I Semester B.Sc(SEP) Mathematics Past Paper PDF

BANGALORE UNIVERSITY DEPARTMENT OF MATHEMATICS I SEMESTER B.Sc.(SEP) MATHEMATICS MATT 1.1: MATHEMATICS -I QUESTION BANK UNIT-1: MATRICES 2 Marks Questions:...

BANGALORE UNIVERSITY DEPARTMENT OF MATHEMATICS I SEMESTER B.Sc.(SEP) MATHEMATICS MATT 1.1: MATHEMATICS -I QUESTION BANK UNIT-1: MATRICES 2 Marks Questions: 1 2 3 1. Reduce the following matrix to row reduced echelon form 𝐴 = [2 4 7] 3 6 9 1 3 −2 2. Find the rank of the matrix by reducing to row reduced echelon form [2 −1 4 ] 1 −1 4 3. Solve the system x + 2y + z = 0 ; 3x + 4y + 4z = 0 ; 7x + 10y + 12z = 0 4. Solve the system x + 3y − 2z = 0 ; 2x − y + 4z = 0; x − 11y + 14z = 0 5. If λ is an eigen value of he matrix A then prove that λ2 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 A2 6. If λ is an eigen value of he matrix A then prove that 1 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 A−1 , provided A is non singular λ 7. If A is any square matrix then prove that A and its transpose A𝑇 have the same eigen values 5 −1 8. Find the eigen values of the matrix A = [ ] 4 9 9. State Cayley- Hamilton theorem 1 4 10. Verify Cayley- Hamilton theorem for A=[ ] 2 3 5 Marks Questions: 1 3 4 3 1. Find the rank of the matrix by reducing to row reduced echelon form [3 9 12 9 ] 1 3 4 1 2. Find the rank of the matrix by reducing to row reduced echelon form 1 1 1 1 A=[1 2 3 4 ] 1 3 5 7 3. Find the rank of the matrix by reducing to row reduced echelon form 1 0 2 A = [ 2 −1 0 ] 2 0 4 4 −1 3 6 𝑎 −1 4. For what value of a, the rank of the matrix A is 3, where A = [2 3 1 ] 3 4 2 5. Find the rank of each of the following matrices by reducing to normal form. 1 −1 2 −3 1 1 −1 1 1 1 1 (i) [2 −3 4 ] (ii) [1 2 3 4 ] (iii) [ 4 1 0 2] 0 3 0 4 3 −2 3 1 3 5 7 0 1 0 2 6. Find the real value of μ such that the system x + μy + 3z = 0;2x + 3y + 4z = 0 ;7x + 13y + 19z = 0 has a non-trivial solution. 7. Find the real value of α such that the system 2x − y + 2z = 0; 3x + y − z = 0 ; α x − 2y + z = 0 has a nontrivial solution. 8. Test the following system for consistency and hence solve. 3x + y + z = 8 ;−x + y − 2z = −5 ; x + y + z = 6 ; −2x + 2y − 3z = −7 9. Solve the system by testing for consistency x + y + z = 4 ; 2 x + y − z = 1 ; x − y + 2z = 2 10. For What values of λ and μ, the equations x + y + z = 6 ; x + 2y + 3z = 10 ; x + 2y + λz = μ have (i) No solution (ii) Unique solution (iii) Infinitely many solutions. 11. For What values of λ , the equations x + y + z = 1 ; x + 2y + 4z = λ ; x + 4y + 10z = λ2 have a solution and solve. 12. Show that the equations − 2x + y + z = p ; x − 2y + z = q ; x + y − 2z = s have solution only if p + q + s = 0 13. Show that the system of equations 3x + 4y + 5z = 𝑎 ; 4x + 5y + 6z = 𝑏 ; 5x + 6y + 7z = c do not have a solution unless 𝑎 + 𝑐 = 2𝑏 −3 8 14. Find the eigen values and eigen vectors of the matrix A = [ ]. −2 7 15. Find the eigen values and the corresponding eigen vectors of the matrix 1 0 −1 A = [1 2 1 ]. 2 2 3 1 2 16. Find the eigen values and eigen vectors of the matrix A = [ ] 5 4 3 1 17. Verify Cayley-Hamilton theorem for 𝑨=[ ] −1 2 1 0 2 18. Verify Cayley-Hamilton theorem for the matrix 𝐴 = [0 2 1] 2 0 3 1 0 2 19. By using Cayley- Hamilton theorem , find inverse of the matrix A=[0 2 1] 2 0 3 4 −1 −2 1 3 20. By using Cayley- Hamilton theorem , find 𝐴 , 𝐴 𝑎𝑛𝑑 𝐴 for A = [ ] 2 5 1 4 and hence find 3 4 𝑎𝑛𝑑 −1 21. Verify Cayley- Hamilton theorem for A=[ ] 𝐴 ,𝐴 𝐴 2 3 UNIT-2: DIFFERENTIAL CALCULUS 2 Marks Questions: 1. Find the nth derivatives of the following functions: 1 1 i) log(x + 3) ii) log (3 − 2x) iii) (x+2)(x−1) iv ) 6x2 −5x+1 v)ex+4 vi)24x+1 vii) e2x cos 3x viii) ex sin 2x 2. Show that D2n (x 2 − 1)n = (2n)! 3. Find the first and second order partial derivatives of 𝑧 = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦. 𝜕2 𝑢 𝜕2 𝑢 4. If 𝑢 = 𝑥 𝑦 show that = 𝜕𝑦𝜕𝑥 𝜕𝑥𝜕𝑦 𝜕𝑟 2 𝜕𝑟 2 5. If 𝑥 = 𝑟 𝑐𝑜𝑠𝜃 and 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃 prove that(𝜕𝑥) + (𝜕𝑦) = 1 𝜕𝑢 𝜕𝑢 𝑦 6. Find and for 𝑢 = sin−1 ( ) 𝜕𝑥 𝜕𝑦 𝑥 𝜕2 𝑢 𝜕2 𝑢 7. Verify that 𝜕𝑥𝜕𝑦 = 𝜕𝑦𝜕𝑥 for 𝑢 = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦 𝑑𝑦 8. If 𝑥 𝑦 + 𝑦 𝑥 = 𝑐, a constant, find 𝑑𝑥 𝜕(𝑢,𝑣) 9. If 𝑢 = 𝑥 2 and 𝑣 = 𝑦 2 find 𝜕(𝑥,𝑦) 𝜕(𝑢,𝑣) 10. If 𝑢 = 2𝑥 − 3𝑦 and 𝑣 = 5𝑥 + 4𝑦 , show that = 23 𝜕(𝑥,𝑦) 𝜕(𝑢,𝑣) 11. If 𝑢 = 𝑥 2 − 2𝑦 and 𝑣 = 𝑥 + 𝑦 , show that = 2(𝑥 + 1) 𝜕(𝑥,𝑦) 𝜕(𝑥,𝑦) 12. If 𝑥 = 𝑟 cos 𝜃 and 𝑦 = 𝑟 sin 𝜃 , show that =𝑟 𝜕(𝑟,𝜃) 5 Marks Questions: 1. Find the nth derivative of sinx. sin2x. sin3x 2. Find the nth derivative of (i)sin2 x cos 3 x (ii) cos 3 x (iii) sin4 x 3. Find the nth derivative of x n logx (n−1)! 4. If y = x n−1 logx show that yn = x th 3 𝑎𝑥 5. Find the n derivative of 𝑥 𝑒 by using Leibnitz’s Theorem 6. If 𝑦 = (𝑠𝑖𝑛−1 𝑥)2 show that (1 − 𝑥 2 )𝑦𝑛+2 − (2𝑛 + 1)𝑥𝑦𝑛+1 − 𝑛2 𝑦𝑛 = 0 2 7. If 𝑦 = (𝑙𝑜𝑔( 𝑥 + √1 + 𝑥 2 )) show that (1 + 𝑥 2 )𝑦𝑛+2 + (2𝑛 + 1)𝑥𝑦𝑛+1 + 𝑛2 𝑦𝑛 = 0 y x n 8. If cos −1 (b) = log (n) , show that 𝑥 2 𝑦𝑛+2 + (2𝑛 + 1)𝑥𝑦𝑛+1 + 2𝑛2 𝑦𝑛 = 0 9. If 𝑦 = 𝑎 𝑐𝑜𝑠(𝑙𝑜𝑔𝑥) + 𝑏𝑠𝑖𝑛(𝑙𝑜𝑔𝑥)  show that 𝑥 2 𝑦𝑛+2 + (2𝑛 + 1)𝑥𝑦𝑛+1 + (𝑛2 + 1)𝑦𝑛 = 0 −1 𝑥 10. 𝐼𝑓 𝑦 = 𝑒 𝑚𝑐𝑜𝑠 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 (1 − 𝑥 2 )𝑦𝑛+2 − (2𝑛 + 1)𝑥𝑦𝑛+1 − (𝑛2 − 𝑎2 )𝑦𝑛 = 0 𝑥 2 +𝑦 2 11. If 𝑢 = log ( ) show that 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 1 𝑥+𝑦 𝑦 𝑥 𝜕2 𝑧 𝑥 2 −𝑦 2 12. If 𝑧 = 𝑥 2 tan−1 (𝑥 ) − 𝑦 2 tan−1 (𝑦) prove that = 𝑥 2 +𝑦 2 𝜕𝑥𝜕𝑦 𝑥 2 +𝑦 2 𝜕𝑢 𝜕𝑢 13. If 𝑢 = sin−1 ( ) prove that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = tan 𝑢 𝑥+𝑦 𝜕2 𝑧 𝜕2 𝑧 14. If 𝑧 = 𝑓(𝑥 + 𝑎𝑦) + 𝑔(𝑥 − 𝑎𝑦) show that 𝜕𝑦 2 = 𝑎2 𝜕𝑥 2 𝜕𝑢 𝜕𝑢 𝜕𝑢 3 15. If 𝑢 = log(𝑥 3 + 𝑦 3 + 𝑧 3 − 3𝑥𝑦𝑧) then prove that + 𝜕𝑦 + 𝜕𝑧 = 𝑥+𝑦+𝑧 𝜕𝑥 𝜕2 𝑟 𝜕2 𝑟 1 16. If 𝑥 = 𝑟 𝑐𝑜𝑠𝜃 and 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃 prove that + = 𝜕𝑥 2 𝜕𝑦 2 𝑟 17. State and prove Euler’s theorem on homogeneous functions 18. If 𝑢 = 𝑓(𝑥, 𝑦) is a homogeneous function of 𝑥 and 𝑦 with degree 𝑛, then prove that 2 𝜕 2𝑢 𝜕 2𝑢 2 𝜕 2𝑢 𝑥 + 2𝑥𝑦 +𝑦 = 𝑛(𝑛 − 1)𝑢 𝜕𝑥 2 𝜕𝑥𝜕𝑦 𝜕𝑦 2 19. Verify Euler’s theorem for 𝑢 = 𝑥 3 + 𝑥 2 𝑦 + 𝑥𝑦 2 + 𝑦 3 𝑥 4 +𝑦 4 𝜕𝑢 𝜕𝑢 20. If 𝑢 = log ( ) , show that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = 3 𝑥−𝑦 21. Verify Euler’s theorem for 𝑢 = 𝑎𝑥 2 + 2ℎ𝑥𝑦 + 𝑏𝑦 2 22. If 𝑢 = 𝑓(𝑥, 𝑦) where 𝑥 = 𝑟 cos 𝜃 and 𝑦 = 𝑟 sin 𝜃, prove that 𝜕𝑢 2 𝜕𝑢 2 𝜕𝑢 2 1 𝜕𝑢 2 (𝜕𝑥 ) + (𝜕𝑦) = ( 𝜕𝑟 ) + 𝑟 2 (𝜕𝜃) 𝜕(𝑢,𝑣,𝑤) 23. If 𝑢 = 𝑥𝑦 2 , 𝑣 = 𝑦𝑧 2 , 𝑤 = 𝑧𝑥 2 , show that = 9𝑥 2 𝑦 2 𝑧 2 𝜕(𝑥,𝑦,𝑧) 24. If 𝑢 = 𝑥𝑦𝑧, 𝑣 = 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥, 𝑤 = 𝑥 + 𝑦 + 𝑧, show that 𝜕(𝑢,𝑣,𝑤) = (𝑥 − 𝑦)(𝑦 − 𝑧)(𝑧 − 𝑥) 𝜕(𝑥,𝑦,𝑧) 𝑦𝑧 𝑧𝑥 𝑥𝑦 𝜕(𝑢,𝑣,𝑤) 25. If 𝑢 = ,𝑣 = ,𝑤 = , show that =4 𝑥 𝑦 𝑧 𝜕(𝑥,𝑦,𝑧) 26. If 𝑥 = 𝑟 cos 𝜃 cos 𝜑 , 𝑦 = 𝑟 cos 𝜃 sin 𝜑 , 𝑧 = 𝑟 sin 𝜃, show that 𝜕(𝑥,𝑦,𝑧) = 𝑟 2 cos 𝜃 𝜕(𝑟,𝜃,𝜑) 𝜕(𝑥,𝑦) 𝜕(𝑟,𝜃) 27. If 𝑥 = 𝑟 cos 𝜃, 𝑦 = 𝑟 sin 𝜃 , find 𝐽 = 𝜕(𝑟,𝜃) and 𝐽′ = 𝜕(𝑥,𝑦). Also verify 𝐽𝐽′ = 1 28. If 𝑢 = 𝑥 2 − 𝑦 2 and 𝑣 = 2𝑥𝑦 where 𝑥 = 𝑟 cos 𝜃, 𝑦 = 𝑟 sin 𝜃 show that 𝜕(𝑢,𝑣) = 4𝑟 3 𝜕(𝑟,𝜃) UNIT-3: INTEGRAL CALCULUS 2 Marks Questions: 𝜋⁄ 1. Evaluate∫0 2 𝑠𝑖𝑛5 𝑥 𝑑𝑥 𝜋⁄ 2. Evaluate∫0 2 𝑐𝑜𝑠 8 𝑥 𝑑𝑥 𝜋⁄ 3. Evaluate ∫0 4 𝑐𝑜𝑠 6 2𝑥 𝑑𝑥  𝜋 4. Evaluate∫ 2𝜋 sin5 𝑥 𝑑𝑥 − 2 𝜋/2 5. Evaluate∫0 sin5 𝑥 cos4 𝑥 𝑑𝑥 𝜋/2 6. Evaluate ∫0 sin8 𝑥 cos6 𝑥 𝑑𝑥 𝜋/2 7. Evaluate∫0 sin7 𝑥 cos5 𝑥 𝑑𝑥 𝜋 8. Evaluate∫04 𝑡𝑎𝑛6 𝑥 𝑑𝑥 𝜋⁄ 9. Evaluate∫0 4 𝑠𝑒𝑐 𝑛 𝑥 𝑑𝑥 10. Write the formula to find the arc length of the curve 𝑦 = 𝑓(𝑥) 11. Write the formula to find the arc length of the polar curve 𝑟 = 𝑓(𝜃) 12. Write the formula to find the surface area generated by revolving about the 𝑥-axis 13. Write the formula to find the surface area generated by revolving about the 𝑦-axis 14. Write the formula to find the volume generated by the curve 𝑦 = 𝑓(𝑥) revolving about the 𝑦-axis 15. Write the formula to find the volume generated by the curve 𝑟 = 𝑓(𝜃)revolving about the initial line 5 Marks Questions: 1. Evaluate ∫ 𝑐𝑜𝑠 7 𝑥 𝑑𝑥 2. Evaluate ∫ 𝑠𝑖𝑛6 𝑥 𝑑𝑥 1 𝑥9 3. Evaluate ∫0 √1−𝑥 2 𝑑𝑥 𝜋 5⁄ 4. Evaluate∫0 (1 + 𝑐𝑜𝑠𝑥) 2 𝑑𝑥 𝜋 𝑠𝑖𝑛4 𝑥 5. Evaluate ∫0 𝑑𝑥 (1+𝑐𝑜𝑠𝑥)2 3𝜋 𝑥 6. Evaluate ∫0 𝑠𝑖𝑛6 (6) 𝑑𝑥 𝜋 7. Evaluate ∫0 𝑥 sin6 𝑥 cos4 𝑥 𝑑𝑥 𝜋/6 8. Evaluate∫0 cos4 3𝜃 sin3 6𝜃 𝑑𝜃 ∞ 𝑥6 9. Evaluate∫0 𝑑𝑥 (1+𝑥 2 )9/2 1 10. Evaluate∫0 𝑥 2 (1 − 𝑥 2 )3/2 𝑑𝑥 𝜋⁄ 11. Evaluate∫0 4 𝑡𝑎𝑛 𝑛 𝑥 𝑑𝑥 𝜋⁄ 12. Evaluate ∫𝜋⁄ 2 𝑐𝑜𝑡 7 𝑥 𝑑𝑥 4 𝜋 13. Find the arc length of the curve 𝑦 = 𝑙𝑜𝑔(𝑠𝑒𝑐𝑥) form 𝑥 = 0 to 𝑥 = 3 13. Find the length of the arc of the catenary 𝑦 = 𝑐 𝑐𝑜𝑠ℎ(𝑥/𝑐) from 𝑥 = 0 and 𝑥 = 𝑎  2 2 2 14. Find the length of the astroid 𝑥 ⁄3 + 𝑦 ⁄3 = 𝑎 ⁄3 15. Find the length of an arch of the cycloid 𝑥 = 𝑎(𝜃 + 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃), 𝜃 = −𝜋 𝑡𝑜 𝜋 16. Find the arc length of the astroid 𝑥(𝑡) = 𝑎𝑐𝑜𝑠 3 𝑡, 𝑦(𝑡) = 𝑎𝑠𝑖𝑛3 𝑡. 17. Find the perimeter of the curve 𝑟 = 𝑎𝑐𝑜𝑠𝜃 18. Find the arc length of the parabola 𝑦 2 = 4𝑎𝑥 cut off by the latus rectum. 𝑥2 𝑦2 19. Find the area bounded by the ellipse + 𝑏2 = 1 𝑎2 20. Find the area bounded by the astroid 𝑥 2/3 + 𝑦 2/3 = 𝑎2/3 21. Find the area of the surface generated by revolving about the 𝑦-axis the curve 𝑦 = 𝑥 3 from 𝑥 = 0 to 𝑥 = 2 22. Find the surface area of a hemisphere of radius ‘𝑎’ 23. Find the surface area of the solid generated when one arch of the cycloid 𝑥 = 𝑎(𝜃 + 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃) revolves about 𝑥-axis. 24. Find the surface area of the solid generated when the cardioid 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃) revolves about the initial line 25. Find the volume of the solid generated by revolving the astroid 𝑥 2/3 + 𝑦 2/3 = 𝑎2/3. 26. Find the volume generated by revolving one arc of the cycloid 𝑥 = 𝑎(𝜃 − 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃) about its base. 27. Find the volume of the solid obtained by revolving the cardioid 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃) about the initial line 4 28. Show that the volume of a sphere of radius ‘𝑎’ is 3 𝜋𝑎3 UNIT-4: ANALYTICAL GEOMETRY 2 Marks Questions: 1. Find the equation of the sphere whose centre is (2, -3, 4) and radius 5. 2. Find the equation of the sphere whose centre is (5, -2, 3) and radius 5. 3. Find the Centre and radius of the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑦 − 4𝑧 − 11 = 0 4. Find the Centre and radius of the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 6𝑥 + 8𝑦 − 10𝑧 + 1 = 0 5. If a right circular cone has three mutually perpendicular generators, show that the semi vertical angle is tan−1 √2 6. Find the equation to the right circular cone whose vertex is at the origin, whose semi vertical angle is 300 and has its axis along y – axis. 7. Find the equation to the right circular cone whose vertex is at the origin, whose semi vertical angle is 300 and has its axis along z – axis. 8. Find the equation of the cone whose vertex is at the origin, semi vertical angle 450 and the axis is x = y = z. 9. Find the equation of the right circular cylinder with radius 2 units and the axis is the line 𝑥 𝑦 𝑧 = 2 = −1 2 10. Find the equation of the right circular cylinder with radius R and the axis is the line 𝑥−𝑥1 𝑦−𝑦 𝑧−𝑧 = 𝑚1 = 𝑛1 𝑙 11. Write the equations of hyperboloid of one sheet with figure. 12. Write the equations of hyperboloid of two sheets with figure. 5 Marks Questions: 1. Find the equation of the sphere passing through the following points (1, 1, 1), (1, 2, 1), (1, 1, 2), (2, 1, 1) 2. Find the equation of the sphere passing through the following points (0, 0, 0), (0, 1, -1), (-1, 2, 0), (1, 2, 3) 3. Find the equation of the sphere having the points (2, 1,-3) and (1, -2, 4) as the ends of a diameter. Find its centre and radius. 4. Find the equation of the sphere having the points (1, 2, 3) and (0, 4, -1) as the ends of a diameter. Find its centre and radius. 5. Find the equation of the tangent plane to the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 + 2𝑥 + 4𝑦 − 6𝑧 − 6 = 0 at (1, 2, 3) 6. Find the equation of the tangent plane to the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 + 𝑥 − 3𝑦 − 2𝑧 − 9 = 0 at (3, 2, 1) 7. Show that the plane 2x + 3y + 4z = 58 touches the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 4𝑥 − 6𝑦 − 8𝑧 = 0 and find the point of contact. 8. Find the tangent planes to the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 4𝑥 + 2𝑦 − 6𝑧 + 5 = 0 which are parallel to the plane 2x + 2y – z = 0 9. Find the value of ‘a’ for which the plane 2x + 2y +z = 9 touches the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑥 − 2𝑦 − 2𝑧 − 6 = 0 10. Show that the spheres 𝑥 2 + 𝑦 2 + 𝑧 2 + 6𝑦 + 14𝑧 + 28 = 0 and 𝑥 2 + 𝑦 2 + 𝑧 2 + 6𝑥 + 8𝑦 + 4𝑧 + 24 = 0 11. If the two spheres 𝑥 2 + 𝑦 2 + 𝑧 2 + 6𝑧 − 𝑘 = 0 and 𝑥 2 + 𝑦 2 + 𝑧 2 + 10𝑦 − 4𝑧 − 8 = 0 cuts orthogonally, find k. 12. Derive the equation of the right circular cone in its standard form 𝑥 2 + 𝑦 2 = 𝑧 2 tan2 𝛼 13. Find the equation of the right circular cone whose vertex is (2, -3, 5) axis makes equal √2 angles with the coordinate axes and with semi vertical angle cos −1 ( 3 ) 14. Find the equation of the right circular cone having vertex at the origin and axes of coordinates as three generators. 𝑥 𝑦 2 15. Find the equation of the right circular cone generated by revolving the line = = 1 2 3 𝑥 𝑦 𝑧 about the line −1 = =2 1 16. Find the equation of the right circular cone through the point (2, 1, 3) with vertex at the 𝑥−2 𝑦−1 𝑧+1 point (1, 1, 2) and axis to the parallel to the line = = 2 −4 3 17. Find the equation of the right circular cone generated when the straight line 2y + 3z = 6, x = 0 revolves about z – axis. 18. Find the equation of the right circular cone, whose vertex is at the point (2, -3, 5) and whose axis is equally inclined to the coordinate axis, given that the cone passes through the point (1, -2, 3) 19. Find the equation of the right circular cone where the vertex is at the origin and passes 𝑥 −𝑦 𝑧 through the point (1, 1, 2) and its axis is 2 = 4 = 3 20. Find the equation of the right circular cone formed by rotating the straight line 2y + 3z = 6, z = 0 revolves about y – axis. 21. Derive the equation of a right circular cylinder. 22. Find the equation of the right circular cylinder for which the radius is 2, axis is the line 𝑥−1 𝑦+3 𝑧−1 = = 2 −2 5 23. Find the equation of the right circular cylinder given that the radius of its normal section is 4 and the axis passes through the points A (1, -2, 3) and B (3, -1, 1) 24. Find the right circular cylinder generated by revolving the line 𝑥−1 𝑦−3 𝑧−5 𝑥+1 𝑦+3 𝑧+5 = = about the line = = 2 2 −1 2 2 −1 25. Find the equation of the right circular cylinder whose generators touch the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑥 + 4𝑦 − 1 = 0 having its generator parallel to the line x = y = z. 𝑥−3 𝑦 𝑧−4 26. Find the equation of the right circular cylinder having its generator = = and the 2 3 6 axis passes through the origin. 27. Explain the equation of ellipsoid with properties. 28. Explain the equation of hyperboloid of one sheet with properties. 29. Explain the equation of hyperboloid of two sheets with properties. 30. Find the equation of the sphere concentric with the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑥 − 4𝑦 − 6𝑧 − 11 = 0 but of double the radius of the sphere. 31. Derive the equation of the tangent plane at any point (x1, y1, z1) on the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 + 2𝑢𝑥 + 2𝑣𝑦 + 2𝑤𝑧 + 𝑑 = 0 *********************