TRIGONOMETRICAL IDENTITY(X)

                 EXERCISE -1

                  ************

1) (1- sin²x)sec²x = 1

2) Sin²x + 1/(1+ tan²x)= 1

3) sinx/(1- cosx)= Cosecx + cotx.

4) (1- sinx)/(1+sinx)= (secx - tanx)²

5) sin²x cot²x+ cos²x tan²x= 1.

6) 1/(1+sinx) + 1/(1- sinx)= 2sec²x

7) (1+sinx)/cosx + cosx/(1+sinx)= 2secx.

8) Sinx/(1+ cosx) + (1+coax)/sinx = 2 Cosecx.

9) {(1+sinx)²+(1- sinx)²}/2cos²x = (1+sin²x)/(1- sin²x).

10) 1/(1+ sinx) +1/(1-sinx)=2sec²x

11) sinx/(1- cosx)= Cosecx + cotx

12) √{1-sinx)/(1+sinx)}= secx - tanx

13) √{(1+sinx)/(1-sinx)}= secx + tanx

14) Sinx/(1- cosx) + tanx/(1+cosx)= secx Cosecx + cotx.

15) (Sinx+ cosx)/(sinx- cosx) + (sinx- cosx)/(sinx+ cosx)= 2/(sin²x - cos²x)= 2/(2sin²x -1)

16) sin/(1+cosx)+ (1+cosx)/sinx = 2 Cosecx.

17) Sinx/(cotx + Cosecx)= 2 + sinx/(cotx - Cosecx).

18) Sin⁴x + cos⁴x= 1 - 2 sin²x cos²x.

19) Sin⁴x - cos⁴x= sin²x - cos²x = 2sin²x -1 = 1 - 2cos²x.

20) Sin⁶x+ cos⁶x= 1 - 3 sin²x cos²x

21) sin²x/cos²x + cos²x/sin²x= 1/(sin²x cos²x) +2.

22) {(1+ sinx)²+(1-sinx)²}/cos²x = 2{1+sin²x)/(1-sin²x)}.

23) (Sinx - siny)/(cosx+ cosy) + (cosx - cosy)/(sinx + siny)= 0.

24) 2(sin⁶x + cos⁶x) - 3(sin⁴x+ cos⁴x)+ 1= 0.

25) Sin⁶x + cos⁶x + 3sin²x cos²x= 1

26) (Sin⁸x - cos⁸x)= (sin²x - cos²x)(1- 2sin²x cos²x).

27) (Sinx + secx)²+(cosx + Cosecx)² = (1+ secx Cosecx)².

28) (1- sinx+ cosx)²= 2(1+ cosx)(1- sinx).

29) (Sinx - cosx)/(sinx+ cosx) + (sinx + cosx)/(sinx - cosx) = 2/(2sin²x -1).

30) {(1+ sinx - cosx)/(1+ sinx + cosx)}² = (1- cosx)/(1+ cosx).

31) Sinx/(secx + tanx -1) + cosx/(Cosecx + cotx -1)= 1

32) Sin²x cos²y - cos²x sin²y= sin²x - sin²y.

33) √{1+sinx)/(1-sinx)}+ √{1-sinx)/(1+sinx)}= 2 secx.

34) (Sinx +1- cosx)/(cosx -1+ sinx)= (1+sinx)/cosx.

35) cos²x (1+ tan²x)= 1.

36) Cos²x + 1/(1+ cot²x) = 1.

37) (1- cos²x)cosec²x = 1.

38) Cosx/(1-sinx)= (1+sinx)/cosx.

39) Cosx/(1+sinx)= (1-sinx)/cosx.

40) (1- cosx)/sinx= sinx/(1+cosx).

41) cos²x/sinx - Cosecx + sinx= 0.

42)a) (1+ cosx)/sinx = sinx/(1-cosx)

43) (1- cosx)/(1+ cosx)= (cotx- Cosecx)².

44)#  Cosx/(cosecx-1) + sinx/(1- cotx)= sinx + cosx.

45) cos²x + 1/(1+ cot²x)= 1.

46) √{(1+cosx)/(1- cosx)}= Cosecx + cotx.

47) √{(1-cosx)/(1+cosx)}= Cosecx - cotx.

48) (1+ cosx+ sinx)/(1+cosx - sinx) = (1+ sinx)/cosx.

49) {cosx Cosecx - sinx secx}/(cosx + sinx)= Cosecx - secx.

50) Cosx/(1- tanx) + sinx/(1- cotx) = cosx + sinx.

51) Cos⁴x - cos²x = sin⁴x - sin²x.

52) cosx/(1- tanx) + sin²x/(sinx - cosx)= 1/(sinx - cosx).

53) Cos²x/(1- tanx) + sin³x/(sinx - cosx)= 1+ sinx cosx.

54) cosx/(1- sinx) + sinx/(1- cosx) + 1 = (sinx cosx)/{1-sinx)(1- cosx)}.

55) Cosx/(Cosecx +1) + cosx/(Cosecx -1)= 2 tanx.

56) (1+ cosx - sin²x) /{sinx(1+cosx)}= cotx.

57) √{(1+cosx)/(1-cosx)}+ √{(1-cosx)/(1+cosx)}= 2 Cosecx.

58) (1+tan²x)(1+sinx)(1- sinx)= 1.

59) tan²x - 1/cos²x = -1.

60) tan²x cos²x = 1 - cos²x.

61) tanx + 1/tanx = secx Cosecx.

62) tanx + cotx = secx Cosecx.

63) tan²x - sin²x= tan²x sin²x.

64) (1+ tan²x)(1- sinx)(1+ sinx)= 1.

65) tanx - cotx = (2sin²x -1)/(sinx cosx).

66) (1+ tan²x)/(1+cot²x)= {(1-tanx)/(1-cotx)}².

67) (Tanx + sinx)/(tanx - sinx) = (secx +1)/(secx-1).

68) tanx - cotx = (2sin²x -1)/(sinx cosx).

69) Tan²x + cot²x = sec²x cosec²x - 2.

70) (1- tan²x)/(cot²x - 1)= tan²x.

71) Tanx/(1- cotx) + cotx/(1- tanx)= 1+ secx cosecx.

72) Tan²x + cot²x+ 2= sec²x cosec²x

73) (Tanx- cotx)/(sinx cosx)= sec²x - cosec²x= tan²x - cot²x.

74) (tanx + secx-1)/(tanx - secx+1)= (1+sinx)/cosx.

75) tanx/(1- cotx) + cotx/(1- tanx) = 1+ tanx + cotx= 1+ secx Cosecx.

76) tan²x - tan²y= (cos²y- cos²x)/(cos²y cos²x)= (sin²x - sin²y)/(cos²x cos²y).

77) (1+ tanx tany)²+(tanx - tany)²= sec²x sec²y.

78) (Tanx + Cosecx)²- (coty - secx)² = 2 tanx coty (Cosecx + secy).

79) {(1+ tan²x)cotx}/cosec²x= tanx.

80) tanx/(1- cotx)+ cotx/(1- tanx)= secx Cosecx +1.

81) (1+ tan²x)+ (1+ 1/tan²x)= 1/(sin²x - sin⁴x).

82) Tan²x/(1+ tan²x) + cot²x/(1+ cot²x)= 1.

83) (Tanx + sinx)/(tanx - sinx)= (secx +1)/(secx-1).

84) Tan³x/(1+tan²x) + cot³x/(1+ cot²x) = secx Cosecx - 2sinx cosx.

85) Tanx/(1+ tan²x)² + cotx/(1+ cot²x)²= sinx cosx.

86) (Tanx + tany)/(cotx + coty)= tanx tany.

87) Tan²x sec²y - sec²x tan²y= tan²x - tan²y.

88) Cot²x - 1/sin²x = -1.

89) (1+ cot²x) sin²x = 1.

90) (1+cot²x)(1-cosx)(1+cosx)=1.

91) Cotx - tanx= (2cos²x-1)/(sinx cosx).

92) (1+ cotx - Cosecx)(1+ tanx + secx)= 2.

93) {(1+cot²x) tanx}/sec²x= cotx.

94) cotx - tanx = (2cos²x -1)/(sinx cosx).

95)(Cotx+ cosecx-1)/(cotx- Cosecx+1)= (1+sinx)/sinx.

96) Cot⁴x - 1= cosec⁴x - 2 cosec²x.

97) Cot²x{(secx-1)/(1+sinx)} + sec²x {(sinx -1)/(1+ secx)}= 0.

98) {(1+ cotx+ tanx)(sinx - cosx)}/(sec³x - cosec³x)= sin²x cos²x.

99)** 1 + cot²x/(1+ cosec²x)= Cosecx

100) (1+ cotx - Cosecx)(1+ tanx + secx)= 2.

101) {Cot²x(secx -1)}/(1+ sinx)= sec²x{(1- sinx)/(1+ secx)}.

102) (1+ cotx + tanx)(sinx - cosx)= secx/cosec²x - Cosecx/sec²x.

103) (Cotx+ tany)/(coty+ tanx)= cotx tany.

104) Cot²x cosec²y - cot²y cosec²x= cot²x - cot²y.

105) (sec²x -1)(cosec²x -1)= 1.

106) (secx + cosx)(secx- cosx)= tan²x + sin²x.

107) secx(1- sinx)(secx + tanx)= 1.

108) 1/(secx -1) + 1/(secx +1)= 2cosecx cotx.

109) (1+ secx)/secx= sin²x/(1- cosx).

110) 1/(secx - tanx)= secx + tanx.

111) (Secx - tanx)/(secx+ tanx)= 1 - 2secx tanx+ 2 tan²x.

112) √(sec²x + cosec²x)= tanx + cotx.

113) Sec⁴x - sec²x= tan⁴x + tan²x.

114) Sec⁶x = tan⁶x + 3tan²x sec²x +1.

115) (secx - tanx)/(secx+ tanx)= cos²x/(1+ sinx)².

116) (Secx - tanx)²= (1- sinx)/(1+ sinx)

117) 1/(secx + tanx) - 1/cosx= 1/cosx - 1/(secx - tanx).

118) {1/(sec²x - cos²x) + 1/(cosec²x - sin²x)} sin²x cos²x= (1- sin²x cos²x)/(2+ sin²x cos²x).

119) (Secx + tanx -1)(secx - tanx +1)= 2 tanx.

120) (Secx - Cosecx)(1+ tanx + cotx)= tanx secx - cotx Cosecx.

121) Sec⁴x(1- sin⁴x) - 2 tan²x = 1.

122) √{(secx -1)/(secx+1)} + √{(secx +1)/(secx-1)}= 2 Cosecx.

123) (Secx -1)/(secx+1)= {sinx/(1+ cosx)}²

124) (Tanx + 1/cosx)² + (tanx - cotx/cosx)²= 2{(1+ sin²x)/(1-sin²x)}.

125) Cosec x √(1- cos²x)= 1

126) (Cosec x - sinx)(secx - cosx)(tanx + cotx)= 1.

127) (Cosecx - sinx)(secx - cosx)= 1/(tanx+ cotx).

128) (Cosecx - sinx)(secx - cosx)(tanx+ cotx)= 1.

129) 1/(Cosecx - cotx) - 1/sinx= 1/sinx - 1/(Cosecx+ cotx).

130) Cosec⁶x = cot⁶x + 3 cot²x cosec²x +1.

131) 1/(Cosecx - cotx) - 1/sinx = 1/sinx - 1/(Cosecx + cotx).

132) Cosecx/(Cosecx -1) + Cosecx/(Cosecx +1)= 2sec²x.

133) (Cosecx - secx)(cotx - tanx)= (Cosecx + secx)(secx Cosecx -2). 

      TYPE -- 2

1) If tanx + sinx = m and tanx - sinx= n, show that m² - n²= 4√(mn).

2) If cosx + sinx =√2 cosx , show that cosx - sinx= √2 sinx.

3) If sinx + cosx = p and secx + Cosecx = q, show that q(p²-1)= 2p

4) If secx + tanx = p show that (p²-1))/(p²+1)=  sinx 

5) If x= a sin k and y= b tan k, then Prove that (a²/x² - b²/y²)= 1.

6) If Cosx/cosy  = m and Cosx/siny = n show that (m²+ n²)cos²y= n².

7) If tanA + sinA = m and tanA - sinA= n, prove that (m² - n²)²= 16mn

8) If Cosecx - sinx= n and secx - cosx = m, show that m²n²(m²+ n²+3)= 1.

9) If x= r sinA cosC , y= r sinA sinC and z= r cosA, show r²= x²+y²+z².

10) If tanA = n tanB and sinA= m sinB, show cos²A= (m²-1)/(n²-1).

11) If x sin³k + y cos³k= sink cosk and x sink = y cosk , show x²+y²= 1

12) If Cosecx - sinx= m and secx - cosx = n, show ³√(m²n)²+ ³√(mn²)² = 1.

13) If cotk + tank = x and seck - cosk = y, show ³√(x²y)²- ³√(xy²)²= 1

14) If a cosx + b sinx= m and a sinx - b cosx= n, show a²+b²= m² + n².

15) If a cosx - b sinx = c, show a sinx + b cosx = ± √(a²+ b³ - c²).

16) If sinx + sin²x= 1, show cos²x + cos⁴x = 1.

17) If sinx + sin²x= 1, find the value of cos¹²x + 3cos¹⁰x+ 3cos⁸x + cos⁶x + 2 cos⁴x + 2cos²x - 2.                  1 

18) If a secx + b tanx + c= 0 and p secx + q tanx + r= 0, show (br - qc)² - (pc - ar)²= (aq - bp)². 

19) If secx +  tanx + = p find the value of secx , tanx  and sinx in terms of p.     1/2(p - 1/p), 1/2(p - 1/p) , (p²-1)/(p²+1)

20) If (sin⁴x)/a + (cos⁴x)/b = 1/(a+b), show (sin⁸x)/a³ + (cos⁸x)/b³= 1/(a+b)³.

21) If tan²x = 1 - a², show secx + tan³x Cosecx= ³√(2- a²)².

22) If sinx + sin²x + sin³x= 1, show cos⁶x - 4 cos⁴x + 8 cos²x= 4.

23) If x= a seck + b tank and y= b seck + b tank ,show x² - y²= a²- b².

24) If x/a cosk + y/b sink = 1 and x/a sink - y/b cosk = 1, then show that x²/a² + y²/b² = 2.

25) If cosec x - sinx = a², secx - cosx = b³, show a²b²(a² + b²)= 1.

26) If a cos³x + 3a cosx sin²x = m,  and a sin³x + 3a cos²x sinx= n, show ³√(m+n)²+ ³√(m-n)²= 2 ³√a².

27) If x= a cos³k, y= + b sin³x show that ³√(x/a)²+ ³√(y/b)²= 1.

28) If 3sin x + 5 cosx = 5, show 5 sinx - 3 cosx = ± 3.

29) If a cosx + b sinx = m and a sinx - b cosx = n, show a² + b²= m²+ n².

30) If cosecx + cotx = m and cosecx -  cotx = n, show mn= 1.

31) If cosA + cos²A= 1, show sin²A+ sin⁴A= 1.

32) If cosx + cos²x= 1, show sin¹²x + 3 sin¹⁰x + 3 sin⁸x + sin⁶x + 2sin⁴x + 2 sin²x - 2= 1

33) If (1+ cosx)(1+ cosy)(1+ cost)= (1- cosx)(1- cosy)(1- cost). Show that one of the values of each member of this equally is sinx siny sinz.

34) If sinx + cosx= a, show sin⁶x + cos⁶x ={4- 3(a² -1)²}/4.

35) If tan²x = 1 + 2 tan²y, show 2 sin²x = 1 + sin²y.

36) If tan²x = cos²y - sin²y, show cos²x - sin²x = tan²y.

37) If (a² - b²) sinx + 2ab cosx= a² + b², find the value of tanx .    (a²-b²)/2ab

38) If cos⁴x/cos²y + sin⁴x/sin²y= 1, show cos⁴y/cos²x + sin⁴y/sin²x = 1.

39) If x= a seck cos m, y= b seck  sinm and z= c tank,  show x²/a² + y²/b² - z²/c² = 1.

40) If a² secx - b² tan²x =c² show sinx = ±√{(c²-a²)/(c²- b²)} .

               EXERCISE --2

                    *******

1) (sec²x -1)/tan²x = 1

2) (sin²x + cos²x)/(sec²x - tan²x) = 1

3) 1 - sin²x -  cos²x) = 0

4) sin⁴x+ sin²x  cos²x)= 0

5) sin⁴x+ 2sin²x  cos²x + cos⁴x = 0

6) (cosec²x - cot²x)  cos²x= cos²x

7) (sinx cosecx tanx cotx)/(sin²x + cos²x)= 1.

8) (sin²30°+ cos²30°)/(sec²57° -  tan²57°)= 1.

9) cosx tanx = 1

10) sin⁴x cosec²x + cos⁴x sec²x= 1

11) sin²x  cot²x + cos²x tan²x= 1

12) cos⁴x - sin⁴x= cos²x - sin²x.

13) (1+ tanx)²+ (1 - tanx)²= 2 sec²x.

14) cot⁴x+ cot²x= cosec⁴x - cosec²x.

15) (secx + tanx)/(cosecx + cotx) = (cosecx - cotx)/(secx - tanx)

16) (1+ sinx)/(1- sinx) = (secx + tanx)².

17) secx cot x = cosecx.

18) tanx + cot x = secx cosecx.

19) cosx/(secx - tanx) = (1+ sinx)

20) (1+ cosx - sin²x)/{sinx(1+ cosx) = cot x 

21) (3 - 4sin²x)/cos²x = 3 - tan²x

22) (tanx + cotx)sinx cosx= 1

23) cosx = cotx/cosecx = cot/√(1+ cot²x).

24) (sin⁴x - cos⁴x)= (sin²x - cos²x)

25) sec²x + cosec²x = sec²x cosec²x

26) sin³x + cos³x = (sinx + cosx)(1- sinx cosx).

27) sin²x(1+ cot²x)= 1

28) (tanx + cotx)²= cosec²x + sec²x.

29) sec⁴x - sec²x = tan²x + tan⁴x.

30) (cosecx - sinx)(secx - cosx)(tanx + cotx)= 1.

31) (cotx + tan y)/(cot y + tanx)= cotx tany

32) sinx/(1+ cosx) + (1+ cosC)/sinx = 2 cosecx

33) 1 + 1/cosx  = tan²x/(secx - 1).

34) (1+ cosx)/(1- cos x)= (cosecx + Cotx)²

35) If tanx + sinx = a and tanx - sinx = b, show that a² - b² = 4 √(ab)

CYLINDER

  

1) The volume of a cylinder is 343π cm³ and height is 7cm. Find its lateral surface area.             7, 308

2) A solid cylinder has a total surface area is 231cm³. It's curved surface area is 2/3 of the total surface area. Find the volume of the cylinder.                        269.5cm³

3) Find the weight of a hollow cylinder lead pipe 26 cm and long 1/2 cm thick. Its external diameter is 5cm.(Weight of 1 cm³ of lead is 11.5 gm).                          2.114 kg

4) he diameter of a roller 120cm long is 84 cm. If it takes 500 complete revolutions to level a playground, determine the cost of levelling at the rate of 30 paise per per m² (π=22/7).                 ₹475.20

5) Two cylindrical pots contain the same amount of milk if their diameters are in the ratio 2:1, find the ratio of their heights.            1:4

6) Find 

A) the volume.                     2310cm³

B) whole surface area.        

of a right circular cylinder whose height is 15cm and radius of the base is 7 cm.                        968cm²

7) The diameter of the base of a right circular is 42cm and its height is 10cm. Find the area of the curved surface and the volume of the cylinder.            1320cm²,13860cm³.

8) Find the volume of a right circular cylinder which has a height a height of 21cm and the base radius of 5cm. Find also the curved surface area of the cylinder.                       1650cm³,660cm²

9) The curved surface area of a cylinder is 1320cm² and its base has a diameter 21cm. Find the height and the volume of the cylinder.                  20cm, 6930cm³

10) The volume of a cylinder is 69300cm³. and its height is 50cm. Find the area of the curved surface of the cylinder.                   6600cm²

11) The volume of a cylinder is 6358cm³and its height is is 28cm. Find its radius and the curved surface area.          1496cm²,8.5cm

12) The height of a cylinder is 10cm, and the ratio of its volume to its lateral surface is 3:2. Find its radius.                                       3cm

13) Find the curved surface area, the total surface area and the volume of the cylinder, the diameter of whose base is 7cm and height being 60cm. Also, find the capacity of the cylinder in litres.                 1320cm²,1397cm², 2310cm³.

14) The area of the curved surface of a cylinder is 4400cm² , and circumference of its base is 110cm. Find the height and the volume of the cylinder.          40cm, 38500cm³

15) A solid cylinder has total surface area of 462cm². Its curved surface area is one third of its total surface area. Find the volume of the cylinder.                        539cm³

16) The ratio between the curved surface area and the total surface total surface area of a right circular cylinder is 1:2. Find the volume of the cylinder, if total surface area is 616cm².                              1078cm³

17) If the radius of the base of a right circular cylinder is halved, keeping the height same , what is the ratio of the volume of the reduced cylinder to that of the original one.                                1:4

18) The radius of two cylinders are in the ratio 2:3 and their Heights are in the ratio 5:3. Calculate the ratio of their volume and the ratio of their curved surfaces.            20:27, 10:9

19) The radius and height of a cylinder are in the ratio 5:7 and its volume is 550cm³. find its radius.                                       5cm.

20) A solid cylinder has total surface area of 462cm². its curved surface area is one third of its total surface area. Find the volume of the cylinder.                           539cm³

21) How many cubic metres of Earth must be dug out to sink a well 14m deep and 4m in diameter? What will it cost to plaster its inner surface area at ₹2.50 per square metre?                                      ₹440

22) A rectangular piece of paper is 71cm long and 10cm wide. A cylinder is formed by Rolling the paper along its length. Find the volume of the cylinder. (Use π= 355/113).                        4011.5cm³

23) Two cylindrical vessels are filled with oil. The radius of one vessel is 15cm. and its height is 25cm. The radius and height of the Other vessel are 10cm. and 18cm. respectively. Find the radius of the cylinder vessel 30cm. in hight, which will just contain the oil of the two given vessels.            15.73cm.

24) The sum of the radius of the base and the height of a solid cylinder solid cylinder is 37m. If the total surface area of solid cylinder is 1628m², find the circumference of its base and the volume of cylinder.                      44m,4620m³

25) The cost of painting the total outside surface of a closed cylindrical oil tank at 25 paise per square decimeter is ₹77. The height of the tank is 3 times the radius of the base of the tank. Find its volume.                          404.25cm³

26) Find the cost of sinking a tubewell 280m deep having diameter 3m at the rate of ₹3.60 per m². Find also the cost of cementing its inner curved surface at ₹1.25 per m².                                           ₹3300

27) A 12m deep well with diameter 3.5 is dug up and Earth from it is spread evenly to form a platform 10.5m by 8.8m. Determine the height of the platform.         1.25m

28) A well with 10m inside diameter is dug 84m deep. Earth taken out of it is spread all around it up 7.5m to form an embankment. Find the height of embankment.                          1.6m

29) A path 2m wide surrounds a circular pond of diameter 40m. How many cubic metres of gravel are required to gravel the path to a depth of 20cm.                      52.8m³

30) The diameter of the roller 120cm. long is 84cm. If it takes 500 complete revolutions to level a playground of 30 paise per m².         ₹475.20

31) A cylindrical road roller made of iron is 1.5m long. Its inner diameter 54cm and the thickness of iron sheet rolled into the roller is 9cm. Find the weight of the roller, if 1cm³ of iron weighs 8 gm.       2138.4 kg

32) Find the number of coins 1.5cm. in diameter and 0.2cm. thick. to be melted to form a right circular cylinder of height 10cm and diameter 4.5cm.                       450

33) 25 circular plates, each of radius 10.5cm and thickness 1.6cm, are placed one above the other to form a solid right circular cylinder. find the curved surface area and the volume of the cylinderso formed.       2640cm²,13860cm³ 

34) The volume of a metallic cylindrical pipe is 748cm³. Its length is 14cm. its external radius is 9cm.find its thickness.          1cm

35) A hollow cylindrical copper pipe is 21dm long. its outer and inner diameters are 10cm and 6cm. respectively. Find the volume of the copper used fo the pipe.  10560cm³

36) A metal pipe has a bore(internal diameter) of 5cm. The pipe is 2mm thick all around. Find the weight in kilograms, of 2 metres of the pipe, if 1cm³,of the metal weight 7.7 gram.           5.033kg.

37) The difference between inside and outside surfaces of a cylindrical tube 14cm long is 88cm². if the volume of the cube is 176cm³. Find the inner and Outer radius of the tube.     1.5cm, 2.5cm

38) A cylindrical bucket 28cm. in diameter and 72cm. high is full of water. The water is emptied into a rectangular tank 66cm long and 28cm. Find the height of the water level in the tank.                   24cm.

39) A rectangular sheet of paper 30cm x 18cm. can be transformed into the curved surface of a right circular cylinder in two ways i.e., either by rolling the paper along its length or by rolling it along its breadth. Find the ratio of the volumes of the two cylinders thus formed.                                     5:3

40) What length of a solid cylinder 2cm in diameter must be taken to recast into a hollow cylinder of length 16cm, external diameter 20cm and thickness 2.5mm?    79 cm

41) A cylindrical container with diameter of base 56cm contains sufficient water to submerge a rectangular solid of iron with dimensions 32cm x 22cm x 14cm. Find the rise in the level of the water when the solid is submerged. 4cm.

42) Water flows out through a circular pipe whose internal diameter is 2cm, at the rate of 6 metres per second into a cylindrical tank, the radius of whose base is 60cm. By how much will the level of water rise in 30 minutes?            3m.

43) A cylindrical water tank of diameter 1.4m and height 2.1m is fed by a pipe of diameter 3.5cm through which water flows at the rate of 2cm/sec. Calculate in minutes the time it takes to fill the tank.                                  2800mins

44) Water is flowing at the rate of 3 km per hour through a circular pipe of 20cm internal diameter into a circular cistern of diameter 10m and depth 2m. In how many times will the cistern be filled? 1hr 40 min

45) Water flows out through a circular pipe whose internal diameter is 2 cm at the rate of 7 metres per second into a cylindrical tank the radius of whose base is 40cm By how much will be level of water rise in half an hour?       787.5 cm

46) A lead pencil consists of cylinder of wood with a solid cylinder of graphite fitted into it. The diameter of the pencil is 7cm. The diameter of the graphite is 1mm and the length of the pencil is 10cm. Calculate the weight of the whole pencil in grams correct to three places of decimals. If the specific gravity of the wood is 0.6 and of the graphite 2.3.                    2.443gm

47) Water flows out through a circular pipe whose internal diameter is 2cm at the rate of 7 metres per second into a cylindrical tank the radius of whose base is 40cm. By how much will the level of water rise of water rise in half an hour?                              787.5ccm

48) A well with 10m inside diameter is dug 14m deep. Earth taken out of it has been spread evenly around it, to a width of 5m.find the height of the embankment so formed.     14/3cm

49) A cylindrical pipe has inner diameter of 7cm and water flows through it at 192..5 litres per minute. Find in kilometre per hour, the rate of flow. (1 litre= dm³).                              3km/hr

50) The difference between inside and outside surfaces of a cylindrical metallic pipe 14cm long is 44cm². If the pipe is made of 99cm³ of metal, find the outer and inner radii of the pipe.   2.5cm, 2 cm

51) A copper wire 4mm in diameter 4mm in diameter is evenly wound about a cylinder whose length is 24cm and diameter 20cm so as to cover the whole surface. Find the length of the weight of the the wire assuming the specific gravity to be 88.8 gram per cm³.      426.24π² gm.

TRIGONOMETRICAL RATIOS OF ANGLES

EXERCISE- A

 Find the value of:

1) sin2760.                                     - √3/2

2) sin 1755                                     - 1/√2

3) sin(-3330).                                      -1

4) sin120.                                          √3/2

5) sin 4530.                                       - 1/2

6) Sin 150°.                                           1/2

7) Sin(-11π/4).                                    -1/√2

8) cosec (-1410).                                     2

9) cosec 675.                                       -√2

10) Cosec(-675).                                    √2

11) Cosec(-660).                                  2/√3

12) Cosec(16π/3).                            -2/√3

13) cos(-1170).                                         0

14) cos 315.                                       1/√2

15) cos 855.                                       -1/√2

16) cos 690.                                        √3/2

17) Cos(5π/2- 19π/3).                        √3/2

18) sec 210.                                      -2/√3

19) Sec(-31π/4).                                   √2

20) sec 11π/4.                                     -√2

21) Sec(15π/4).                                   √2

22) Sec 15π/4.                                      √2

23) tan(-1755).                                        1

24) Tan(-1125).                                        -1

25) tan(-17π/4).                                     -1

26) Tan(3π/2+ π/3).                         -1/√3

27) cot 1230.                                        -√3

28) cot(-870).                                        √3

29) cot 330.                                          -√3

30) Cot 840.                                     -1/√3

31) Cot(16π/3).                                1/√3

32) cot(-1575).                                        1

EXERCISE - B

 Express in terms of TRIGONOMETRICAL ratio of a positive angles less than 45°:

1) sin 139°29'30".               sin 40°39'30"

2) sin(-7495).                                   Cos 25

3) sin 194.                                        - sin14

4) sin 348.                                        - sin 12

5) sin (-1785).                                   sin 15

6) cosec(-830).                              -sec 20

7) cosec(-1324).                            Sec 26

8) cosec(-7498).                           Sec 28

9) cos 189.                                      - cos 9

10) sec(-227'5°).                  - cosec 42'5°

11) sec(-1875).                           Cosec 15

12) tan 615.                                    cot 15

13) tan 305.                                    -cot 35

14) tan 3598.                                   -tan 2

15) cot(-1952).                               Cot 28

16) cot(-1358).                              Tan8

EXERCISE - C 

 Express in terms of the Trigonometrical ratios of x:

1) sin(x- 450).                                    - cosx

2) sin(π+x).                                       - sinx

3) cosec(x-3π/2).                              Secx

4) cos(x - 450).                                    Sinx

5) tan(-π/2 - x).                                    Cotx

6) cot(540 - x).                                 -cot x

EXERCISE - D

Find the values of:

1) sin135 cos315+ sin420 cos320.    √3/2

2) √3sin(1380)+tan²(-240)- cos²(405).   1

3) sin 330 + tan 45 - 4 sin²120+ 2 cos² 135 + sec²180.                             -1/2  

4) sin480 cos690 +cos780 sin1050.   1/2

5) sin²120 + cos²150 + tan²120 + cos 180 - tan 135.                                        9/2

6) sin²(-300)cos³(120) + cos²(-240) sin³(390).                                           -1/16

7) sin750 cos300 + cos1470 sin (-1020).      1

8) {sin150 - 5cos 300+ 7tan 225}/(tan 135 +3 sin 210).                            -2

9) {sin(90-x)sec(180-x)sin(-x)}/ {sin(180+x) cot(360-x)cosec(90+x)}.                           Sinx

10) m² sin π/2 - n² sin 3π/2 + 2mn sec π.                      (m-n)²

11) {sin(270+A) cos(90- A)}/{sin(180-A) cos(180- A)}.                                        1

12) cos570 sin510+ sin(-330)cos(-390).      0

13) cos 24+ sin 55 + cos125 + cos 204 + cos(300).                                1/2

14) {cos(90+x) sec(-x) tan(180-x)}/{sec(360+x)sin(180+x)cot(90-x)}.    -1

15) cosx/{sin(90+x)} + sin(-x)/{sin(180+x)} - {tan(90-x)}/cotx.          3

16) cos(-x)/{sin(90+x)}.                            1

17) (cos 3x - 2 cos 4x)/(sin 3x + 2sin 4x), when x= 150.                           -1/2(1+√3)

18) {sec(270+x)sec(90-x)+ tan(270-x) tan(90+x)}/{cotx + tan(180+x) + tan(90+x)+ tan(360-x) + cos180}.          -1

19) tan π/4+ tan 3π/4 + tan 5π/4 + tan 7π/4.                                                0

20) tan(-x)/{sin 540)}.                       secx

EXERCISE - E

 PROVE:

1) sin780 sin480+ sin30 cos120= 1/2.

2) sin780 cos 390- sin330 cos(-300)= 1.

3) sin420cos390+ cos(-390)sin(-330)=1.

4) (sin250 + tan 290)/(cot 200+ cos340)= 1.

5) 3[sin⁴(3π/2 - x) +sin⁴(3π+x)] - 2 [sin⁶(π/2 +x)+ sin⁶(5π-x)]= 1.

6) sin420 cos390+ cos(-300)sin(-330)=1.

7) (sin150 - 5cos300 +7 tan225)/(tan 135 + 3 sin 210) = -2.

8) sin²54 - sin²72 = sin² 18 - sin²36.

9) Sin(π/2 + x) cos(π- x) cot(3π/2 + x) = sin(π/2- x) sin(3π/2 - x) cot(π/2+ x).

10) cos 306+ cos234+ cos162+cos18= 0. 

11) {cos²(π/2 +x)}/[{sec²(π+x)} - 1] + {cos²(2π -x)}/{cosec²(π+x)}- 1]= 1.

12) 4 cos²210 - tan 315 + 4 cosec 90 = 8.

13) cos 24 + cos 55 + cos 125 + cos 204 + cos 270 + cos 300= 1/2

14) tan 225 cot 405+ tan765 cot 675 = 0.

15) tan130 tan 140 = 1.

16) {1+ cotx - sec(π/2+ x)}{1+ cotx + sec(π/2+ x)}= 2 cotx.

17) cot(9/-x) cotx cos(90-x) tan(90- x) = cosx.

                                  

EXERCISE - F

Show that following are are independent of x:

1) {sin(π/2 -x) sin(3π/2 - x) cot(π/2+ x)}/(sinx cos x).

2) sin(π/2+ x) sin(π+ x)+ cos(π/2+ x) cos(π- x).

3) {sin³(2π - x)}/{cos²(3π/2 +x)}. {Cos³(2π- x)}/{sin³(2π+x)} . {tan(π-x)}/{cosec²(π-x)}.{sec²(π+x)}/sinx.

4) 3{sin⁴(3π/2 - x) + sin⁴(3π + x)} - 2{sin⁶(π/2+ x) + sin⁶(5π- x)}.

5) cotx + tan(180+x)+ tan(90+x) + tan(360-x).

EXERCISE - G

1) In any triangle ABC, Prove that:

a) sin A cos (B+C)+ cosA sin(B+C)= 0.

b) sin(B+ C)+ sin(C+ A)+ sin(A+ B)= sinA + sinB + sinC.

c) {sin(B+ C)+ sin(C+ A)+ sin(A+ B)}/{sin(π+A) + sin(3π+B) + sin(5π+C)= -1

d) Sin(A+ B) - cosC = cos(A+ B)+ sinC.

e) cos(A+B)/2= sin C/2.

f) cos C = - cos(A+B).

g) Cos A cos C + cos(A+ B) cos(B+C)} /{cosA sin C - sin(A+B) cos(B+C) =  cot C.

h) tan A/2= cot{180 + (B+C)/2}.

i) {tan (B+C)+tan (C+A)+ tan(A+B)} /{tan(π+A)+ tan(3π+B)+ tan(5π+C) = -1

j) tan (B+C)/2 = cot A/2.

k) tan (B+C)+tan (C+A)+ tan(A+B)}/ {tan(π-A)+ tan(2π-B)+ tan(3π-C) = 1.

l) cot A tan (B+C)- cosA sec(B+C)= 0

2) If A, B, C, D are the angles of a quadrilateral, prove:

1) cos(A+B)/2 + cos(C+D)/2= 0.

2)  cos (A+B)/2 + cos(C+ D)/2= 0.

3) cos ((A+C)/2 + cos(B+ D)/2= 0.

4) tan (A+ B)/2 + tan (C+D)/2= 0.

3) If A, B, C, D are the four angles, taken in this order, of a cyclic quadrilateral, prove that:

1) cos A + cosB + cosC + cosD= 0.

2) tan(A+B)+ tan(C+ D)= 0

3) cot A + cotB + cotC + cotD= 0

EXERCISE - H

If n is any integer, find the value of:

1) sin{nπ +(-1)ⁿ π/3}.                         √3/2

2) {1+ sin (π- x) cos nπ}{1+ sin (π+x) cos nπ}                                      cos² x

3) Sin(-1230) - cos{(2n +1)π +π/3}.        0

4) cosec{nπ +(-1)ⁿ π/4}.                    ± √2

5) cosec{nπ/2 +(-1)ⁿ π/6}.         ±2, ±2/√3

6) cos nπ.                                             (-1)ⁿ

7) cos{nπ +(-1)ⁿ π/3}.                        ±1/2

8) Cos(nπ+ x).                            (-1)ⁿ cosx

9) tan{nπ +(-1)ⁿ π/4}.                             1

10) tan{nπ +x}.                        (-1)ⁿ tanx. 

11) Tan(nπ - x).                               - tanx

EXERCISE - I

1) If tan 25= a, then show that (tan 205 + tan 295)/(tan 65- tan 335) = (a² -1)/(a² +1).

2) If tan 25= x, then show that (tan 155 - tan 115)/(1+tan 155. tan 115) = (1-x²)/2x.

EXERCISE- J

Prove:

1) sin²π/8 + sin² 3π/8+ sin² 5π/8 + sin² 7π/8 =2.

2) sin²π/4 + sin² 3π/4 + sin² 5π/4 + sin² 7π/4 =2.

3) cos²π/4 + cos²3π/4+ cos²5π/4 + cos²7π/4 = 2.

4) cos²π/8 + cos²3π/8+ cos²5π/8 + cos²7π/8 = 2.

5) tan π/16 tan 3π/16 tan 5π/16 tan 7π/16 = 1.

6) tan π/12 tan 5π/12tan 7π/12 tan 11π/12 = 1.

7) cot π/20 cot 3π/20 cot 5π/20 cot 7π/20 cot 9π/20 = 1

EXERCISE - K

Solve:

1) 2 Sinx -1= 0.                                     30

2) 2 sin 3x =1.                                       10

3) 3 Sinx = 2 cos²x.                              30

4) 2 sin 2x = 1/√3.                                30

5) 2 sin 2x = √3.                                       30

6) 2 sin²x =1/2.                                        30

7) Sin²x - 2 cosx + 1/4 = 0.                   60

8) 4 sin²x - 3 = 0.                                      60

9) Sin²x - 1/2 Sinx = 0.                     0 , 30

10) Sinx/(1- cosx) + Sinx/(1+ cosx)= 4.    30

11) Sinx= coax.                                       1

12) 2 sin²x = 1/2.                                    30

13) 2 cosx = 1.                                       60

14) 2 cos 3x = 1.                                     20

15) Cosx/(1- Sinx) + cosx/(1+ Sinx)= 4.    60

16) Cosx/(cosecx +1) + cosx/(cosec -1)= 2.    45

17) 2 cos²x = 1/2.                                    60

18) 2 cos²x -1= 0.                                   45

19) 2 cos²x + Sinx - 2=0.                   0 , 30

20) Cos²x - 1/2 cosx = 0.               90,60

21) (Cos²x - 3 cosx +2)/sin²x = 2.         90

22) Cos²x/(cot²x - cos²x) = 3.               60

23) Tan 5x = 1.                                          9

24) 3 tanx + cotx = 5 cosecx 60.

           

25) 3 tan²x -1= 0.                                   30

26) 4 tan²x =12.                                      30

27) Tan²x + 3 = 3 Secx.                       0,60

28) 3 tan²2x -1= 0.                                  15

29) Tan²x + 3= 3 Secx.                       0,60

30) Tan²x + cot²x = 2.                              45

31) Tan²x - (√3 +1)+ √3 = 0.            45, 60

32) Sec²x - 2 Tanx = 0.                           45

33) Sec²x + tan²x = 5/3.                       30

34) Cosec²x - cotx (1+√3)+ √3 -1= 0.       45, 60

   

Mg. A- R.1

1) Find the value of:

a) cos(-1170°).                                     0

b) cos(-870°).                                    √3

c) tan(-1755°).                                        1

d) cot 660° + tan(-1050°).                     0

e) sec(-945°).                                     -√2

f) cosec(-840°).                              -2/√3

g) sin 135° cos 210° tan 240° cot 300° sec 330°.                                         1/√2

h) cos 24°+ cos 55° + cos 125° + cos 204° + cos 300°.                                 1/2

Mg. A- R.2 

Find the value of:

1) tan π/12 tan 5π/12 tan 7π/12 tan 11π/12.              1

2) tan 1° tan 2° tan 3°....... tan 87° tan 88° tan 89°.              1

3) sin²120+ cos²120+ tan²120+ cos180 - tan 135.             9/2

4)  If A, B, C, D are the successive angles of a cyclic quadrilateral then prove 

i)cosA+ cos B + cos C + cos D = 0.

ii) tan(A+ B)+ tan(C + D)= 0.

5) If cos x - sin x =√2 sin x then show cosx + sinx =√2 cosx.

6) If x= r cos k cos m, y= r cos k sin m and z= r sin k then show x² + y² + z² = r².

7) If tan⁴x + tan²x = 1, then show cos⁴x + cos²x = 1.

8) If x= a sec k cos m, y= b sec k sin m and z= c tan k, then show x²/a² + y²/b² - z²/c² = 1.

9) if tan x= (siny - cos y)/(sin y + cos y) then show sin y + cos y= ± √2 cos k.

Mg. A- R.3

1) If x be an angle of fourth quadrant and sec x= 5/3 then find the value of (6tan x + 5 cosx)/(5 cotx + cosecx). 1

2) If tanA + sin A= m and tanA- sin A= n, then show m² - n² = 4√(mn).

3) If 3 sinx + 4 cosx = 5, then show sin x= 3/5.

4) If tan x + sinx = m, tan x - sinx = n, then show mn= tan²x. sin²x and 4√(mn)= m² - n².

5) show 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2 + x) + sin⁶(5π - x)] independent of x.

6) If sinx + cosecx = 2, then show sin⁷x + cosec⁷x = 2.

7) If 1 + sin²A = 3 sinA cosA then find the value of tan A.                 1, 1/2

8) A, B, C are the three angles of an acute angled triangle and cos(B+ C - A)= 0, sin(C+ A- B)= √3/2. Find the value of A, B, C.           45, 60, 75

Mg. A- R.4

1) Find the value of m² sin(π/2) - n² sin(3π/2) + 2mn sec π.           (m - n)²

2) tan(π/4)+ tan(3π/4)+ tan(5π/4)+ tan(7π/4).                                                 0

3) If 6x= 11π, the value of 2 cosx + 3 tan x is

a) 1 b) 0 c) √3 d) 2√3

4) If cot x= cos 60+ sin 30, The value of cosx + cos(x -90) is

A) √2 B) 1/√2 c) 1 d) 0

5) If 3 sin²x + 5 cos²x= 4 and π/2< x < π, find the value of sin 2x.                       -1

6) If 3 tanx = -4/3 , find the value of sin x.                   -4/5 or 4/5

7) If sinx = -2/3 and 270°< x < 360°, find the value of sin (x-270°) tan(360° - x).     2/3

Mg. A- R.5

1) If A, B, C, D are the four angles of a Quadrilateral, show that Sin(A+ B)+ sin(C+ D)= 0

2) A, B, C are the angles of a triangle, show that tan (C- A)/2 = cot (A + B/2)

3) If A+ B =60, show that, sin(120- A)= cos(30- B).

4) Find the value of tan(nπ +π/4).        1

5) If x=100°, determine the sign of the expression Sinx + cosx.             Positive

6) If sin(A- B)=√3/2 and sinA=1/√2, find the positive value of B.                       75°

7) If cosx= cos y (x≠ y), find the possible value of cos(x+ y).                                  1

8) Prove that, tan 1° + tan 2°.... Tan 88° tan 89° = 1

9)Find the value of cos 1 + cos 2 + cos 3+ .....+ Cos 180.                                  -1

Mg. A- R.6

1) Prove that sec(-1680). Sin 330=-1.

2) If A, B, C, D are the four angles of a cyclic Quadrilateral, show that cosA+ cos B + cos C+ coa D= 0.

3) If tan 25= a, prove that (tan 155 - tan 115)/(1+ tan 155. Tan 115)= (1- a²)/2a

4) If A, B, C are the angles of a triangle, show that {Sin(B+ C)+ sin(C+ A) + sin(A+ B)}/{Sin(π+A)+ sin(3π+B)+ sin(5π+C)}= -1

5) If Secx =√2 and 3π/2< x < 2π, find the value of (1+ tanx + cosecx)/(1+ cotx - cosecx).                                                  1

6) If tan x= 0.4, when x lies between 0° and 360°, write down the possible values of x and Sinx.           21°47', 201°48', 0.3714, -0.3714

l) If cos²x - sin²x = tan²y, then show cos²y - sin²y = tan²x.

m) Show: 4(sin⁶x + cos⁶x) - 6(sin⁴x + cos⁴x)= - 2.

n) If x sin³a + y cos³a= sina and x sina - y cosa = 0, then prove x² + y² = 1.

o) If a sinx = b cos x = (2c tanx)/(1- tan²x), prove (a² - b²)²= 4c²(a² + b²).

p) If a sinx + b cos x = c then show a cosx + b sinx = ±√(a²+ b²+c²)

q) If 4x secA = 1+ 4x² then show secA + tanA = 2x or 1/2x.

r) If cos⁴x + cos²x = 1 then prove tan⁴x + tan²x = 1.

s) If p tanx = tan px then show sin²px/sin²x = p²/{1+ (p² -1)sin²x}

t) Eliminate x: tanx - cotx = a, cosx + sinx = b.

u) If tanA= n tan B, sinA= m sin B, then show cos²A = (m² -1)/(n² -1).

IDENTITY AND ASSOCIATED ANGLE

Miscellaneous Chapter-2

1) Eliminate x:

A) a= c(secx + tanx) and b= c(secx - tanx)

B) a sinx + b cosx = c and a cosx - b sinx = d.

C) tanx+ cot x= a, cosx + sinx =b.

D) eliminate x and y from a cot²x + b cot²y = 1, a cos²x + b cos²y = 1 and a sinx = b sin y.           (a+ b)(a³ + b³)= ab(a+ b+1)(a+ b -1).

2) The angles of a triangle are in AP and the greatest angle is twice the smallest. Find the angles in radians.

3) If secx + tanx =√3 find secx - tanx .                                            -5/4

4) Prove tan 130 tan140 = 1

5) If cosx = -4/3 find secx and cosecx.        

6) Evaluate: tan 1. tan 2. Tan 3. ....tan 89.                                        1

7) If cosx - sinx =√2 sinx show cosx + sinx =√2 cosx.

8) If secx - tanx = p, find sinx.

9) if a cosx - b sinx =c express a sinx + b cosx in terms of a, b, c.                    ±√(a²+ b²- c²)

10) If cos⁴x + cos²x =1. Prove tan⁴x + tan²x = 1.

11) Find maximum and minimum value of 9 tan²x + 4 cot²x.

12) If x= r cos a cos b, y= r cos a sin b, z= r sin a, show x²+ y²+ z² = r².

13) Prove: (1- sinx)/(1+ sinx)= (sec x - tanx)².

14) If A, B, C, D are the angles of a cyclic quadrilateral, then prove cos A+ cos B+ cos C + cos D= 0.

15) If cos²x - sin²x = tan²y show that cos²y - sin²y = tan²x.

16) If x+ y= 90 and 2(cos²x - cos²y) = 1 then find the value of x.      30°

17) Value of 4(sin⁶x+ cos⁶x)- 6(sin⁴x + cos⁴x).                                         -2

18) value of sec(-1680)sin 330.     1

19) If cosx + sinx =1, then find cosx - sinx.                      ±1

20) If tanx = -4/3 find sinx.       ±4/5

21) The cotangent of the angles π/6, π/4, π/3 are in AP or G. P or HP or no sequences.                    GP

22) The equation (a+b)² sec²x = 4ab is possible when

a) a=0 b) b= 0 c) a= b. d) a≠ b

23) If z= cos²x + sec²x then the value always

a) z< 1 b) z= 1 c) 2>z>1 d) z≥2.

24) If sin²x + sinx = 1 then the value of cos¹²x + 3 cos¹⁰x + 3 cos⁸x + cos⁶x - 1.           0

25) Which of the following are true

a) tan 1°< tan 1 b) tan< tan 1° 

. c) tan 1< tan 2 

26) If tan⁴x + tan²x = 1. Show cos⁴x + cos²x = 1.

27) If sinx + cosecx = 2, show sin⁸x + cos⁸x = 2, also sinⁿx + cosⁿx = 2.

28) If x cos³a + y sin³s = sina cosa and x cos a - y sin a = 0 prove x² + y² = 1.

29) If tan x + sin x= m, tan x - sin x= n show m² - n² = √(4mn).

30) If x = cosec a - sin a and y= sec a - cos a then show x²y²(x² + y²+3) = 1.

31) If (sec a+ tan a)(sec b+ tan b) (sec c+ tan c)= (sec a- tan a)(sec b - tan b)(sec c+ tan c) then show each side = ±1.

32) If bx sin m = ay cos m and ax sec m - by cosec m = a² - b² show that x²/a² + y²/b² = 1.

33) If a sin x = b cos x = 2c {tanx/(1- tan²x)} show (a² - b²)² = 4c²(a² + b²).

34) If sin⁴(x/a) + cos⁴(x/b)= 1/(a+ b) prove sin⁸(x/a³) + cos⁸(x/b³)= 1/(a+ b)³.

35) If x= π/4n show sin²x + sin²3x + sin²5x + ..... to 2n terms = n.

36) If m² + m₁²+ 2mm₁ cos x = 1, n²+ n₁² + 2nn₁ cos x = 1 and mn + m₁n₁ + (mn₁ + m₁n) cos x= 0, then prove cosec²x = m²+ n².

COORDINATE GEOMETRY ( DISTANCE FORMULA) - (IX)

A) FILL THE BLANKS:

a) If a point lies on x-axis, then y coordinate is_____

b) If a point lies on y-axis, then x coordinate is ____.

c) If two points have same abscissa, the line joining them is parallel to____.

d) If two points have same ordinate, the line them is parallel to_____.

e) The line joining (2,3) and (-2,-3) passes through the_____.

1) Find the distance between:

a) (-2,3), (6.9).                   10 units

b) (5, 18),(-4, -22).            41 units

c) (-7, -24),(0,0).                  25 units

d) (2/5, 2/5), (-2/5,-1/5).       1 unit

e) (5/10, 1/10),(-1/10,-7/10).      1

f) (√3,1),(0,0).                                2

g) (-2,3),(6,-3).                               10

h) (2,3),(-2,3).                         4 units

i) (3,5),(3,1).                           4 units

j) (2,-2),(5,2).                          5 units

k) (3,7),(-2, -5).                      13 units

2) Find:

a) A is a point on the Y-axis whose ordinate is 5 and B is the point (-3,1). Calculate the length of AB.  5 units

b) The distance between A (1,3) and B(x,7) is 5. Find x.           4 or -2

c) Calculate the distance between A(7,3) and B on the x-axis whose abscissa is 11.                             5

d) Find the coordinates of the points on the y-axis Which are at a distance of 13 units from the points (12,9).                            (0,4),(0,14)

e) Find the coordinates of the points on the x-axis Which are at a distance of 5 units from the points (5,4).                            (8,0),(2,0)

f) What point/s on the Y-axis are at a distance of 10 units from the point (8,8), (0,2).                  (0,14)

g) Find point/s which are at a distance of √10 from the point (4,3), given that the ordinate of the point/s is twice the abscissa. 

(1,2)

h) A point P is at a distance of √10 units from the point (4,3). Find the coordinate of P, it being given that its ordinate is twice its abscissa.                           (3,6),(1,2)

i) If A is (4,2) and B(1,y), find the possible values of y so that AB= 5.            -2, 6

3) 

a) Show that the point (4,4) is equidistant from the points A(1,0) and B(-1,4).

b) What point on the x-axis is equidistant from A(5,4), B(-2,3). (2,0)

c) Find a point equidistant from the points A(6,2), B(-1,3), and C(-3, -1). (2,-1)

d) If A is (2,5) and B is (x, -7), find the possible values of x so that AB= 13.                                                -3, 7

e) What points on the x-axis are at a distance of 17 units from the point A(11,-8).                 (26,0) and (-4,0)

f) What points on the y-axis are at a distance of 10 units from the point A(-8, 4).                    (0,10) and(0,2) 

4) show that the points form a right angled triangle ::

a) A(6,6), B(2,3), C(4,7).

b) A(60), B(-2,6), C(12,8).

c) A(0,3), B(-2,1), C(-1,4).

d) A(-3,-4), B(2,6), C(-6,10).

e) A(3,3), B(9,0), C(12,21).

5) Show that the points form an equilateral triangle..

a) A(1,1), B(-1,-1), C(-√3,√3).

b) A(2a, 4a),B(2a, 6a),C(2a +√3a, 5a).

6) Show that the points form an isosceles right angled triangle.

a) A(7,10), B(-2, 5), C(3, 4).

7) Show that the given points are collinear:

a) A(-2,3), B(1,2), C(7,0) 

b) (3, -2), B(5,2), C(8,8) 

c) A(1, 1), B(--2, 7), C(3,-3).

d) (-1,-1), B(2, 3), C(8, 11).

8) If A(7,5), B(2,4) and C(6,10), show that AB= AC. Assign special name to the triangle ABC.  Isosceles triangle

9) Show that Isosceles triangle

a) A(2,-4), B(7,-1), C(5, 1).

10) Find the circumference of the triangle whose vertices are (-2,-3), (-1,0), (7, -6). (3,-3)

11) Show that P(11,2) is the centre of the circle which passes through A(1,2), B(3,-4), C(5, -6).

12) The centre of a circle is (x+2, x-1). Find x if the circle passes through (2,-2),,(8,-2)..                     3

13) Show that the points are the vertices of a rectangle.

a) (2,-2),(8,4),(5,7),(-1,1)

14) Show that the points form rhombus..

a) A(7,3), B(3,0), C(0,-4) (4, -1).

b) A(0,5), B(-2,-2), C(5,0) (7, 7)

15) Prove for Parallelogram:

a) A(1,2), B(-2,-1), C(1,0) (4, 3).

 

16) Find a if the triangle formed by A(8,10), B(7, -3), C(0, a) is right angled at B.                                    -4

17) Show that it is square:

a) (2,1), B(0,3), C(-2,1) (0, -1).

ANNUITY

                  ANNUITIES

                  -----------------

DEFINATION:

An annuity is a series of equal payments made at EQUAL interval of time. This interval usually a year but it may be a half-year, a quarter- year, a month and so on. 

Examples: Payment for rent, premium of life insurance, recurring deposits in a bank etc.

The interval between two successive payments is called PAYMENT PERIOD. The interval between the beginning of the first payment period and the end of the last payment period is called the TERM or STATUS of the annuity and is measured in years.

The sum payable in each payment period is called PERIODIC PAYMENT or PERIODIC RENT and the total sum payable in a year is called ANNUAL RENT.

TYPES:

A) ANNUITY CERTAIN: An annuity in which payments begin and end at fixed dates is...

B) PERPETUITY: An annuity in which payments begin at a fixed date but continue for ever is...

C) CONTINGENT: It an annuity in which the payments are dependent on some conditions.

*According to their time of payment

A) ORDINARY or IMMEDIATE ANNUITY: If the payment are made at the end of each payment period is...

B) ANNUITY DUE: If the payment are made at the beginning of each payment period, 

C)* DEFERRED ANNUITY: The payment of which commence after a certain period. If an IMMEDIATE ANNUITY is deferred for m years, the first payment is to be made at the end of (m+1)th year, but if an ANNUITY DUE is referred for m years, the first payment is to be made at the beginning of (m+1)th year.

**NOTE

Unless otherwise stated or implied

A) annuity will mean ordinary or immediate annuity and

B) payment period will be one year.

FORMULA USED

------------------------

1) The amount of an immediate annuity or, simply, annuity is given by

M= A/i{(1+ i)ⁿ -1}

Here, 

A is paid at the end of every year for n years.

n is number of years

i is rate/100.    

NOTE: If the number of payments made in a year be m and the total amount paid in a year be A, 

If the payment is made at the end of half yearly i.e, twice a year, then

M= A/i {1+ i/2)²ⁿ - 1}

REMEMBER: A is always the total payment made in a year.

2) The amount of an annuity due is:

M= (1+ i). A/i {(1+ i))ⁿ -1}

NOTE: If the number of payments made in a year be m and the total payment paid in a year be A, then

M= (1+ i/2) A/i {(1 + i/2)²ⁿ -1}  

* PRESENT VALUE OF AN ANNUITY

The present value of an annuity is the sum of the present values of all the payment of the annuity.

V= A/i {1- (1+ i)⁻ⁿ}.

NOTE: If the payment is made m times a year and the total payment in a year be A, then

V= A/i {1- (1+ i/2)⁻²ⁿ}

** The present value of an annuity due , when paid beginning of each year.

V= (1+ i)A/i {1- (1+ i))⁻ⁿ}

NOTE: If the payment is made m times a year and the total payment in a year be A,

V= (1+ i/2)A/i {1- (1+ i/2))⁻ⁿ}

** The present value of an immediate perpetuity or, simply, perpetuity is given by:

V= A/i

             EXERCISE -1

1) Find the amount of an immediate annuity of rupees 100 p.a left unpaid for 10 years, allowing 5%p.a C. I.

A) 1000 B) 1257.80. c) 1500 D) 1698.30

2) Find, correct to nearest rupee, the amount of an annuity of ₹100 in 20 years allowing compound interest 9/2%, Given log 1.04= 0.0 191163 and log 24 117= 1.3823260.

A) 3100 B) 3400 C) 3137. D) 4200

3) Find the amount and the present value of annuity of ₹150 for 12 years, reckoning compound interest at 3.5 % per annum.

A) 1200,1430  

B) 2190.28, 1449.50.

C) 3000, 320 D) none

4) What sum will buy an annuity of ₹1050 payable for 4 years, the rate of interest 3.5 % per annum compound ?

A) 3846. B) 4300 C) 5200 D) 6300

5) find the amount of the annuity of ₹150 in half yearly installments for 15 years at 4% p.a. interest also payable half-yearly.

A) 2041.30 B) 3041.25.

C) 4320 D) 6700.43

6) calculate the amount and present value of annuity of annual value of ₹400 payable at the end of each of 3 months for 5 years at 4% C. I compounded quarterly.

A) 2000, 1500 B) 2201, 1804.

C) 2100, 1370 D) 3200, 1400

7) A man decided to deposit ₹300 at the end of each year in a bank which pays 3% p.a.(compound interest). If the installments are allowed to accumulate, what will be the total accumulation at the end of 15 years ?

A) 5500 B) 5560 C) 5580 D) 6000

8) If the present value of an annuity for 10 years at 6% p.a is ₹2500, find the annuity.

A) 300 B) 350 C) 340. D) 400

9) If the present value of an annuity for 25 years at 5% per annum is 50000, find the annuity.

A) 1000 B) 1046.90 C) 1500 D) n

10) what annuity should be paid for 10 years if 10,000 is paid now, the first payment of annuity being made in one year time and the subsequent payment in yearly installments at the end of each subsequent year ? compound interest is to be taken at 5% p.a

A) 1100.94 B) 1200.94

C) 1294.91 D) none

11) A loan of ₹40000 is to be paid back by 30 equal installments. Find the amount of each such installment to cover principle and compound interest at 4% per annum.

A) 2316. B) 2512 C) 2814 D) N

12) A loan of ₹10000 is to be repaid in 30 equal installments of ₹ P. find, P if the compound interest charge is at the rate of 4% p.a. (Annuity is an annuity immediate) given (1.04)³⁰=3.2434

A) 500 B) 525.30 C) 578.40. D) n

13) A loan of ₹40000 is to be repaid equal installments consisting of principal and interest due in course of 30 years. find the amount of each installment reckoning Interest @4%.

A) 2121 B) 2315. C) 2400 D) 2500

14) A company borrows ₹10000 on condition to repay it with compound interest 5% p.a. annual installments of ₹1000 each. In how many years will the Debt be paid off ?

A) 10.2yrs B) 11.2yrs C) 14.2yrs D) n

15) S. Roy borrows ₹20000 at 4% compound interest and agrees to pay both the principal and the interest in 10 equal installments at the end of each year. find the amount of these installments.

A) 2000.15 B) 2300.75

C) 2470.15. D) 2550.75

16) A loan of Rs4000 is to be repaid on equal half-yearly installments in four years. If the rate of compound interest be 10% per annum. find the value of each instalment.

A) 618.80. B) 720.30

C) 830.60 D) 900.50

17) A firm borrows ₹1000 on condition to repay it with compound interest at 4% per annum by annual installments of rupees 100 each. In how many years will the debt be paid off.

A) 13.1 years .      B) 14.3 yrs

C) 15yrs                D) none

18) A person borrows ₹4000 on the condition that he will repay the money with compound interest at 5% per annum 6 equal annual installments, the first one being payable at the end of 1st year. find the value of each installment.

A) 441.44       B) 666.66

C) 787.71.      D) 991.91

19) A Overdraft of ₹50000 is to be paid back in equal installment over a period of 10 years. find the value of this Payment reckoning compound interest at 5% p.a

A) 6,499. B) 6785 C) 7500 D) N

20) A motorcycle is purchased on installment basis, such that ₹3400 is to be paid on the signing of the contract and four yrly installments of ₹2400 each payable at the end of 1st ,2nd, 3rd and 4th years. If interest is charged at 8% per annum. what would be the cash down price? Given that (1.08)⁴= 1.36

A) 11341.18.       B) 11911.50

C) 12651.50        D) 12000.35

21) A man buys a house for ₹40000 on the following conditions. He will pay ₹10000 cash down and the balance in 10 equal annual installments, the first to be paid one year after the date of purchase. Calculate the amount of each installment, compound interest being calculated at the rate of 5% p.a.

A) 3885. B) 4338 C) 4600 D) n

22) S. Roy wishes to purchase a printing machinery valued ₹17000. He is prepared to pay now ₹9000 and the balance in 8 equal annual installments, if interest is calculated at 7/2% p.a., how much should he pay annualy

A) 2137 B) 1167. C) 2100 D) 3200

23) A government constructed housing flat cost ₹136000; 40 % of which is to be paid at the time of possession and the balance, reckoning compound interest @9%p.a., is to be paid in 12 equal annual installments. find the amount of each such instalment. Given 1/(1.09)¹²= 0.3558.

A) 10000 B) 11400. C) 12000 D) n

24) A person deposits his whole fortune of ₹20000 in a Bank at 5% compound interest and settles to withdraw ₹1800 per year for his personal expenses. if he begins to spend from the end of the first year and goes on spending at this rate, find the time he will be ruined his savings.

A) 12  B) 14  C) 16 D) 17th year.

25) On his 48th birthday, a man decides to make a gift of ₹5000 to a hospital on his 60th birthday. He decides to save this amount by making equal, annual payments up to and including his 60th birthday to a fund which gives 7/2% compound interest, the first payment being made at once. Calculate the amount of each annual payment (answer to the nearest paise).

A) 219.90 B) 310.90. C) 400 D) n

26) A person retires at the age of 60. He is entitled to get a pension of ₹200 per month payable at the end of every six months. He is expected to live up at the age of 70. if the rate of interest be 12% per annum payable half-yearly, what single sum he is entitled to get at the time of his retirement equivalent to his pension? (Given (1/(1.06)²⁰= 0.3119).

A) 12762 B) 13762. C) 21000 D)n

27) find the sum of money received by a pensioner at 58 if he wants to commute his annual pension of ₹1200 for present payment when compound interest is reckoned at 4% p.a., and the expectation of his life is assessed at 10 years only.

A) 9000 B) 9500 C) 9717. D) n

28) A man retires at the age of 60 years and is employer gifts him pension of ₹3600 a year paid in half yearly installments for the rest of his life. If the expectation of his life is taken to be 10 years and interest is @6% per annum payable half-yearly, determine the present value of the pension.

A) 25000 B) 26724. C) 30000 D)n

29) A sinking fund is to be created for the purpose of replacing some machinary worth ₹100000 after 20 years. How much money should be set aside each year out of the profits for the sinking fund, if the rate of Compound interest is 5% per annum.

A) 3000 B) 3021. C) 3500 D) n

30) A sinking fund is created for replacing some machinery worth ₹ 54000 after 25 years. The scrap value of the machine at the end of the period is ₹4000. How much should be set aside from profit each year for the sinking fund when the rate of compound interest is 5%p.a.

A) 1046.90 . B) 12000 C) 13900 d) n

31) A machine costs a company ₹65000 and its effective life is estimated to be 25 years. A sinking fund is created for replacing the machine at the end of his life, when its scrap realises a sum of ₹2500 only. Calculate what amount should be provided every year from the profits earned for the sinking fund, If it accumulates 7/2% p.a. compound (given that (1.035)²⁵= 2 358)

A) 15000 B) 16000 C) 1610.82. D) n

32) A sinking fund to created for the redemption of debentures of ₹100000 at the end of 25 years. how much money should be set aside out of profits each year for the sinking fund, if the investment can earn interest at 4% per annum ?

A) 2408.09. B) 2500 C) 3200 D) n

33) A company buys a machine for ₹1 lakh. Its estimated life is 12 years and scrap value is ₹5000. what amount is to be retained every year from the profits and allowed to accumulate at 5% Compound interest for buying new machine at the same price after 12 years.

A) 5959.85. B) 6000 C) 6500 D) n

34) An equal sum is provided out of profit by the company each year for the replacement of a machinery after 15 years. Present cost of the machinery is ₹90000 and the cost of machine at the time of time of Replacement is 20% more than the present cost. find the sum if it can earn interest at 5% per annum compound interest.

A) 5000. B) 6000 C) 7000 D) n

35) An equal sum is set aside every year for 25 years to pay off a debenture issue of ₹1 lakh. If the fund accumulates at 2% per annum compound interest, find the value of this annual payment. Given that the amount of annuity of rupees 1 p.a. for 25 years at 2% per annum is equals to 32.03030

A) 3122. B) 4122 C) 5122 D) 5000

36) What sum should be invested every year at 8% per annum compound interest for 10 years, to replace plant and machinery, which is expected to cost then 20% more than its present cost of ₹50000?

A) 3212 B) 4121 C) 4142. D) n

37) the cost of a machine is ₹80000. The estimated scrap value of the machine at the end of its life time of 10 years is ₹₹12000. find the amount of each equal annual installments to be deposited at 9% per annum compound interest annually just sufficient to meet the cost of new machine after 10 years assuming an increase of 40% of the price of the machine then. The first installment is to be paid at the end of the first year. (Given (1.09)¹⁰ = 2..368)

A) 6500 B) 6578.35. C) 4500 D) N

39) A man propose to make an endowment on 1st July 1969 to the university by depositing a sum of money to its banking account stipulating 

a) the payment of scholarship of ₹1000 p.a. for 10 years 

b) the award of book prizes to the value of ₹500 every year for 20 years. the money deposited together with compound interest @ 5% p.a is to exhaust itself at the end of said 20 years. assuming that the payments and the awards stipulated are made at 1st July every year commencing with the year 1970, determine the amount of endowment required to be made on 1st July 1669 for this purpose.

A) 12000 B) 13000 C) 13975 D) n

40) A man aged 40 years take an insurance policy for ₹50000 for which he is expected to make equal annual payment of ₹ x to the insurance Company commencing now and going on until his death. If the expectation of life of a man aged 40 years is 25 years, find the value of x if the insurance company agrees to pay interest at 4% p.a. compound.

A) 1000 B) 1157.78 C) 1221.31 D) n

41) The annual rent of a perpetuity is ₹4000. find its value, the interest being compounded at 5% per annum .

A) 60000 B) 70000 C) 80000 D) 90000

42) the value and the annual rent of a perpetuity are ₹15000 and ₹900 respectively. find the rate of compound interest..

A) 4% B) 5% C) 6% D) 7%

43) The annual rent of a freehold estate is ₹1000. if the rate of compound interest be 10% per annum. find the price of the estate.

A) 10000. B) 11000

C) 12000 C) 13000

44) the price of freehold estate is ₹50000. If the rate of interest be 9% per annum. what would be the property was worth 12500 if the annual rent?

A) 4000 B) 4500 C) 5000 D)5500

45) A free-hold property was worth ₹12500. If the annual rent of the property be ₹1000, find the rate per cent per annum.

A) 8%. B) 9% C) 10% D) 11%

46) The annual subscription for the membership of a club is ₹₹240 and a person may become a life member by paying ₹8000 at a time. find the rate percent per annum .

A) 3% . B) 4% C) 5% D) 8%

47) which is better--an annuity of ₹150 to last for 10 years or the reversion of a freehold estate of ₹ 79.20 p.a., to commence 7 years hence, the rate of interest being 5% per annum.

A) the first one . B) 2nd one

C) both. D) no comment

48) For endowing an annual scholarship of ₹12000 a man wishes to make 3 equal annual contributions. the first award of the scholarship is to be made three years after the last of his 3 Compounded annually.

A) 1300000 B) 150000.

C) 200000 D) none

49) how many years purchase should be given for a freehold estate, if 5% interest be desired

A) 10 B) 20. C) 30 D) 40 yrs

50) The number of years purchase of a property is 12. reckoning compound interest at 6% per annum find the nearest to the duration of lease.

A) 22. B) 24 C) 26 D) 30 years

51) how many years purchase should be given for a free home estate, if 3% compound interest be desired?

A) 33.33 years. B) 44.44yrs

C) 55.55yrs D) 22.22 yrs

52) what perpetuity can be purchased by investing ₹15000 at 5/2% per annum compounded intrest?

A) 375 . B) 475 C) 575 D) 675

EXPANSION: FIND THE VALUE::

TYPE-1

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1) If x+ y= 2, xy= 1 find x²+ y².        2

2) If x+ y= 4, xy=6 find x²+ y².         4

3) If x+ y=6, xy= 10 find x²+ y².     16

4) If 2x+ y= 4, xy= 4 find 4x²+ y².    0

5) If x+ 3y= 7,xy= 8 find x²+ 9y²      1 

6) If 3x+ y= 10, xy= 6 find 9x²+ y².   64

7) If 5x+ y= 12, xy= 8, find 25x²+ y². 64

8) If 7x+ y= 13, xy= 0 find 49x²+ y². 169

9) If x+ 2y= 4, xy= 1 find x²+ 4y².  12

10) If x+ 6y= 25, xy= 50 find x²+ 36y².        25

11) If x+ 9y=5 , xy=1 find x²+ 81y². 7

12) If 2x+ 3y= 5, xy= 2 find 4x²+ 9y². 1

13) If 4x+ 3y= 9, xy= 3 find 16x²+ 9y².      9

14) If x/2+ y=5 , xy= 9 find x²/4+ y². 16

15) If x+ y/3= 11, xy= 63/2 find x²+ y²/9                            100

16) If x/3+ y/5= 12, xy= 345/2 find x²/9+ y²/25 .      121

16) If x/5 + y/2= 1, xy= 1 find x²+ y². 4/5

17) If x+ y= 13, xy= 72, find 2(x²+y²). 50

18) If x+ y= 3, xy= -7/2, find 4(x²+y²).  64

19) If x+ 1/x= 2, find x²+1/x².   2

20) If x+ 1/x= 4, find x²+1/x².     14

21) If 2x+ 1/x= 2, find 4x²+1/x².     0

22) If x+ 1/3x= 3, find x²+1/9x². 25/3

23) If 2x+ 1/2x= 3/2, find 4x²+1/4x². 5/4

24 If 2x+ 1/3x= 4/3, find 4x²+1/9x². 4/9

25) If 4x+ 1/x= 3, find 16x²+1/x². 1

Type - 2

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1) If x - y = 5, xy= 3. Find x²+y². 31

2) If 2a - b= 6, ab= 2. Find 4a²+b². 44

3) If a - 4b= 6, ab= 5. Find a²+16b². 76

4) If 3a - 4b= 10, ab= 3. Find 9a²+ 16b². 172

5) If 2x - y/3= 1, xy= 9. Find 4x²+y²/9. 13

6) If 5x - 4y= 7, xy= 2. Find 25x²+16y². 129

7) If x - 1/x= 2, find x² + 1/x². 6

8) If 2x - 1/x= 5, find 4x² + 1/x². 29

9) If 3x - 2/x= 2, find 9x² + 4/x². 16

10) If 4x - 1/3x= 10, find 16x² + 1/9x². 308/3

11) If 5x - 1/5x= 5, find 25x² + 1/25x². 27

12) If x - 1/x= 7, find 4(x² + 1/x²) 205

13) If x - 1/3x= 10, find x² + 1/9x². 302/3

14) If 2x - 2/x= 1, find x² + 1/x². 9/4

15) If 5x - 5/x= 20, find x² + 1/x². 18

Type--3

******** 

1) If x+ y= 4, xy= 3 then find the value of x - y. ± 2

2) If a+ b= 7, xy= 10 then find the value of x - y. ± 3

3) If 2x+ y= 4, xy= 2 then find the value of 2x - y.                                0

4) If a+ 3b= 10, ab= 8 then find the value of a - 3b.                              ±6

5) If 2x+ 3y= 7, xy= 2 then find the value of 2x - 3y.                            ± 1

6) If 5x+ 2y= 15, xy= 3 then find the value of 5x - 2y.                            ± 2

7) If x- y= 4, xy= 5 then find the value of x + y.                               ± 6

8) If x - 2y= 1, xy= 12 then find the value of x +2 y.                              ± 7

9) If 2x- 3y= 11, xy= 1/6 then find the value of 2x + 3y.                 ±5√5

10) If x+ 1/x = 3. Find x- 1/x.    ±√5

11) If 2x - 1/x= 3. Find 2x +1/x.   ±1

12) If 2x + 5/x= 7. Find 2x - 5/x.  ±3

13) If 2(x - 1/x) = 3.Find x +1/x ±5/2

Type -4

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1) x+ y= 2, xy= 3, find x⁴+ y⁴