PRACTICE MATH AFER SEE

25. What is the area of the square whose diagonal is 4 cm?

a. 16 cm b. 8 cm

c. 12 cm d. 6 cm

26. If 2/3 of a log of wood is 5.6 m what will be the length of the
S/7 of it?

a. 5 m b. 6m

C. 7 m d. 8 m

27. The length of the rectangle is doubled its breadth and has a
perimeter of 12 cm. Find the length of the rectangle.

a. 6 cm b. 4 cm

C. 5 cm d. 3 cmn

28. A circular cylinder has the perimeter of the base equal 44

cm and has a height of 6 cm. What is the curved surface
area of the cylinder?

a. 132 cm? b. 528 cm

C. 264 cm d. 375 cm?

29. The total surface area of a solid sphere made of earth is 616
sq cm. Two hemisphere are formed when it is cut into two
equal parts, find the total surface area of each hemisphere?

a. 316 cm b. 346 cm?

C. 416 cm d. 462 cm?

30. Find the surface area of the volley ball of radius 21 cm.

a. 231 cm b. 5544 cm

C. 5244 cm? d. none

31. Cuboid has its dimension 4 cm x3 cm x 2.5 cm. What is its

volume?

a. l cm b. 60 cm

C. 15cm d. 30 cm

32. A prismn with a square base of side 6 cm has a height of 5
cm. The lateral surface area of the prism is

a. 120 cm b. 30 cm

C. 60 cm dk d.none

33. Cuboid has a length 5 cm, breadth 4 cm and height 3 cm.
What is the total surface area of the cuboid?

a. 60 cm? b. 30 cm

C. 94 cm d. none

50

34. A sphere has the diameter of 7 cm. VWhat is its volume?

a. 329 cm? b. 179.67 cm?

C. 49 cm? d none

35. The radius of a circular plate is 7 cm. What is the

circumference of the plate?

a. 28 cm Nb. 14cm

C. 214 cm d. 44 cm

36. What is the volume of a right pyramid standing on the
square base of the edge 4 cm and height 5 cm?

a. 20cm b. 26 cm

C. 20 cm d. none

37. If the perimeter of the rectangular room of 5 m high was 26
meters, what will be the area of four wall of the room?

a. 130 m? b. 140 m

18 C. 175m d. 190 m²

38. Find the curved surface area of a drum whose height is 25

cm and diameter 14 cm.

a. 1000 cm b. 1100 cm

C. 1050 cm² d. 2200 cm

39. What are the area of the 4 walls and a floor of the room?

a. 2h(l + b) (| x b) b. 2h(l + b) + 2(1 x b)

c. 2h(l x b) + (| x b) d. 4h(| + b)

40. Find the area of the 4 walls and floor of the room whose

length is 4 m, breadth is 2 m and height 2 m.

a. 30 sq.m b. 32 sq.m
C. 28 sq.m
sri d. 40 sq.m

41. The area of the 4 wall of the room is 72 sq. If its length is 8

m and its breadth 4 m, find its height?
stifes4 i b. 4 m
a. 3m

C. 2.5 m St d. 8 m

42. Give the formula to calculate the area of 4 walls, floor and

the ceiling of the room.

a. 2h(l + b) b. 2h(l + b) + 2 (| x b)

C. 2h(l x b) + 2(l x b) d. 2h(l x b) + 2(1 x b)

51

43. The number of the square bricks required paving a floor of

length 8 m and breadth 6 m is 40. Find the area of each

brick.

a. 1.2m b. 4.8 m

c. 0.83 m? d. 1.6 m?

44. If the circumference of the pond is 264 m. Find its diameter.

a. 829.7 m b. 84 m

C. 42 m d. 94 m

45. Hari can see up to 14 km away. What is the total area of the

vision?

a. 615 km? b. 616 km?

C. 620 km d. 655 km

46. Find the area of the circle when the difference between

circumference and diameter of which is 60 km.

a. 616 km b. 317.1 kmti,2r

C. 620 km d. 516 km

47. The area of the circle and a square are equal. If the radius

of the circle is 14 cm. Find the side of the square.

a. 23.62 cm b. 24.819 cm

C. 32.83 cm d. 23.83 49 cm

48. If the circumference of the circle is divided by its diameter

what is the result called?

a. area b e b.radius t6
d. arc
C. pie

49. The curved surface area of the cylinder is S and height h.
S500 Whatdoes h Means
b.diameter
a. radius

C. total surface area d. volume

50. The curved surface area of the drum is 154 cm. Its height is

6m. Find the diameter.

a. 8.16 cm b. 4.08 cm

C. 10.26 cm d. 9.97 cm

51. The curved surface area of the cylinder is S and radius r.

What is the formula to calculate its height?

a b
S
C2S d
d.

52

52. The total surface area of the cylinder is 154 sq em and

radius 3.5 cm Find its height.

a. |4 cm b. 19cm

C. 10 cm d 3.5cm

53. What is the volume of the cylinder of radius 6 cm and
height 8 cm?

a. 226.28 cc b. 452.57 cc

C. 905.14 cc d. 895.28cc

54. The area of a square pond is 100 sq. meters. A path of
uniform width is surrounded outside the pond and its area
is 300 square meter. Find the outer length of the path.

a. 36 m b. 20 m

C. 39 m d. 42 m

55. The volume of the cylindrical tank is 2200 m and its height

is 25 m. What is its diameter?

a. 26 m b. 27 cm

C. 28 m d 29m

56. The area of the path of width Sm, surrounded outside the
square pond is 900 square m. Find the length of the pond.

a. 40 m b. 45 m

c. 48 m d. 52 m

57. What is the surface area of sphere of diameter 1 em?

a. 'cm b. cm

C. 2 Cm d. T cm

58. Which of the formula is not the formula to caleulate the
volume of the sphere?

a. b. 288 3

4 d.

C.

59. If the radius of the lemon is 1 cm, what is its volume?

a b. 1ce

C. 4n cc d. cc

53

60. The volume of the tennis ball is r/6 cc. What is its diameter?

a. 0.1 cm b. 0.5 cm
d. 1 cm
C 0.8 cm

61. What is the volume of the apple whose surface area 4n sq
cm?

3 b. rce

a. Tcc

C. 7n cc d. 6T cc

62. What is the volume of the wall that can be prepared by 500

bricks each of dimensions 8 cm x 4 cm x 4 cm?

a. 8000 cc b. 64000 cc
C. 4000 cc
d 80000 cc

63. How many one Rs coins of volume 15 cm' can be made from

a bulk of copper of volume 3000 m°?

a. 2x 10 b. 2 x 108

C. 2 x 108 d. 2x 1o

64. Find the area of the circle whose circumference is x.

xb. 4n

C. 4-4se beoadsyed.TX“

65. One side of the rectangle is x inches. If the perimeter is p

inches, what is the length on the other side?

a. p-X b. p-2x

C. P-2 d. p-2x

2

66. The length of rectangle is increased by 50%. By what

percent would the width have to be deereased to maintain

the same area?

a. 33 % b. 50%

C. 66% d. 150%

67. Ifr=7 cm, h=21 cm then what is the volumeof thecone?

a. 1070cm b. 1080 cm

C. 1078 cm d. 544cm

54

68. If the area of the four wall of the room is 420 cm and the
height of the room is 10 cm. Find the perimeter of the room.

a. 42 m b. 40 m

C. 45 m d. 105 m

69. If the area of the circular track is "s" m² then what is its
diameter?

Vna. b.

4S d. YSmIm oos

C Tm

70. What is the volume of the pyramid of which the base is a
square of the side 9 cm and the height of 12 cm?

a. 423 cm b. 432 cm

C. 433 cm d. 324 cm

71. The perimeter of the rectangular hall is 112 m. If its length
and breadth are in the ratio 5:3. Find its length.

a. 35 cm b. 25 cm

C. 30 cm d. 24 cm

72. A square garden has area 6400 m“. If two path 2m wide are
running midway and intersecting each other inside the
garden, find the area of path

a. 24 m² b. 28 m

C. 32 m d. 316m

73. Find the volume of the given prism:

a. 596 cm Sem 3Cly14 cm
C. 496 cm
4cm

b. 588 cm

d. 456 cn

55

74. A path of width 2m rt2ou5tnaspl eacr orsosqutunafdorer otmuhteestirdece.onasstrquuuccattiioroennfieoeflldthoef
length 8m. find the

path at the rate of Rs

a. Rs 1575 b. Rs 1660

C. Rs 1975 d. Rs 2000

75. A man runs at the rate of 30 m per min and goes round the
he
circular field in 7 min. Find its radius.

a. 66.81 m b. 33.40 m

C. 72.43 m d. 74.98m

76. The area of square is 900 cm“. The length of it's diagonalis

a. 2042 cm b. 2043 cm

c. 3042 cm d. 3043 cm

77. Roshan can see upto 14km far. There are of land the hecan
see around is

a. 612 sq.km b. 614 sq.km

C. 651sq.km d. 61s6q.km

78. A metatic cylinder has an inside radius r and outsideradius

R and the length l. Find the volume of the metal.

a. (R°- b. rIR?-)

C. rİR'- o d. RRI

79. A cube has an edge of four inches long, It the edge is

increased by 25% then how much the volume will be

increased approximately.

a 25% b. 48%

C. 73% d. 95%

80. A metallic sphere is melted into a solid right ular

cylinder whose height is twice the radius of itsbase. I
radius of the sphere and cylinder are 'r' andd 'R'
respectively, then Ris

a. b.

C. d.

56

81. Two right circulat cones X and Y are made X having three
times the radius of Y and Y having half the volume of X.
Then the ratio of height of X and Y will be

a. 9:1 b. 1:9

C. 2:9 d. 9:2

82. A car travelling at 60 km/hr takes 30 minutes for a journey.

How long will the car takes travel the same distance if it is

at 25 km/hrs.

a. 72 minutes b. 90 minutes

C. 30 minutes d. 30 minutes

83. If the curved surface area of a cylinder is numerically equal
31S to the volume of the cylinder then the area of the base for

cylinder is

a. 4 b.
c.) )a g(atb+e)
C. 27 d. 3T

84. IfV and S are volume and total surface area of cuboid a x b

xe then is

85. The radius of the wheel of vehicle is 70 cm. The wheel

makes 10 revolutions in 5 seconds, then the speed of the

wheel is

a. 32.72 km/hr b. 36.25 km/hr

C. 31.68 km/hr d. 29.46 km/hr

86. The circumference of the second circle is 2 feet longer than
the circumference of the first circle. How many feet longer
is the radius of the second than the of the first?

a 1 b.

2T

C. 2T: d.

57

87. The curved surface area of a right circular cone is 2310cm
and the radius of the base is 21 cm, then the slant height of
the cone is

a. 53 cm b. 35 cm

C. 42 cm d. 24 cm

88. A square of 4 inches on a side is cut up into smaller squares
of 1 inch on a side what is the maximum number of such
squares that can be formed.

a. 4 b. 8

C. 16 d. 64

89. What is the area of a rectangle whose length is twice its
width and whose perimeter is equal to that of a square
whose area is 1?

a. 1 b. 6

c. d.

Answers: a 3. C4. b7. a 8.

L. a2. c 13. d 14. b c 6. c 17. a 18. c 19. ac1200..b

11. a12. c 15. a 16.
21. c 22.c 23. d 24. b 25. b 26. b 27. b28. c29. d 30
31. d 32. a |33. bb4355.. bd 4366.. ab 4377. .ba4388.. d 39. c 40. b
b55c5634.
41. a 42. b 43. 49. a 50. a
S1. a 52 c44.
61 c62 53 c I54 a d 58 a 59 d 60 d

71 a 72 b 63 b 64 b 65 d 66 a 67 c68 a 69 b 70 d
b 74 d 75 b 76 c77 d 78 b
d 73 b 79 d 80
81 c 82 a 83 a 84 b85 c 86 b 87 b 88c 89 d

58

Chapter-8

ALGEBRA

1. Some Useful formula

a-b'= (a+b)(a - b)

a+ b= (a+b) - 2ab= (a-b' +2ab

(a+ b)' =a +2ab +b?

n 2 (a+ b) =a+3ab +3ab?+ bn solimesiz
- a+b +3ab(a+ b)
(a-by' = a-3a'b+3ab-b

=a-b-3ab(a-b) oat ss2tdV?

a'+b= (a+b)(a' -ab +b)

a-b =(a -b) (a' +ab+ b)

(a+b+c)'= a' +b +d+ 2ab +2bc+ 2ca

(a-b-c=a+b +c-2ab +2be–2ca

(a+b+c'-a'+b'+ c+ 3(b + c) (c + a)(a + b)

2. L.C.M. and H.C.E.

L.C.M. = Common factor

H.C.F. = Common factor x Remaining Factor

Note: L.C.M. x H.C.F. = Product of two numbers.

(-0
Objective questions

1, Thesquareroot of 1 - 8a+ 16a*is

a. 11+4a b. 1-4a

c. 1+8a? d. 1-8a( a

1

2, Ifr+*=3 then thevalueof x' + is

a. 7 b. 9
d. 12
C. 3

59

a-b3. The simplified form of the expressionB2(2+ b) S

a+b b. 2(aa+-bb))

a. 2(a -b) a+b

a-b C a-b

C. a+b

4. = 1 then the value ofx:

a. 1 b. 2 d s)=+

C. -1 d. -2

5. What should be added to (m - n)m to get the square of (m-

n)?

a. m-mn b. m'n-n

C. n-mn d. n'm- m

6. Whatisthefactorizedfornof x- 4y+ 2x-2x'y'?

a, (x'+2y) (x+2) b. (x'+2y)«-2)

c. (*'- 2y)(+2) d. (²- 2y'( +2)

7. Ifx+-2 thenthevalueofy +is

a. tV2 b. 12

c. 2

8. The produt of H.C.F.andL.C.Mof r - 1and 3x+4x +1

is

a. (x- 1)(x + 1)(3x + 1) b. (x1)(x+ 1)

c. (3x' +4x + 1)(* - 1) d. (*+1) (x+ 1)
9. TheL.C.Mof a' + b' and a+ab +b° is

a. (a+ bX(a- b)(a+ ab +b)

b. (a+ b)Xa- ab + b’)\a +ab + b’)

. (a- b) (a' - ab +b (a' + ab + b')

(- ab+ b(+ab + b

10. H.C.F. of 30(a + 4ab + 3b) and 48(a + 3b) is

a. 12(a+ b) b. 9(a+3b)

C. 6(a+ 3b) d. 6(a + b)

60

11. Ifa+bte=7 andab + be+ ca =12,find thevalueof a" +

b' +e.

a 25 b. 16

C. 19 d 36

.(e))12. Theexpression 1 sequal to
a
b

G)
13. Thesimplestform of 4x+4y

a. b.

C. d. 2xy

14. The product of twoexpressionsis (a + 1' (a-1) and LC.M.
is (a + 1) (a -1). Then their H.C.F. is

a. a+1 b. a-1

a+l dd at1

C. a-1

15. The HCF and LCM of two expressions are x(x + a) and

12x(3+ a) (-a) respectively.If firstexpressionis 4x(x +

a', findsecondexpression.

a. 3x(** + a) b. 3(:* + a)

c. 3x(* - a) d. 3x(a'- x')

16. Simplifiefdornof GHG)

a. 1 b. -1

c. d. :2

17. Ir?-7a- a=0 thenthevalueof a

a. 4 b. 5
d. 7
C. 6

61

18. From whatmust x +-x+1 be subtracted to get 2x 42
b. x'-*+x -4
a. x+x+x+4
c. x'-x-x+4 d. x'-x-X - 4

19. If2a = 3b=4c= 24thenthevalue ofa - ab- is

a. 9 b. 10

C. 11 ap d. 12 ERw

20. Ifa = 2b= 3ethenthevalue of, is

a. b.

C. 2 d. 5

2

21. Ifx+y=3 thenthevalueof x +y' +9xy is

a. 27 b. 9

C. 18 d. 24

22. Ifa+ b=5, b+e= 7 and c + a =6 then the value of a is

a. 2 b. 3

C. 4 d6

23. If ab =6, be =6 and ca = 4 then the value of a is

a. 2 b. 4

C. 6 isC. 8
b 4ab
24. Simplified form of:
a-b'
ab

nz ziM a-b

4ab ab

C. 5-a d. B-a

25. IIP= nd Q=+then find P+ Q-PQ is

a. x+3 b.x-*+12

c. d x-1

Answers:

1. b2., 3. 14. 5. c 166.. bd177.. cd 188. . ac91.9. bd1200..

1. 112, b 13.b14.
21. 22. 123. 124. b25. 1

62

Chapter – 9

SURD AND INDICES

1. Basic facts
a. a" x a"= amt(a" x a' x aP= amtnt).

b. (a"= amn

am

C.

d.

e. a= 100°= 1

m

f. a" =Vam

m

h.

i. Ifa=b' thena =b*

j. If a" =a thenm =n 2tniSR isetls sviti2o4

k. Vaxya=a b sabureow lsT.

2. Surds: t0c 3 l t02 t adt

IfVa or n" root of a is irational numbers and a positive rational

number, then Va is called a surd. Thus a root of a positive rational
number, which is an irrational number is called a surd, for eg:

V2, v3,2, 6 aresurds(.wheren isdegreeofsurd).
a. Law of radicals

If n, m and p are positive integers and a, b, c are positive

()-rational numbers, then

63

b. Pure and mixed surd

A surd having a rational factor except one is called a
mix-surd and having no rational factor except 1 is called
a pure surd.

E. g. 4/5, 32 are mixedsurds.
V5.\3 arepuresurds.

C. Comparison of surds.

Surds of same degree can be compared.

Let a positive integer m and positive rational numbers

bea & bwherea > b,then Va> b.

Eg (a>F and 5<Ta)

For positive integer m and n, mn, then for any

positiverationanlumbera: Va<a.

Taketwosurds,Vaand Vb. If m=n and b#a,then

they can be compared by making their root sign of
equal degree of making the radical equal.

d. Fundamental operations of surds.

Surd having the same order and radicals are called

2 like surds. E.g 345, 245, 5\5 all like surds.

Similarly, xp yVp, zVp all also like surds.These

surds can be added subtracted, multiplied or divided.

e. Rationalization of single term surds.
If the product of two surds is a rational number then

these surds are called rational factors of each other. A
surd may have more than one rational factor.

f. Rationalization of a binomial surd

64

When a binomial surd is multiplied by its conjugate,
it becomes a rational number. Therefore a surd and its

conjugate are rational factors of each other.

Objectives Questions b. 3
d. 9
1. Thevalueo3f*42x-33-3a1t5i
Ye isq
a.2
b. 1
C.3
d. 1
2. Simplifiefdormof:

a. 0

C. 2

3. The expression + 1+ isequal toatz toauleto

a. 4 b. 2
C. 0
d. 1

4. Thevalueof is

a. 3b. 31

28 d

C. 31 2n

5. SSiimmpplliieeddffoorrmm oof fg243)x3gna+-31 is
b24
•a 6 b. 8
'C. a5
C. d. 27 20181 isaiaT siC
96. 9dT

The numerical value s10+nx 259n -is

of

a. b. 15
C. 25
d. 35

7. Simplifed form of

a. b b.
d.
a

65

8. Simpliedformof 16 yis

X ue X l 6alb

a.

y

y d. Xy

9. Simplify:V405 - 3125 +\45

a. b. 2\5 ot
C. d. -45

10. Simplifiedform of 3\32 x 48 is

a. b. ss

c.

11. Thevalueofx in: y -7=x -1is

a. 16 b. 15
C. 12
d. 9 b

12. The value of- is

a. 4 b3

C. 2 d. 1

13. Simpliedform ofVr&Ns

a. x' b. v

C. d.

14. The simple rational factor of 16 is

a. b.

c. d. &

15. Theconjugte of3-2 is

a. V3+2 b. 3-2

2C. d. none

66

16. What is the value of x when 22* = 2*? bue S$

a. b. 2 SE
C. 3 d. 4
0
17. The value of

a. b.

c.

18. If a=b thenthevalueof(

-a. b. go

C. d. b-1ss.s
19. If ab + be +ca = 0then xax gb xyc is
1a 0
b.

C. 2 d.

20. The value of x in 2*

a. b. ±2

C. 0 d. ±1

21. Ifx=(ab- (ab)then x +3xİs so90s+ z

a. a ab ab ab

C. ab 1d. ab-ab
2

22. If 1176 =2P.39.7"then value of p + q + ris

a. 7 b. 6
c. 5 d. 3

23. A surd is

a. a rational number b. an integer

C. an irrational number d. a whole number

67

24. Ifr-27 anda =, finda

a. 32 b. 16

C.8 d. 64

25. The value of x in 3*+3 4 = 162 is o

a. 0 b. 2
d. 3
1C.

26. Ifr=y=t andx, y, z areinG.P.thenvalueof +
is equal to

a. 2b b. b

c. 2

d.

27. aIfa=0 bf, b=, c=a, thenthevalueof abe=

a , b. 1

c. d. -1

28. Ifa'=b=e,andb=act,hen1+ Z

a. 2 by

y

cy d. 2y

29. Ifx= 2 +2°thenthevalueof -6x=t
(a-ds1a.10
c. 6 b. 8

d. 4

30: Simplifiedform of

a b.

c. a. (pa

68

31. Ifx=2+ 2 thenthevalueof 2r- 6xis

a.3 b. 4

C. 5.. d. 6

32. Theresultof yx'v is

b. a

C. d. 4yx

33. The value of x in 3.81" = g*+4 is

7 b. 7
a.

C. d.

34. Ifa = 10;, b= 1°' and a'b = 100, then xy =

a. 2 b. 0

C. 2 d. 1

1

35. Ifr"=y=z andxyz=1thenthevalueofm+ ntp=

a. 0 b. 1

C. -1 d. 2

36. Ifa =b anda = 2bthenthevalueofa +b is

a. a'-a b. ata

C. b-b d. b' +b

37. Ir2+3 =7and2 -3=1 thenthevalueofx + y is

a. 2 b. 3
C. 4
d. 5

38. Ifa=6-35 thenthevalue ofa +

a. 8 b. 10
C. 12 d. 14

39, The rational factor of 3/5 is

a. b. s

c. d.

69

40. Which pair has like surds?

a b. 2,5

c. V7/28 d.

41. Which is the correct statement?

C. 6. 5<5

42. Thevalueof 2+5 a i8<i44

is

a. 4

C. d.

Answers: d 4. d 5. d 6. c7. c8. d 9. d 10. c

1. 2. b 3. c 14. c 15. b16. b17. d 18. c19. b20. d
c 24. d |25. c 26. c27.b 28. a 29. c 30. d
|11. a 12. d 13.
d34. d 35. a 36. b 37. b 38. c39. c H0. c
|21. d 22. b23.

4311.. cc4322.. bb 33.

70

Chapter - 10

QUADRATIC EQUATION

The equation of the form ax + bx +c= 0(a 0) is called
quadratic equation

If a# 0, b0 thenax +bx +c=0 isaffectedquadratic

equation.

If a # 0, b=0 thenax + c=0is purequadraticequation.

Remarks:

1. Roots of quadratic equationareb+yb2'a-4ac

2. b-4ac isdiscriminantof the quadraticequation.
3. Ifb- 4ac >0, then rootsarereal andunequal.

4. If bʻ - 4ac > 0 and perfect square then roots are rational

and unequal provided that a, b, c are rational.

5. Ifb-4ac = 0 then roots are real and equal.

6. Ifb - 4ac < 0, then the roots are imaginary and distinct.

Relation between roots and coefficient.

Let a and B be the roots of quadratic equation ax' + bx +

C=0 then

a+p-&a.p=

Formation of quadraticequation

x- (a +B)x+ a.ß =0whererootsa, Baregiven.

Special case
Ifa and B be the roots of equation ax + bx+c= 0.
i. Roots are equal in magnitude but opposite in sign.

a +B=0

b= 0

71





ii. Reciprocal roots

aß=1

a

C=a

i. One root zero

c=0

iv. Both roots zero

b=c=0.

Note:

i. In a quadratic equation with rational coefficients
irrational roots always occur in conjugate pair.

ii. In a quadratic equation with real coefficienty

imaginary (complex) roots always occur inconjugat:
pair.

ii. Rational numbers either terminating or recurring.

iv. Irrational numbers neither terminating nor recurring.

Objective questions:

1. Which of the following equation is quadratic?

a. y3-x=2x? b. Ix+y+z=0

C. 3x +8x +7=0 d. d3-x =4x

2 Which of the following equation with roots - 3 & 2?

a. x'-x+6=0 b. x'-x-6=0

c. x*+x- 6=0 d. x²+x+6= 0

3. If thequadraticequatlon + 2(k + 2)x + 9k =0 hasequal

roots then k=

a. 1,4 b 2,4

C. 3,4 d. 4, 4

4. The quantity b' - 4ac in a quadratic equation ar + bx + e=

0 Ls called

a. determinant b. conjugate
c. discriminant d. rational factor

72

5. Theminimumvalueof r' +is

a. 2 b. 0

C. 1 d. 3

6. The roots of equation 2x - 6x + 7=0 are

a. real and equal b. real & unequal

C. Rational C. Imaginary

7. For what value of p will theequation 5r- px + 45 =0 have

equal roots?

15 b. 10
d. 90
C. 120

8. If oneroot ofquadraticequationis 2 1 then other roots

is

a. b +1

C. d 2-1

V2+1

9. The quadratic equationwhoseone root is y5 -2 is

a. x*+4x +4 =0 b. x 4x- 1=0

C. x- 4x + 4 = 0 d. x-4x+ |=0

10. A two digit number is four times the sum and three times

the product of its digit. Then number is

a. 24 b. 25

C. 26 d. 28

11, The number of two digits is equal to 4 times the sum of the
digits. If the sum of digits is 9, find the number.

a. 36 b 46

C. S6 d. 66

12. Thvealueofxinyx +t13 T9i1is

a 33 b. 35

C. 34 d. 36

13. ThevalueofxsatistyingtherelationVVE+25 +-5is

a. 144 b. 169
C. 225
d. 625

73

14. If one root of the quadratic equation is then other root

will be 0

a 25

C. 3-4i al d. 3+4icei

25 oups adilir e sisv 32hW,

15. Thequadraticequationwhoseonerootis y +yis

2x - 6x + 1 =0 b. 2x'+ 6x- 1=0

C. 4x +6x +9 =0 d. 2x +6x +l =0

16. If oneroot ofequation x+ 2kx+3=0 is 2thenvalueof k is

a. i b.

c. d.

17. Ifa, ß are the roots of 3x + 4x +7= 0, then the value of

1

a. 7

7 d. 4

c.

18. Ifx + 3x + 4 is divided by x - 2 then remainder is

a. 0 b. 20z ant2!

C. 14 d. -14

19. If1 +\5 is a root of theequation x+ px + q=0, wherepP,q
arerealthenp tq=

a. 3

c. 6 d. 11

20. What valueof k for3x +7x+6-k=0 hasone rootequalto

zero.

a. 3 b. 5
C. 6 d. 8

74

A

i-



21. If roots of equation 4x - 17x + k= 0 has the reciprocal roots
then k is

a. 0 .2

c. 4 d. 6

22. The roots of equation 2x + (4- k)x - 17 = 0 are equal in
magnitude but opposite in sign then k =

a. 0 a e b. 2

.4 d. 5

23. If one root of equation be 3 + 2i then its equation becomes.

a. x- 6x + 13 =0 b. x+x+ 13 =0

C. x- 6x- 13 =0 d. x +6x + 13 =0

24. The degree of bi-quadratic equation is

a. 5 b. 4

C. 3 d. 2yin sit

25. The value of m in 3x- 6mx+ 8m =0 has one root = 4

a. -1 b. 1

c. 3 d. 4

26. The product of roots of the quadratic equation x* + y3x -6
=0 is

a. -16 b. 6

C. -6 d. 16

27. Which of the following is a quadratic equation with roots of

nd?

a. 8x' + 2x - 3 = 0 b. 8x* +5x=3

C. 8x+5x +3 =0 d. 8x-2x -3 =0

28. What should be the value of k so that 9x + kx + is a

perfect square? b. V3

a. 6

C. 3 d.

AnswerS:

1, C 2. C3. a 4.c5.a6. c7.a 8.c9. b10. a
11.a 12. d 13. a14: c 15. a 16. b17, b |18. b|19. c 20. c
21. c 22. c 23. a 24. b |2s. c 26.c 27. a 28.e

75

Chapter-11

RATIO AND PROPORTION

Ratio:

Let a and b are two quantities of the same kind. Then the

fraction is called the ratio of a to b. Similarly the fraction- is

called the ratio b to a. The ratio is also denoted by a:b.

Note: Duplicate, triplicate, sub-duplicate and sub-triplicate
ratios:

If; is the given ratiothen

Duplicateratio=

Triplicartaetio --)otle

Sub-duplicateratio= =

Sub-triplicraatteio=e -8

Proportion

If two ratios are equal then four quantities composing them are

said tobeproportional.If -or a:b:c:d, a,b, canddare

called first, second, third and fourth proportion respectively

where a and d are called extremes while b and c are called

means.i. e.;

ad = bc

Note: product of means = product of extremes

Continued proportion

76

then a, b, c are said to be in continued proportion.

Similarly

Ir then a, b, c, d ande are said to be in

continued proportion,

Mean proportion

If a and b are two quantities then their mean proportion is given

byyab.

Note:

2 isalsodenotedbya: b:c=x:y :z

Propertiesof proportion

1I.nvertendoI:f- henba C

2A. lternend:Iro - then-

3 Componend:1ofğ- hen

4 Dividend: o If-then-

5. Componnendaonddividendo: If=i theana+-bb cc-+dd

6. Addendo: then abt+cd++ff

Note: -+ở+f

Objective questions:

1. 7 then what value of 6a - 4b?

a. 0 b. 8
C. 10 d. 6

0:

77

2. Irthenthvealueoyfi-xs

a. 3 b. 8

C. 10 d. 6

3. If 4a =6b then triplicate ration of a : b is

a. 125 b.

c. 16 d.

4. r then

a. b.

c. d. To

5. Find the mean proportional of and5

4 1

a. b. 100

C. d. 10

6. ,find thevalueofa++b+'+d

a. b.

c. d.

7. If we divide 247 in three parts such that they bear the ratio

.:2. Whatisthevalueosf econdpart?

a. 180 b. 114

C. 76 d. 120

8. If 125, a and 5 are in continued proportion then value of a is

a. 20 b. 25

C. 30 d. 35

78

9. Arrangteheratios609 7 11
and in the ascending order

of magnitude

a b.

6 7 9 11 1 116,9.

10. If 3a = 2b, 7b =6c and8c =7d, find a :d

a b.

c. d,

11. If x:y = 2:3 and y:z = 4:7 then x:y:z will be

a. 1:5:7 b. 8:12:21

c. 3:5:1 d. 8:12:28

12. If (3x+ 1): (5x +3) isthetriplicateratioof 3:4 thenvalue

of x will be

a. 17 b 17

57

C. 57

13. Two quantities are in the ratio 7 :4. If the greater quantity

is 24.5, then the smaller quantity will be

a. 14 b. 15

C. 10 d. 18

14. In a mixture of 35 liters, the ratio of milk to water is 4: 1.

r Another 7 liters of water is added to the mixture. Then the
ratio of milk to water in the resulting mixțure will be:

a. 2:1 b. 3:5

C. 10:13 d. 3:4

15. The sum of two numbers is 40 and difference is 10. Find the

ratio of two numbers.

a. 5:4 b. 5:3

C. 3:2 d. 2:5

79

16. The ratio of side of square to it's diagonal is

a, 1:2 b. I:2t ohbimese

C. I:3 d. V2:3

17. If a and b arepositivenumberwith a =3 and a= 12b

What is the ratio of a to b?

a. 2:3 b. 3:2

C. 2:1 d. 1:2

18. Four persons are engaged in the business in the ratio 18 : 10

:8:4. What percentage does the person occupies who have

the least share of the whole business.

a. 10% b. 4%

C. 20% d. 30%

19. What number should be subtracted from each of the

numbers : 54, 71, 75 and 99 so that the remainders may be

proportional?

a. 3 b. 7

C. 2 d. 5

20. In a fourth proportion, the product of extremes is equal to

a. Product of all the four

b. Product of the means

C. Product of the first and the fourth

d. Product of the first and the third.

21. If 2a =y2 =V2, in which of the following a, b and e

arranged in descending order of value?

a. a, b, c b. c, b, a

C. b, c, a d. b, a, c

22. In a kilometer race A beats B by 40 meters and B beats C
by 50 meters. How much does A beats C?

a. 81 m b 88 m

C. 90 m d. 92 m

80

23. Ifx:y:z=3:4:5, thenthevalue o2fx;-y+ 4zis

a. b. -1

C. -2 d.

24. The ratios of salt and water in the salt water solution taken

in 3 beakers are 1:2, 3 :5 and 4 : 9. If all three solutions

are poured in to a single vessel, then the ratio of salt and
water in the vessel is

a. 3:5 D19y b. 1:2
d. 3:5
C. 4:9

Answers:

. 2. d3. d4. a5. d6. a7. c 3. b 9 10. b

11. b 12. b 13. a 14. a 15. b| 16. b17. c 18. a 19. a 20. b

21. a 22. b 23. a 24. b

32111

81

Chapter- 12

SEQUENCE AND SERIES

Sequence

A list of numbers arranged in a definite order determined bya

rule is called a sequence. Each member of sequence is called

term ofsequence.

E.g. 1, 3, 5, 7 and 9 are 1, 2nd,3rd, 4h and 5 term of sequence.

Series

A series is the term of sequence in which each term is
connected by plus or minus.

E.g. 1 +1+5+7+9is a series.

Types ofseries:
a) Arithmetic Series (Progression)

A series is said to be arithmetic series (Progression) if every

term is greater or less than the preceding term by a fixed
number. The fixed number is called common difference It is
denoted by "d". Generally term of A.P. are in the form : a, a +

d, a + 2d, .. a + (n- 1)d.
Important results of A.P.

The n" term(general)or lastterm (t) =a+ (n– 1)d,
Wherea = First term, d= Common difference.

If a, b, c, are in A.P., then b is arithmetic mean between a & c

given by : A.M. (b) =Te

SumofntermsinA.P:S, -H2a + (n-)4]-(a+)

Where, a = First term, d = Common difference, n = Number of

terms, = Lastterm

The common difference (d) = i wherefirstterm(a),Last

term () and n is number of means between them are given.
b. Geometric Series (Progression):

A series is said to be geometric if the ratio of any term to

preceding term is constant. The constant number is called

common ratio. It is denoted by'r. Generally term of G.P are in

the form; a, ar, ar.... ar

82

Important result of G.P.

The n" (last) term ofgeneral term () = ar

Where, a = First term

r=Common ratio
n=Numbers ofterm.

Geometric mean between two positive numbers a and c is given

byG.M.(b)=Vac

Sum of n-term of G.P:

S,= ar--1) iTfirfr##|l

S,= na ifr= 1

S.= (if I<1for infiniteseriesisverylarge)

Commonratio of G.P. (r) if firstterm.

(a),lastterm (b) & number ofmeans (n) are given.
Relation between A.M. and G.M.

i) A.M. 2 G.M.

2ii) for any two positive real number 2 b Vab ifa#band b

=Vabif a=b>0

Note

Sum of the first n natural numbers:

S,=1+2+.. +nterms

-}(n+1)

Sum of the first n even numbers:

S,=2+4+6+...+n terms

=n(n+1)

Sum of the first n odd numbers :

S,=1+3+5 + ... +nterms =n

Sum of the square of first n natural numbers:

S,=1'+2'+....+ n

S nO t l)(2n+1)
6

83

Sum ofthecube of first n naturalnumbers:

S,= 1+2' +..tn'

number not equal

Note: If a, b, c, d, ... be in A.S. and k be a constant number not

1.

to zero, then

a. a+k, b +k, c +k, d + k, ..

a-k,b-k,c-k,d-k, e.eee.e

b.

ak,bk,ck,dk, ..

C.

d. a kkb'kc'kd' ......e*e*

are each on A.S. (arithmetic sequence) equal

If a, b, c, d, ... be in G.S. and k be a constant number noteon

2.

to zero, then

a ka,kb,kc,kd, ...*

b. a kbkck d

a, b, c d...are eachinG.S.

C

3. Selection ofterms in an A.P. Common

No.ofterms Terms difference

3 a -d, a, a+d 2d

4 a -3d, a-d,a + d, a+ 2d

5 a- 2d, a -d, a, a + d, a+ 2d

4 Selection of terms in G.P. Common difference

No. of terms Terms

3 r: a

4 Fa f;ar,ar'

5 a 4a, ara, r

84

ObjectiveQuestions

1. Ifa =7, d =-3 and n = 12,then Sı=?

a. -54 b. -114

C. 48 d. -s0

2. Find the value of (3m -1)
m-0

a. 25 b. 29

C. 39 d. 35

3. The series -1+2-3+4-5+6 can be expressed as :

6

b. Ž (-I2n

n=l

c. d. none

n=1

4. If the 2d term and the 5th term of a G.P. are 4 and 32

respectively. Find the 8" term.

a. 256 b. 250

C. 265 d. 260

5. Find the sum of first 30 natural numbers.

a. 465 b. 456 s
. c. 445
d. 545

6. If the third and thirteenth terms of A.S. are 40 and 0
respectively. Which term is 28?

a. 4th b. Sth

sd c. 6th d. 7th

7. Find the value of a and b if3, a , b, -9 are in A.P.

a. 1&5 b. -1 &5

35a c. 1&4n d. 1&-5

8. Thesumofseries:1+3+9+ ...+8(terms) is

a. 3280 b. 3240

C. 3020 d. 3380

9. What is the sum of series ; 1+3+5+ ...+ 19?

a. 105 b. 100
C. 95 d. 90

85

10. The sum of G.P. with first term 5 and the last term 160is

315. Find the common ratio.

a. 2 b. 3

C. 4 d. 5

11. What is the product of A.M. and G.M. between 4 and 16

a. 50 b. 20

C. 60 d. 80

12. The n" termofsequenceis3n +5 then ts ty=

a. 25 b. 26

C. 27 d. 30

13. The sum of even numbers between 1 to 30 is

a 240 b. 230

o C. 220 d. 210

14. What is the value of k, so that k + 2, 4k -6, and 3k -2 are

three consecutive terms of A.P.

a. -1 b. -2

C.0 d. 3 3

15. The arithmetic and geometric mean are 25 and 20

respectively. what are the two numbers.

a. 10 & 40 b. 20 & 20

C. 30 & 20 d. 50 & 8

16. How many terms of the sequence -12, -9, -6, -3, ... mustbe

taken to make the sum 54?

a. 5 b. 9

C. 12 d 16

17. In A.S. if common difference is 2 & 5th term is 17. Find the

first term ofseries.

a. 6 b. 7

C. 8 9d.

18. If5, p, q, 9 are in A. P., then value of p and qg are

b.

17 21 d 12 17

C. 3 3 34

86

3

What&che rutt afA.M. and GM, between d and l6? 20,
A 20
21.
d.
22.
TheaermosfoquenesJn+$thent -t=
23.
26
24.
d 30
25.
he samafevenaumdersbeween I to 30 is
b 230 26.

220 d 210 27.
a
-What thevalueofk,sothat k+2,4k-6, and 3k 2 a
C
hreecoasecutiveterns of A.P.
28.
2 a
C
d. 3

T arithmetic and geometric mean are 25 and 20

respective what are the two numbers.

a 10&40 b. 20 & 20

c 30& X0 d 50 & 8

Howmanytermsofthesequence-12, -9, -6, -3, . mustbe

takea to make the sumS4

b. 9
12

d. 16

fihrsAt tSermiicoofmsemrieosndifferenceis 2 & sh term is 17. Find the
6

b. 7

d9
IISA49are inA P

thenvalueof p and q are

b3

86

A. A.M. G.M b. A.M. <G.M
d. A.M. < G.M.
0. A.M. > G.M,

For what value of a vwill (a - 5), (a + 1), (a + 5) be in G.P.?
a.-13
b, -2

C, 12 d. 10

Findthesumof 1' +2+3+....+. 12

a. 1850 b. 1950
d, 650
C. 1900

TwomeansofG.S.betwecn1/4and16are.... 9

a. 4,2 b. 4,3

c. 4,-1 d. 1,4

What isthesum of 1+2+ 0eeeeeee

a. 684 b. 874
C. 784 d. 690

Find the sum of odd numbers between 100 and 200.

a S000 b. 7500

C 10,000 d. 12,500

Thevalueof 9gl g/27 0eeooeeeee 1S

a. 1 b. 3

C. 5 d. 9

The common ratio of the geometric progression: 1, 0.1, 0.01,
0.001 is

a. 10 b. 100

C. 10 d. 1 o nrst o

The sum of three numbers in A.P. is 21 and the sum of their

squares is 155 then the common difference of the A.P. is

2a. b. -2
C. +2 d. 3

The two A.M. between 160 and 172 as

a. 168, 164 b. 164, 168

C. 165, 170 d. 148, 184

87

19. Which of the following statement is true.

a. A.M. G.M b. A.M.< G.M

c. A.M.> G.M. d. A.M. < G.M.

20. For what value of a will (a –5), (a + 1), (a + 5) be in G.P.?

a. -13 2b.

C. 12 d. 10

21. Findthesumof 1' +2+3+... 12²

a. 1850 b. 1950
C. 1900 d. 650

22. Two means of G.S. between 1/4 and 16 are1a io9 bh

a. 4, 2 b. 4,3
d. 1,4
C. -4, -1

23. Whatisthesumof 1' +2'+.... +?

a 684 b. 874

C. 784 d. 690

24. Find the sum of odd numbers between 100 and 200.

a. S000 b. 7500

C. 10,000 d. 12,500

25. Thevalueof g"gg/27 ee. IS

1a. b. 3

C. 5 d. 9

26. The common ratio of the geometric progression: 1, 0.1, 0.01,

0.001 is

a. 10 b.

1 d. 1 osst31 dy3a!

C. 10

27. The sum of three numbers in A.P. is 21 and the sum of their

squares is 155 then the common difference of the A.P. is

a. 2 b. -2 et

C. +2 d. 3

28. The two A.M. between 160 and 172 as

a, 168, 164 b. 164, 168

C. 165, 170 d. 148, 184

87

, 20,

If thesequence5, x, y is proportional to thesequence

29. 32 which of the following is y?

b. 12

a. 16 d. 10

C. 14 termoftwoAP's-19 - 12 -5+2. and1+664+111

30. Thaerne equal. The value of nbis.

eer 10 't e

a. 9 d. 12

C. 11

The first term of GP is 16 and sum up to intinity is 32,then

31.

the common ratio is

a. 2 b.

d. 1
C.

32. Ifa, b, c are in AP as well as GP, then

b. b'<ac
a. b>ac
d. b + ac =0
C. b=ac

Find the three numbers in AP where sum is 21 and product

33.

is 315. b. 9,7, 6
a. 9, 7,5 d. 11,9, 2

C. 8,7, 6

34. Thesumoftheseries4 +4+4+.. is

1a. b.

c. d. 4

35. The seventh term of the following sequence 6, 12, 24,48,

is b. 96

aC1..8068i t 1 d. 384 neds

36. Determine the common ratio of the geometric serieswhose
first term and fifth term are 2 and 512 respectively.

a b. 3

C. d. 5

88

29. If thesequence5, x, y is proportional to the seqquueenncoe x,20,
32 which of the following is y?

a 16 b. 12

C. 14 d. 10

30. The nh term of two AP's -19- 12-5+2.,.. and 1 +6+11

are equal. The value of n is

a. 9 b. 10

c. 11 d. 12

31. The first term of GP is 16 and sum up to infinity is 32,then
the common ratio is

a2 b.

C. d. 1

32. Ifa, b, c are in AP as well as GP, then

a. b>ac b. b'<ac

c. b= ac d. b + ac = 0

33. Find the three numbers in AP where sumn is 21 and product
is 315.

a. 9,7, 5 b. 9, 7, 6
C. 8,7, 6
d. 11,9, 2

34. Thesumoftheseries 4+4+4+.... s

1a.

2 d. 4

C.

35. The seventh term of the following sequence 6, 12, 24,48, n

is

a. 86t b. 96

C. 108 D nd. 384 9

36. Determine the common ratio of the geometric seriesWhose
first term and fifth term are 2 and 512 respectively.

a.2 b. 3

C. 4 d. 5

88

37. Ritu borrows Rs 4368 which she promises to pay in 6
annual installments, each installment being treble of the
preceding one. Then the first installment is

a. Rs 12 b. Rs 16

C. Rs 24 d. Rs 28

Answers:

. b2. c3. 24. a 5. a 6. c 7. b 8. a 9. b |10. a

1. d 12. c 13. d14. d1S. a 16. c17. d 18. b 19. a20. a

21. d 22. d 23.c 24. b 25. b 26. c 27. c 28. b 29. a 30.

31. b 32. c 33. a 34. b 35. d 36. c 37. a

20

Chapter -13

TRIGONOMETRY

Sexagesimal system Centesimal system

60" = 1 100=1'
60 = 1°
100 = 18
90° = 1 right angle
1008 = 1 right angle

P-(0)- -G 5057.29585=°7"1745°

and 1°=(00 3.14159= 63.6620"= 636620"

I- ,D-D)

0-(4 where,

= angle subtended by arcs at the centre in radian

l=Length ofarc
r=radius ofcircle

Trigonometric Ratios

sin =P cos = tan-

cOsec = h sec@- cot
p
tan
1 cos 1Sec 1= cot 0
cot
sin = = tan 0
=1
cOsec

tan9 Sin 0 cot0 CosiSn

cos 0 cosec9 -coto

sin 0 + cos' 0 =1

sec'e -tane =1

90

Chapter- 13

TRIGONOMETRY

Sexagesimal system Centesimnal system
60" = 1' 100"=1'
60 = 1° 100' = 18
90° = 1 right angle 100= 1 right angle

r-0)-( = 57.2958°= 571745°

(200 3.14159= 63.6620" = 63#6620"

and1° =

- where,

e= anglesubtended by arcs at the centre in radian
l=Length ofarc
r=radius ofcircle
Trigonometric Ratios

sin8- - cos tan9 -

cosec 0 sec 9 = cot 0

sin 1cos = tan9 ot01
cosec 9 sec 0

tan9 Sin 9 cot 0 cose cot 0 tan
sin0
cos 0
cosec'0-cote =1
sin 0 +cos =1
sec' -tan'e =1

90

Note:

1. -1ssin s1

2. -1s cos s 1

3. -0< tan < o

Table ofvalues of ratios

Angles 0° 30° 45° 60° 90°120° 135° 150° 180°
0
Ratios 0 2 1
sin 1 2
cos 1 1:
0 1 -1
tan 2
1 V3 -V3 0
sec 2
V3 2 2-2 -1

cosec 12

0 1 -)

cot

Compound angles
sin (A + B) = sin A cos B + Cos A Sin B
sin (A - B) = sin A.cos B- cos A.sin B
cos (A + B) = cos A. cos B- sin A.sin B
cos (A- B) = cos A cos B + sin A sin B

tan A +tanB
tan (A + B) =1-tan A. tan B

tan A - tan B
tan (A - B) = 1+ tan A tanB

cotA.cotB- 1

cot (A + B) = cot B + cot A

cotA ,cotB+1
cot(A - B) = cot B- cotA

sin (A+B). sin (A - B) = sin A - sin' B = cos' B- cos?A
cos (A+B), cos (A - B) = cos A – sin' B cos B- sin' A

Transformation of products into sums or difference
2sin A .sin B = sin (A + B) + sin (A- B)

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