1 / 121

இலக்கு : 100% அன்பார்ந்த சென்னை மாவட்ட , 10ம் வகுப்பு பயிலும் மாணவச் செல்வங்களே !

இலக்கு : 100% அன்பார்ந்த சென்னை மாவட்ட , 10ம் வகுப்பு பயிலும் மாணவச் செல்வங்களே !

turi
Download Presentation

இலக்கு : 100% அன்பார்ந்த சென்னை மாவட்ட , 10ம் வகுப்பு பயிலும் மாணவச் செல்வங்களே !

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. இலக்கு: 100% அன்பார்ந்தசென்னைமாவட்ட, 10ம் வகுப்புபயிலும்மாணவச்செல்வங்களே! அரசுபொதுத்தேர்வுஎழுதத்தயாராகிக்கொண்டிருக்கும்இவ்வேளையில்மாணவர்கள்சிறந்தமதிப்பெண்கள்பெறவேண்டும்என்றநோக்கத்தில்கணிதப்பாடத்திற்கானஇந்த PowerPoint file (ppt) தயாரிக்கப்பட்டுள்ளது. மாணவர்கள்இதனைமுழுமையாகபயன்படுத்திசிறந்தமதிப்பெண்கள்பெறவாழ்த்துகின்றேன். இப்படிக்கு, த. இராஜேந்திரன்.M.A.,B.Ed., M.Phil., முதன்மைக்கல்விஅலுவலர், சென்னை.

  2. Practical geometry and 5 marks questions with answers PREPARED BY: R.RAJENDRAN. M.A., M.Sc., M.Ed., Headmaster (IC) K.C.SANKARALINGA NADAR. HR. SEC. SCHOOL, CHENNAI-21.

  3. Draw two tangents from a point which is 10cm away from the centre of a circle of radius 6cm. Also measure the lengths of the tangents. Length of the tangent segment A 8 6 10 M O P 8 B

  4. Draw a circle of diameter 8cm. Draw two tangents to the circle from a point P at a distance of 9cm from the center. CONSTRUCTION Length of the tangent segment A 4 8.1 9 P O M B

  5. Draw a circle of diameter 8cm. Draw two tangents to the circle from a point P at a distance of 9cm from the center. CONSTRUCTION Length of the tangent segment A 4 8.1 9 P O M B

  6. Construct the Cyclic Quadrilateral ABCD with AB = 8cm, BC = 7cm, AC = 6cm, AD = 4cm. Rough figure C D C 6 7 4 D 8 A B 6 7 4 8 A B

  7. Construct a cyclic quadrilateral ABCD with AB = 7cm, BC = 5cm, AC = 6cm, BD = 6.5 Rough figure D C 5 6 6.5 C D 7 A B 6 6.5 5 B 7 A

  8. Construct a cyclic quadrilateral ABCD with AB = 7.2cm, mABD = 45°, mBAD = 100° and BC = 4cm. D Rough figure D C 4 100° 45° A B 7.2 C 100° 45° A B

  9. Draw the graph of x2 – x – 8 = 0 and hence find the roots of x2 – 2x – 15 = 0. In x axis 1cm = 1 unit y axis 1cm = 2 units. x2 – 2x – 15 = 0 (x + 3)(x – 5) = 0 x = –3, 5 x -4 -3 -2 -1 0 1 2 3 4 5 x2 16 9 4 1 0 1 4 9 16 25 -x 4 3 2 1 0 -1 -2 - 3 -4 -5 -8 -8 –8 –8 –8 –8 –8 –8 –8 –8 –8 y 12 4 -2 -6 -8 -8 -6 -2 4 12

  10. Solution set = {– 3, 5}

  11. A cyclist travels from a place A to a place B along the same route at a uniform speed on different days. The following table gives the speed of his travel and the corresponding time he took to cover the distance.

  12. Draw the speed-time graph and use it to find (i) the number of hours he will take if he travels at a speed of 5 km / hr (ii) the speed with which he should travel if he has to cover the distance in 40 hrs.

  13. If x = 5, y = 24 Time in hr If y = 40, y = 2.4 Speed in km/hr

  14. 1. Prove That A – (B  C) = (A – B)  (A – C) (1)BC (2) A-(BC) (3) A – B A B B B A A C C C (4) A – C (5) (A – B)  (A – C) B A B A C C From the diagrams (2) and (5) A – (B  C) = (A – B)  (A – C)

  15. 2. Prove that A – (B  C) = (A – B)  (A – C) using Venn diagram (2) A – ( BC) (3) A – B (1) BC A B A B B A C C C (4) A – C (5) (A – B)(A – C) A B B A C C From the diagram (2)and(5) A – (BC) = (A – B)(A – C)

  16. 3. Prove that A  (B  C) = (A  B)  (A  C) using Venn diagram. B A A B A B C C C A  (B  C) A B B C A B A B From the figures 2 and 5 A(B  C)=(AB)(AC) C C A C (A B)(A  C)

  17. 4. Prove that A  (B  C) = (A  B)  (A  C) using Venn diagram. B A A B A B C C C A  (B  C) A B B C A B A B From the figures 2 and 5 A(BC)=(AB)(AC) C C A C (AB)  (AC)

  18. 5. Prove that (A  B)’ = A’  B’ using Venn diagram. U U U A B B A B A 2. (AB)’ 3. A’ 1. AB U U B A B A From the diagrams 2 and 5 (A  B)’ = A’  B’ 5. A’  B’ 4. B’

  19. 6. Prove that (A  B)’ = A’  B’ using Venn diagram. U U U A B B A B A 2. (A  B)’ 3. A’ 1. A  B U U B A B A From the diagrams 2 and 5 (AB)’ = A’B’ 5. A’  B’ 4. B’

  20. 7. Given, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8}, show that (i) A (B C) = (A B) C. (ii) Verify (i) using Venn diagram. Solution (i) B C = {3, 4, 5, 6}  {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8} ` A  (B C) = {1, 2, 3, 4, 5}  { 3, 4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}………… (1) A B = {1, 2, 3, 4, 5}  {3, 4, 5, 6} = {1,2,3,4,5,6} `(A B) C = {1,2,3,4,5,6}  {5,6,7,8} = {1, 2, 3, 4, 5, 6, 7, 8}……………. (2) From (1) and (2), we have A  (B C) = (A B) C.

  21. A A B Using Venn diagram, we have A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6} and C = {5, 6, 7, 8} B 3 3 4 1 4 5 5 2 6 6 8 8 7 7 C C (1) B  C (2) A  (B  C)

  22. 8. Let A = {a,b,c,d}, B = {a,c,e} and C = {a,e}. (i) Show that A  (B C) = (A B) C. (ii) Verify (i) using Venn diagram. Solution Given A = {a,b,c,d}, B = {a,c,e} and C = {a,e}. B C = {a,c,e}  {a,e} = {a,e} A  (B C) = {a,b,c,d}  {a,e} = {a}…… (1) A B = {a,b,c,d.}  {a,c,e} = {a,c}. (A B) C = {a,c}  {a,e} = {a}…….. (2) From (1) and (2) A  (B C) = (A B) C.

  23. A A B B A = {a,b,c,d}, B = {a,c,e} and C = {a,e}. a a e C C (1) BC (2) A(BC)

  24. A A B B A = {a,b,c,d}, B = {a,c,e} and C = {a,e}. c a a C C (1) AB (2) (AB)C from (2) and (4) , it is verified that A  (B C) =(A B) C

  25. 9. Let A = {0,1,2,3,4}, B = {1, - 2, 3,4,5,6} and C = {2,4,6,7}. (i) Show that A  (B C) = (A B) (A C). (ii) Verify using Venn diagram. Solution (i) B C = {1, - 2, 3, 4, 5, 6}  {2, 4, 6, 7 } = {4, 6}; A  (B C) = {0,1, 2, 3, 4}  {4, 6} = {0,1,2,3,4,6}…….(1) A B = {0,1,2,3,4}  {1, - 2, 3,4,5,6} = {- 2, 0,1, 2, 3, 4, 5, 6}, A C = {0,1,2,3,4}  {2,4,6,7} = {0,1, 2, 3, 4, 6, 7}. (A B)  (A C) = {- 2, 0,1, 2, 3, 4, 5, 6}  {0,1, 2, 3, 4, 6, 7} = {0,1, 2, 3, 4, 6}. …….(2) From (1) and (2) ,we get A  (B C) = (A B)  (A C).

  26. A A B B A = {0, 1, 2, 3, 4}, B = {1, –2, 3, 4, 5, 6},C = {2, 4, 6, 7}. 3 1 0 4 4 6 6 2 C C (1) BC (2) A(BC)

  27. A = {0, 1, 2, 3, 4}, B = {1, –2, 3, 4, 5, 6},C = {2, 4, 6, 7}. A A -2 B B 1 3 3 5 1 0 0 4 6 4 6 2 2 7 C C (3) AB (4) A C from (2) and (4) , it is verified that A  (B C) =(A B) C

  28. A B 3 1 0 4 6 2 C (5) (AB)(AC) from (2) and (5) , it is verified that A  (B C) = (A B)  (A C)

  29. 10. Given that U = {a,b,c,d,e, f,g,h}, A = {a, b, f, g}, and B = {a, b, c}, verify De Morgan’s laws of complementation. U = {a, b, c, d, e, f, g, h} A = {a, b, f, g} B = {a, b, c}. De Morgan’s laws (AB)’ = A’  B’ (AB)’ = A’  B’ VERIFICATION A  B = {a, b, c, f, g} (A  B)’ = {d, e, h}……..(1) A’ = {c, d, e, h} B’ = {d, e, f, g, h} A’B’ = {d, e, h}……….(2) From (1) and (2) (A  B)’ = A’B’ A  B = {a, b} (A  B)’ = {c, d, e, f, g, h}....(1) A’ = {c, d, e, h} B’ = {d, e, f, g, h} A’  B’ = {c, d, e, f, g, h}….(2) From (1) and (2) (A  B)’ = A’  B’

  30. 11. Verify De Morgan’s laws for set difference using the sets given below: A = {1, 3, 5, 7, 9,11,13,15}, B = {1, 2, 5, 7} and C = {3,9,10,12,13}.. A = {1, 3, 5, 7, 9,11,13,15}, B = {1, 2, 5, 7} C = {3,9,10,12,13}.. De Morgan’s laws (1) A\(BC) = (A\B)(A\C) (2) A\(BC) = A\B)  (A\C) VERIFICATION B  C = {1, 2, 3, 5, 7, 9, 10, 12, 13} A\(BC) = {11, 15}……..(1) A\B = {3, 9, 11, 13, 15} A\C = {1, 5, 7, 11, 15} (A\B)(A\C) = {11, 15}……….(2) From (1) and (2) A\(BC) = (A\B)(A\C)

  31. 12. Verify De Morgan’s laws for set difference using the sets given below: A = {1, 3, 5, 7, 9,11,13,15}, B = {1, 2, 5, 7} and C = {3,9,10,12,13}.. A = {1, 3, 5, 7, 9,11,13,15}, B = {1, 2, 5, 7} C = {3,9,10,12,13}.. BC = { } A\(B C) = {1, 3, 5, 7, 9,11,13,15}……..(1) A\B = {3, 9, 11, 13, 15} A\C = {1, 5, 7, 11, 15} (A\B)  (A\C) = {1, 3, 5, 7, 9,11,13,15}……….(2) From (1) and (2) A\(B C) = (A\B)  (A\C)

  32. 13. Let A = {10,15, 20, 25, 30, 35, 40, 45, 50}, B = {1, 5,10,15, 20, 30}and C = {7, 8,15,20,35,45, 48}. Verify A\(B C) = (A\B)  (A\C) A = {10,15, 20, 25, 30, 35, 40, 45, 50}, B = {1, 5, 10, 15, 20, 30} C = {7, 8, 15, 20, 35, 45, 48} BC = {15, 20} A\(B C) = {10, 25, 30, 35, 40, 45, 50}……..(1) A\B = {25, 30, 35, 40, 45, 50} A\C = {10, 25, 30, 40, 50} (A\B)  (A\C) = {10, 25, 30, 35, 40, 45, 50}……….(2) From (1) and (2) A\(B C) = (A\B)  (A\C)

  33. Let A = { 6, 9, 15, 18, 21 }; B = { 1, 2, 4, 5, 6 } and f : A " B be defined by f(x) = . Represent f by (i) an arrow diagram (ii) a set of ordered pairs (iii) a table (iv) a graph . ARROW DIAGRAM A B 6 9 15 18 21 1 2 4 5 6 SET OF ORDERED PAIR f = {(6, 1), (9, 2), (15, 4), (18, 5), (21, 6)}

  34. Table Graph – 7 – 6 – 5 – 4 – 3 – 2 – 1  (21, 6)  (18, 5)  (15, 4)  (9, 2)  (6, 1) | | | | | | | | | 3 6 9 12 15 18 21 24 27

  35. 15. Let A= { 0, 1, 2, 3 } and B = { 1, 3, 5, 7, 9 } be two sets. Let f : A B be a function given by f (x) = 2x + 1. Represent this function as (i) a set of ordered pairs (ii) a table (iii) an arrow diagram and (iv) a graph. Given A= {0, 1, 2, 3}, B = {1, 3, 5, 7, 9} f(x) = 2x + 1 f(0) = 2(0) + 1 = 0 + 1 = 1 f(1) = 2(1) + 1 = 2 + 1 = 3 f(2) = 2(2) + 1 = 4 + 1 = 5 f(3) = 2(3) + 1 = 6 + 1 = 7 (i)Set of ordered pairs {(0, 1), (1, 3), (2, 5), (3, 7)} (ii) Table 0 1 2 3 x 1 3 5 7 f(x)

  36. 4 6 8 10 ARROW DIAGRAM B A 3 4 5 6 7 – 7 – 6 – 5 – 4 – 3 – 2 – 1 Graph  (10, 6)  (8, 5)  (6, 4)  (4, 3) | | | | | | | | 0 2 4 6 8 10 12 14 16

  37. 16. A function f: [1, 6)R is defined as follows f(x) = Find the value of (i) f(5), (ii) f(3), (iii) f(1), (iv) f(2) – f(4) (v) 2f(5) – 3f(1) Since 5 lies between 4 and 6 f(x) = 3x2 – 10 f(5) = 3(5)2 – 10 = 75 – 10 = 65 Since 3 lies between 2 and 4 f(x) = 2x – 1 = 2(3) – 1 = 6 – 1 = 5 Since 1 lies in the interval 1  x < 2 f(x) = 1 + x f(1) = 1 + 1 = 2 Since 2 lies in the interval 2  x < 4 f(x) = 2x – 1 = 2(2) – 1 = 4 – 1 = 3 Since 4 lies in the interval 4  x < 6 f(x) = 3x2 – 10 f(4) = 3(4)2 – 10 = 48 – 10 = 38

  38. f(2) – f(4) = 3 – 38 = – 34 2f(5) – 3 f(1) = 2 (65) – 3(2) = 130 – 6 = 124

  39. 17. A function f: [-3, 7)R is defined as follows Find the value of (i) f(5) + f(6) (ii) f(1) – f(– 3), (iii) f(–2) – f(4) (iv) (i) Since 5 lies between 4 and 6 f(x) = 2x – 3 f(5) = 2(5) – 3 = 10 – 3 = 7 Since 6 lies in the interval 4 < x  6 f(x) = 2x – 3 f(6) = 2(6) – 3 = 12 – 3 = 9 f(5) + f(6) = 7 + 9 = 16

  40. 18. A function f: [-3, 7)R is defined as follows f(x) = Find the value of (i) f(5) + f(6) (ii) f(1) – f(– 3), (iii) f(–2) – f(4)(iv) (ii) Since 1 lies in the interval -3  x < 2 f(x) = 4x2 – 1 f(1) = 4(1)2 – 1 = 4 – 1 = 3 Since –3 lies in the interval-3  x < 2 f(x) = 4x2 – 1 = 4(-3)2 – 1 = 36 – 1 = 35 f(1) – f(-3) = 3 – 35 = – 32

  41. 19. A function f: [-3, 7)R is defined as follows f(x) = Find the value of (i) f(5) + f(6) (ii) f(1) – f(– 3), (iii) f(–2) – f(4) (iv) (iii) Since -2 lies in the interval -3  x < 2 f(x) = 4x2 – 1 f(1) = 4(-2)2 – 1 = 16 – 1 = 15 Since 4 lies in the interval 2  x  4 f(x) = 3x – 2 = 3(4) – 2 = 12 – 2 = 10 f(-2) – f(4) = 15 – 10 = 5

  42. 20. A function f: [-3, 7)R is defined as follows f(x) = Find the value of (i) f(5) + f(6) (ii) f(1) – f(– 3), (iii) f(–2) – f(4) (iv) (iv) Since 3 lies in the interval 2  x  4 f(x) = 3x – 2 f(3) = 3(3) – 2 = 9 – 2 = 7 Since -1 lies in the interval -3  x < 2 f(x) = 4x2 – 1 = 4(-1)2 – 1 = 4 – 1 = 3 f(6) = 9, f(1) = 3

  43. 21. A function f: [-7, 6)R is defined as follows Find the value of (i) 2f(-4) + 3f(2) (ii) f(-7) – f(– 3), (iii) (i) Since -4 lies in the interval -5  x  2 f(x) = x + 5 f(-4) = – 4 + 5 = 1 Since 2 lies in the interval -5  x  2 f(x) = x + 5 f(2) = 2 + 5 = 7 2f(-4) + 3f(2) = 2(1) + 3(7) = 2 + 21 = 23

  44. 22. A function f: [-7, 6)R is defined as follows f(x) = Find the value of (i) 2f(-4) + 3f(2) (ii) f(-7) – f(– 3), (iii) (ii) Since -7 lies in the interval -7  x < -5 f(x) = x2 + 2x + 1 f(-7) = (–7)2 + 2(–7) + 1 = 49 – 14 + 1 = 36 Since -3 lies in the interval -5  x  2 f(x) = x + 5 f(2) = -3 + 5 = 2 f(-7) – f(–3) = 36 – 2 = 34

  45. 23. A function f: [-7, 6)R is defined as follows f(x) = Find the value of (i) 2f(-4) + 3f(2) (ii) f(-7) – f(– 3), (iii) (iii) Since -3 lies in the interval -5  x  2 f(x) = x + 5 f(-3) = -3 + 5 = 2 Since -6 lies in the interval -7  x < -5 f(x) = x2 + 2x + 1 f(-6) = (–6)2 + 2(–6) + 1 = 36 – 12 + 1 = 25 4 lies in the interval 2 < x < 6 f(x) = x – 1 f(4) = 4 – 1 = 3 Since 1 lies in the interval -5  x  2 f(x) = x + 5 f(1) = 1 + 5 = 6

  46. 24. A function f: [-7, 6)R is defined as follows f(x) = Find the value of (i) 2f(-4) + 3f(2) (ii) f(-7) – f(– 3), (iii) (iii) f(-3) = 2 f(-6) = 25 f(4) = 3 f(1) = 6

  47. LHS = RHS  (AB)T= BTAT .

  48. 26. If then show that A2 – 4A + 5I2 = O A2 – 4A + 5I2 = O

  49. 27. Find X and Y if 2X + 3Y =

More Related