QUESTION BANK FOR II SEMESTER UNIT- I GROUP THEORY I. TWO MARKS QUESTIONS: 1. A binary operation * is defined on the set Z of integers by a*b = 1 + ab, ∀ a, b ∈ Z. Show that * is commutative but not associative. 2. A binary operation * is defined on the set Z of integers by a*b = a + b – 1. Show that 1 is an identity element and inverse of 2 is 0. 3. In the set R0 of non zero real numbers the binary operations * is defined by a*b = 푎푏 ,∀ 푎, 푏 ∈ R0. Find the identity element and the inverse of the element 4. 2 4. Define semi group , group and abelian group . 5. Prove that the identity element of a group is unique. 6. Prove that the inverse of an element of a group is unique. 7. If a is an element of a group (G, *) then prove that (a-1)-1 = a 8. Show that (N, x), where N is the set of natural numbers is a semi group with identity element. 9. Show that the set N is not a group under multiplication. 10. In the set G of all rational numbers except -1, the binary operation * defined by a*b = a + b + ab. Solve 3 * 4 * x = 0. 11. If every element of a group G is its own inverse, show that G is abelian. 12. If in a group (G, *), a*a = a ,∀ 푎 ∈ G, show that a = e, where e is the identity element. 13. Show that the groups of order 1, 2, 3 are abelian. 14. Show that the set of even integers including 0 is a subgroup of additive group of the set Z of integers. 15. Show that the set of square roots of unity is a subgroup of the group of fourth roots of unity under multiplication. 16. If f=(1 2 3;3 1 2) then find 푓−1 . 17. If g=(1 2 3;1 3 2) ,h=(1 2 3;2 1 3) then find gh,hg,gℎ−1,푔−1ℎ, 푔−1 ℎ−1. II. FIVE MARKS QUESTIONS: 1. Let (G, *) be a group and a, b ∈ G. Then prove that (a*b)-1 = b-1 * a-1. 2. In a group (G, *), for all a, x, y ∈ G (i) a * x = a * y x = y (ii) x * a = y * a x = y 3. Given two elements a and b of the group (G, *),show that there exists a unique solution for a*x = b and y*a = b 4. Prove that the set C of all complex numbers is a group under usual addition of complex numbers. 5. Show that the square roots of unity form a group under usual multiplication. 6. Show that the cube roots of unity form a group under multiplication. 7. Show that the set of fourth roots of unity form an abelian group under multiplication. 8. Show that (Q+, *), where Q+ is the set of all positive rational numbers, is a group 푎푏 where * is defined by a*b = 5 9. Let G be the set of rational except -1 and * be the binary operation on G defined by a * b = a + b + ab, show that (G, *) is a group. Is it an abelian group? 10. Show that the set of all 2x2 real non singular matrices form a group under matrix multiplication. 11. Show that a group G is abelian if and only if (ab)2 = a2b2 ,∀ 푎, 푏 ∈ G 12. Show that (Z6, +6), where Z6 ={0,1,2,3,4,5} is a group. 13. Show that G ={1,2,3,4} is an abelian group under multiplication modulo 5. 14. Assuming the associative law holds, show that the set G = {2,4,6,8} is a group under multiplication modulo 10 15. Prove that the set S3 of all permutations of degree 3, is a finite non-abelian group of order 6,under the product of permutation. 16. Prove that a non-empty subset H of a group (G, *) is a subgroup of G if and only if (i)a, b ∈ H→ 푎 ∗ 푏 ∈ 퐻 (ii)푎 ∈ 퐻 → 푎−1 ∈ 퐻 18. Prove that a non-empty subset H of a group (G, *) is a subgroup of G if and only if ∀푎, 푏 ∈ 퐻, a * 푏−1 ∈ 퐻 19. Show that H ={0,2,4} is a subgroup of the group (G,+6), where G={0,1,2,3,4,5} UNIT- II DIFFERENTIAL CALCULUS: I. TWO MARKS QUESTIONS: 1. Find the angle between the radius vector and the tangent to the curve r=a푒휃푐표푡훼 2. For the curve r=a(1+cos휃), show that 휑=휃/2 3. Find the angle between the radius vector and the tangent to the curve r=a(1+sin휃) at 휃 = 휋/6 휋 휃 4. For the curve r=a(1+cos휃), show that 휑 = + 2 2 5. Find the angle between the radius vector and the tangent to the curve 푟푛=푎푛cosn휃 6. Prove that the polar sub tangent to the curve r=asec2휃 푠 1/2푎푐표푠푒푐2휃 7. Find the polar sub-normal for the curve r=a(1-cos휃) 푎푡 휃 = 휋/3 8. Show that in the curve r=a푒휃푐표푡훼, the polar sub-tangent = rtan훼 푎푛푑 polar sub- normal=rcot훼 9. Show that in the curve r=a휃,the polar sub-tangent varies as the square of radius vector and the polar sub-normal is a constant. 푑푠 푑푠 10. For the curve y=alogsec(x/a), find and 푑푥 푑푦 푑푠 푑푠 11.Find and for the cardioid r=a(1+cos휃) 푑휃 푑푟 12. Find 푑푠 for the curve x=a(t – sint), y=a(1 – cost) 푑푡 푑푠 푟 푑푠 푟2 13. With usual notation show that (i) = (ii) = 푑푟 √푟2−p2 푑휃 푝 푑푟 14. Show that the radius of curvature in the pedal form is 휌 = 푟 푑푝 15. Show that the radius of curvature of the catenary y=c cosh(x/c) is y2/c 16. Find the radius of curvature at any point (p,r) on the curve r3=a2p 17. Find the envelop of the family of circles whose centres lie on the x-axis 18. Find the envelop of the family of straight lines y=cx+c2 ,where c is a parameter 19. Find the envelop of the family of lines xcos훼 + ysin훼 = p, where 훼 is a parameter 20. Find the asymptotes parallel to the co-ordinate axes of the curve y2(2a – x) = x3 21. Prove that the asymptotes of the curve y2(x2 –a2) = x parallel to co-ordinate axes are y=0 and x=±a 22.Show that the origin is a node for the curve x4+y3-2x2+3y2 = 0 23. Show that the origin is a cusp for the curve (x2 + y2)x – 2ay2 = 0 24. Find the tangents at the origin to the curve y2(a-x)=x3 II FIVE MARKS QUESTIONS: 푑휃 1. With usual notations prove that tan휑 = r for the polar curve r = f(휃) 푑푟 2. Find the angle of intersection of the curves a2 = r2sin2휃 and b2 = r2cos2휃 3. Find the angle of intersection of the curves r=a and r=2acos휃 4. Show that the angle of intersection between the curves r=b and r2=a2cos2휃 +b2 is 푎2 tan−1( ) 푏2 푎 5. Find the angle of intersection of the curves r = and r = a휃 휃 6. Find the angle of intersection of the curves r = 푎휃 and r = 푎 1+휃 1+휃2 7. Show that the curves r = a(1+cos휃) 푎푛푑 푟 = 푏(1 − 푐표푠휃) cut orthogonally 8. Show that the curves r = asec2휃 and r = bcosec2휃 2 2 1 1 1 푑푟 9. With usual notation prove that (i) p = rsin휑 (ii) = + ( )2 푝2 푟2 푟4 푑휃 10. Show that the length of the perpendicular from the pole to the tangent to the curve r = a(1+cos휃) is r3 = 2ap2 11. Find the pedal equation of the circle x2 +y2 = 2ax 12. Find the pedal equation of the curve 푥2/3 +푦2/3 = 푎2/3 2 2 13. Find the pedal equation of the ellipse 푥 +푦 = 1 푎2 푏2 14. Find the pedal equation of the curve rm = am cosm휃 푙 15. Find the p-r equation of the conic = 1+ecos휃 푟 16. With usual notation prove that the radius of curvature of the curve y=f(x)is 휌 = 3/2 (1+푦12) 푦2 17. Derive an expression for the radius of curvature in parametric form 18. Derive an expression for the radius of curvature for the polar curve r=f(휃) 푑푟 19. Show that the radius of curvature in the pedal form is 휌 = 푟 푑푝 20. Find the radius of curvature of the curve ay2=x3 at (a/4,a/8) 21. Show that the radius of curvature 휌 for the cycloid x=a(휃 − 푠푛휃), 푦 = 푎(1 − 푐표푠휃) at 휃 any point 휃, 0 ≤ 휃 ≤ 2휋 is given by 휌 = 4asin ( ) 2 22. Find the radius of curvature at any point on the curve x2/3 +y2/3=a2/3 and hence show that 휌 =3p,where p is the length of the perpendicular from the pole to the tangent 휌2 23. For the cardioid r=a(1+cos휃), show that is a constant 푟 24. If 휌1 and휌2 be the radii of curvature at the extremities of any chord of the cardioid 2 2 2 r=a(1+cos휃) which passes through the pole, provethat 9(휌1 +휌2 ) = 16a 푥2 푦2 25.