4.  dx dy R represents (a) Area of the region in polar form (b) Area of the region in Cartesian form (c) Both (a) & (b) (d) None of these 5. Volume of a solid is equal to (a) ∫ ∫ ∫ 𝑑𝑥 𝑑𝑦 𝑑𝑧 (b) ∫ ∫ 𝑑𝑥 𝑑𝑦 (c) ∫ ∫ 𝑥 𝑦 𝑑𝑥 𝑑𝑦 (d) None of these 6. Volume of a solid (in polars) obtained by the revolution of a curve enclosing an area A about the initial line is given by _____. (a)𝑉 = ∬𝐴 2πr 2 sinθ dr dθ (b) 𝑉 = ∬𝐴 2πr 2 cosθ dr dθ (c) 𝑉 = ∬𝐴 2πr 2 dr dθ (d) 𝑉 = ∬𝐴 2πr 2 tanθ dr dθ ∞ 7. The integral ∫0 𝑥 𝑛−1 𝑒 −𝑥 𝑑𝑥 is defined as _____. (a) Gamma function (b) Beta function (c) Proper function (d) Gamma and Beta function 8. In terms of gamma function, 𝛽(𝑚, 𝑛) is _____. 1 1 Γ(𝑚+ )Γ(𝑛+ ) Γ(𝑚)Γ(𝑛) Γ(𝑚)Γ(𝑛) Γ(𝑚)Γ(𝑛) (a) 2 2 (b) (c) (d) Γ(𝑚−𝑛) 2Γ(𝑚−𝑛) Γ(𝑚+𝑛) 2Γ(𝑚+𝑛) 9. 1 The value of Γ (2) = _____. √𝜋 (a) 2√𝜋 (b) (c) 3 √𝜋 (d) √𝜋 2 10. 1 3 The value of Γ ( ) Γ ( ) = _____. 4 4 (a) 𝜋√4 (b) 𝜋√2 (c) 3 √𝜋 (d) 5√𝜋  Engineering Mathematics –I Subject Code : 19MA1ICMAT Module – 5 (Ordinary Differential Equations) Q.No. Questions 1. 𝑑𝑦 2 3 𝑑2 𝑦 2 The order and degree of the differential equation [1 + (𝑑𝑥 ) ] = 𝑐 2 (𝑑𝑥 2 ) is ________ a) 2 , 2 b) 2 , 3 c) 3,2 d) 1,3 2. 𝑑3 𝑦 𝑑2 𝑦 The order and degree of D.E ( 3 )2 + ( 2 )6 + 𝑦 = 𝑥 4 , respectively are _________ 𝑑𝑥 𝑑𝑥 a) 6 , 2 b) 3 , 2 c) 2 , 6 d) 2 , 3 3. 𝑑𝑦 The differential equation of the form 𝑑𝑥 + 𝑃𝑦 = 𝑄𝑦 𝑛 is called as ____________ equation. a) Exact b) Homogeneous c) Bernoulli’s d) None of these 4. In the differential equation 𝑑𝑥 + 𝑃𝑥 = 𝑄𝑥 𝑛 , P and Q are functions of _______variable 𝑑𝑦 a) 𝑥 b) 𝑦 c) 𝑥 & 𝑦 d) None of these 5. For what value of b, the D.E (𝑥 + 𝑦 2 )𝑑𝑥 + (𝑦 2 − 𝑦 + 𝑏𝑥𝑦)𝑑𝑦 = 0 is exact? a) 1 b) 2 c) 3 d) 4 6. The necessary condition for the differential equation to be Exact is _____________ 𝜕𝑀 𝜕𝑁 𝜕𝑁 𝜕𝑀 𝜕𝑀 𝜕𝑁 a) 𝜕𝑦 = 𝜕𝑥 b) 𝜕𝑦 = 𝜕𝑥 c) 𝜕𝑦 ≠ 𝜕𝑥 d) None of these 7. If two family of curves are such that every member of one family intersects every member of the other family at right angles then they are said to be ___________ of each other a) Evolute b) Involute c) Orthogonal Trajectories d) None of these 8. 𝑑𝑦 In orthogonal trajectories , replace 𝑑𝑥 = ________ 𝑑𝑥 𝑑𝑥 𝑑𝑦 a) b) − 𝑑𝑦 c) − 𝑑𝑥 d) None of these 𝑑𝑦 9. A D.E. of the first order but of second degree , solvable for p, has the solution as _____ a) 𝑓1 (𝑥, 𝑦, 𝑐) 𝑓2 (𝑥, 𝑦, 𝑐) = 0 c) 𝑓1 (𝑥, 𝑦, 𝑐) 𝑓2 (𝑥, 𝑦) = 0 b) 𝑓1 (𝑥, 𝑦) 𝑓2 (𝑥, 𝑦, 𝑐) = 0 d) 𝑓1 (𝑥, 𝑦) 𝑓2 (𝑥, 𝑦) = 0 10. A governing first order differential equation by Kirchhoff’s law is given by _______ where L is the inductance, R is the resistance and E is the electromotive force. 𝑑𝑖 𝑑𝑖 𝑑𝑖 𝑑𝑖 a) 𝐿 𝑑𝑡 + 𝑅𝑖 = 𝐸 b) 𝐸 𝑑𝑡 + 𝑅𝑖 = 𝐿 c) 𝑅 𝑑𝑡 + 𝐿𝑖 = 𝐸 d) 𝐸 𝑑𝑡 + 𝐿𝑖 = 𝑅