PRACTICE MATH AFER SEE

2cos A. sin B = sin (A + B)- sin (A +B)
2cosA.cos B=cos (A +B) + cos (A-B)
2sin A .sin B = cos (A -B)– cos (A + B)
Transformation ofsums or difference into products

C+D C-D

sin C + sin D= 2sin 2 cos 2

sin C - sin D= 2cos C+DsinC-2D

cos2C+D C-D

cos C + cos D = 2cos

cos C- cos D=2 sin D:C
2

Multiple Angles cos2A =cosA- sinA
sin 2A =2sinA. COSA =1-2sin'A
=2cos'A -1
2 tan A
cot 2A =co2tcAot-A1
tan 2A =1- tan A
sin2A =12+tatannA'A cos2A =1+tatan'nAA

sin 3A =3sinA - 4sin'A cos 3A = 4cos'A - 3cosA

tan 3A =3t1a-n3At-atann'AA cosA = cos-sin A

Sub multiple angles

sin A = 2sin cos

A =1-2sin
2 tan
=2cos-1
tanA =
1- tan' sin A =3sin -4 sin

92

A

3 tan -tan

COSA=4 cos-3 cos , tanA=
1-3 tan

IfA+B+C=1

then,sin (A + B) = sin C

cos (A + B) =- cos C

tan (A + B) =- tan C

Areaof the triangle ABC =;be sin A A

1 b

ca sinA

1 a

-ab sinC

Note: 6-\42 =cos 75°

1. sin15°=V2/32

2. cos159 N3+1_N6+2 = sin 75°

242 4

3. tan15°= =2-\3 = cot 75°

44. sin18° = cos72° =

5. cos18° Vi0+42V3= sin72°

5-16. tan18° = = cot 72°
V10 + 2y5

Note: sin(90° + 0) = cos 0
sin(90° - 0) = cos
cos(90° -0) = sin 0 cos(90+ 8) =- sin 0
tan(90°- 0) = cot tan(90°+ 0) = - cot

93

sin(180°- 0) = sin® sin(180°+0) = -sine
cos(180° - 8) = - cose cos(180°+) = -cos0
tan(180°- 8) = -tan tan(180° +0) = - tan

sin(270°- 6) = -cos® sin(270°+ 0) =-cose
cos(270°-0) =- sing cos(270°+0) =sine
tan(270°+0) = - cot
tan(270°- 8) = cot®

sin(360° - 6) = - sin® sin(-0)= -sin
cos(360°- 6) = cos cos(-0) = cos
tan(360°- 0) =-tan tan(-0) = -tan

Objective questions

1. 1 is equal to b. (180
a.

c. d.

2. 1grade is equal to b. (180
a.

d. (200

3. D° is equal to

a.

c. 0 xD d. botha and c

4. Which one is correet? (Where 8) angle sub tended by arel

a. b. o-9

c. d.

94

5 The chord AB of a circle subtends an angle of 60° at the
center. What is the ratio of the chord AB to the arc AB?

a. 22:21 b 21.:22

C. 1:1 d. 1:2

6. Find the angle made by two hands of the clock at 2:30 o
clock in the circular measure.

a. b

C.

7. Which one is true? b. sec'45- tan'e = 1
a. sec'45 - tan'45 =1 d. botha and c

C. sec'e- tan'e = 1

8. Find the value of sin 30°. cos 60° - cos 30. sin 60°.

a1 b..0

c. 342 d.

9. Which one is correct? b. 0a=0

a d. both a and b

0

C

10. A tower is 50/3 m high. Find the angle of elevation of its

top from a point 50m away from its foot?

a. 45° b. 30°

C. 75° d. 60°

11. What is the value of b. 0 DC
d. none
a.
b. 2sin
C. 1

12. Value of cos C - cos D.

a. 2sin +D2

C. 2cos .sinC2-D d. 2cos P. sin D

95

Ifsin9 =,13. is an obtuse angle then tan 0 is equal to

a. 0 b. 1
C. -1
d. -y3

14. If sin'x - cos'x = 1 then the value of sinx - cosx =

a. 0 b. -1

1c. d. 2
15. The value of cos 15° + sin 15° is

a. b. 2
d. 2
3
C.

V2

16. The value of cos 15° - sin 15° is

a y2 b. 1 s

c. d. 0

17. V11+-cocso6ssiseeqquuaalttloo

a (cosec0 t cot 0) b cosec0 t cot 0

C. (cosec - cot 0) d. cosec - cot0

18. The value of cos 75° - cos 15° is

a. y2 b2

c. d.

sin .cos19. is equal to

a sin b. jsin'A

C. ¿sin'A 1b. ;sin'A

20. cos0 + sec' 0 is b. =1
d. >1
a. <l

c. 22

96

21. Thevalueof tan.sint sin + cos.sin will be

a. b. ît2
2

C. 2

22. sin 0 +cosec 0 is b. >1
d. 1
a. <1

C. 22

23. As observed from the top of the house, the angle of elevation
to the top of the tower is 30, find the height of tower, if the
height of the house is 10m and the distance between the

tower and house is 60m.

a. (20 +\2)m b. 10(243 + 1)m

c. (2/3+1)m d. 20/3 m

24. If sin a + cos a = 1, the value of sin“ a is

a. 0 b. 1/2

C. 1 d. none

25. Ifrsl-csosinfind thevalueof l/x.

l- sin0 b. 1-cos

a. cos 0 sin

1+ sin 0 d. 1- sin0
cos
C.

cos

26. The greatest value of sin x. cos x is

1a, b. 2

C. V2 d. 1/2

27. The lowest value of sin x. coS x is

a. v2 b. -2

C. 1/2 d.

97

28. There are two post of height 27m and 9m respectively. 34
the foot of thesecondthe angle of elevation of the top of the 35
3
first is found be 60 °. What is the angle of elevation of
b
second from the foot of the first? 3

a. 30° b. 45°
C. 60° d. 90°

2 tan30°
29. 1- tan' 30 İSequal to

a. 3 b.

sthic. 1:n got tc.

sin30° - cos30°
30. sin30° + cos30 İS equal to

b. -vNõ-1

a. 2\2 l a'aizosulry1+y3

d. NB+1
V3-1

31. Find the length of the shadow of a building of height 25 m
when theshadow of5 m tall man is 75 m. ti

a. 45m b. 625m

C 3N20 m d. 20/3 m

32. If cosec q= -2 and q is in the third quadrant,then thevalue
of tan q is

a. b. -3

c. 3 d.

33. Findthevalueoftan , where =

b. 5a. y3 1

c. -V3 d,

98

In an angle of elevation object is
4.

a. below the observer
b. above the observer
C. observer is above the object
d. both a and c

5. If sin (x- y) =cos (x+ y) =$, thevalueof x and y are

a. X= 135°, y = 45° b. x=45°, y = 135°

c. x= 165°, y = 15° d. x= 165°, y = 135°

36. If A, B, C and D be the angle of cyclic quadrilateral, find
the value of cos A + cos B+ cos C + cos D.

a. 0 b. 1

C. 2(cosA+ cos B) d. none

37. If cos A - secA =2 then cosA - sec'A isequalto

38. a. 8 b. 6
C. 2 d. 14
bu
3ri Find the valud of sin 18°

5-1 b V2-V2

a, 4 2

C. 10:42/5 d N2+2

2

39. If sin 30= cos0 then thevalueof0.

a. 45° b. 30°

C. 224 d. 15°

40. The area of the triangle ABC is
1a. ,asin B. sin C
b. jab. sinC

C. Ga sin B. cos C 1

d. ; a.be

41. Find the value of sin 70° + sin 40°

a. 2 sin 55°, cos 15o b. 2 cos 5S°. sin 1s°

C. 2 sin 55. sin 15° d. 2 cos 45°. cos 15°

99

42. Ife is an acute angle and sin =4/5. Find the vaalluuee ofSeOs
0 +3tan 0.

a. 5 b. 6
C. 7
d. 8vc

43. In a triangle ABC, right angled at C, if tan A 1A

tan B =y3,find the value of sin A.cos B t cos A, sin B
0a.
b. 1

C. d. 2

4. A man is 2m high and the length of his shadow in the
Sun is

2/3 m. what is the altitude of the sun?

a. 45° b. 60°

C. 90° d. 30°

45. When the height and shadow of objects are equal then the
altitude of sun is

a. 90° b. 45°

C 30° d. 43°

46. An aeroplane takes off at an angle of 45° with the ground
level. It covers 2000m at what height will it reach from the

ground?

a. 1400m b. 1415m

C. 2000m d. 1475m

47. Isf in(90-°0)=CBCA then sin 0 is equal to

VBC- CA b VCA-BC

CA CA

VBC+CA d CA
BC
C. CA

Answers:

2.1. a 3. d 4. a 5. b 6. b 7. d 8. dc 919.. d |0. d
d 12, b |13. d 14. c 15. a 16. c 20. c
c 17. b 18.
21. b 22. c 23. b 24. c 25. c |26. d 27. d 28. a 29. b 30. b
31. b 32. d 33. c 34. b 35. d 36. a 37. d 38. a 39. c 40. b

41. 2 42. c 43. b 44. d 45. b 46. c 47. b

100

Chapter- 14

SLOPE AND STRAIGHT LINES

1. Distance formula: If A(X1,yi) andB(*%,y2)be two points on the
plane then the distance between them is denoted by 'd' and is
given by

d=V(g- x) +(ya-y

2. Slope: The slope of a line is defined as the tangent of the
angle made by the line with positive x - axis. Thus if0 is the
angle made by the line with positive x-axis, then slope of line =
tan (slope of line is denoted by m)

Note:
i. Slope of line joining two points A(x1, y) & B(x, y) is

Xy - X1

ii. Slope of line equally inclined to co-ordinateaxes is t1.

ii. Slope of x-axis is zero.

iv. Slope of y-axis is not defined.

3. Section formula:

Internal division: If C(x, y) be a point dividing the line joining

the points A(X1, y) and B(x2, y2) internally in the ratio of

m:m2, then x = mx> + m2Xı and y = miy, + m;yı
mị + ma
mị + m

External division: If C(x, y) be a point dividing the line joining

the points A(X1, yi) and B(X2, y) externally in the ratio of

m:m2 them x = 1andvmỳ) -mYı
mị - m2 mị - m2

Note:

If C(x, y) is the middle point of the line AB, then x=*X +X,

2

andy= t2

4. Centroid formula: Let A(X1, y), B(x2, y) and C(X3, ys) be the
vertices of triangle. If G(x, y) the centroid of the triangle then

x=** andy= Ytyt

101

5. Three standard forms of equation of straight lines:

a. Slope intercept term:

y= mx + c (When m be the slope &c be y - intercentof
line)
Note: If linePassesthroughoriginthen y mx.
b. Double Intercept form:

+-1(x -intercepat &y -intercepbt )

C. Normal form or perpendicular form:

x cos a +y sin a = p (p be the length of perpendicular &
a be angle made by perpendicular at origin)

6. Equation oflines isspecial cases.nt
a. Point slope form:

y-y1=m(x - xi) (slope 'm' & point(%1, y) is given)

b. Two point form: y - yı (-x)

Note: More than two points are collinear if they lines on the
same line
1. More than two lines are concurrent if they intersect at

- 2. a same point.
Any line parallel to the lineax + by + c=0 is ax + by
3.
4. +k=0.

5. Any line perpendicular to the line ax + by tc=0 is

bx -ay + k=0.

Slope of line ax + by + c=0 is- a

6. Two lines are parallel if their slopes are equal.
otherwise they are intersect.

7. Two lines are perpendicular each other if product of
their slopes is -1.

Objective Questions:

1. Find the slope of line which made an angle 45 with negative
direction ofx- axis.

a. 1 b. -1

C. d. y3

102

2. What value of k, when lines 2x + 3y + 7 =0 and kx+ 2y = 4
are perpendicular each other?

a. 2 b.

C. -3 d. 3

3. The two straight lines ajx + biy + c =0 and ax + bxt ;=
0 will be perpendicular if

a. aj az + bjb, =0 b.

b C. ajā, - bib, =0

C. bị

4. Two lines ax + biy+ e =0 & apx+ by + q =0 will be
parallel if

a. ajb- azbj =0 b. aja, + bjb, =0

c. ajb, + ab, =0 d. ajby - azb, =0

+=1s5. e slope of line

a. b.

C ab C. ab

6. The x-intercept of line 2x +3y =6 is

a. 2 b. 3

C. 6 d. 4

7. Find the area of triangle formed by line x +y=4 and co-

ordinate axes.

a. 4 sq. unit b. 6 sq. unit

C. 8 sq. unit d. 16 sq. unit

8. Which point is equidistance from two points A(2, 4) and

B(0, -2)?

a. (2, 0) b. (3, 1)

C. (1, 1) d. (1, 2)

9. Theslope of line x+ my +3 =0 and 2x+3y = 4 is samethen
value of m is

a b.

c.

103

10. Whatisthevalueof k ifx-interceptof line 2x+ 3y 0is

6?

a. 6 b. 2

C. - 12 d. 18

11. Slope of line equally inclined to both axes is

a. 1 b. -1

1

C. £1

12. The centroidof a trianglewhosevertices are (1, 2), (4,3)

and (0, 1) is

s6 a. (2, 1) b. (2, 3)

C. (3,2) d. (-1, 2)

13. The distance between two points (2, 4) and (5, 8) is

a 4 unit b. 3 unit

C. 25 unit d. 5 unit

14. Which one is parallel to line x +y=4?

a. y=-x+3 b. x-y=7

C. 2x+ y-7=0 d. y-xX-7=0

15. Which line is perpendicular to line 2x + 3y = 4?

a. 2x-3y = 5 b. 2x-3y+ 7=0

C. 3x+2y =8 d. 3x-2y + 7=0

16. The equation of straight line passing through (2, 4) and
having slope -2 is

a. 2x +y=8 b. 2x-y=4

C. x+2y =8 d. x-2y=4

Answers: b 10.

I1. cb |22. dc 13.3. adL44. . aa15S. . bd166. . a

104

Chapter - 15

ANGLES AND TRIANGLES

1. Types of angle:

a. Acute angle: An angle less than 90° and greater than 0°.

b. Obtuse angle: An angle greater than 90° and less than
180°.

C. Right angle: An angle of 90°,

d. Straight angle: An angle which is 180° is called straight
angle. It is also known as two right angle.

e. Reflex angle: An angle greater than 180° and less than
360° is called reflex angle.

f. Adjacent angles: Two angles are said to be adjacent if they
have common vertex anda common arms.

bel'e Complementary angles: Two angles are said to be
complementary, if their sum is 90° (right angle)

h. Supplementary angles: Two angles are said to be

supplementary,if theirsumis 180º.

i. Vertically opposite angles: Two angles are said to be
vertically opposite angle if they have common vertex but
ita o no any common sides.

2. Parallel lines:

Two lines Lị and Lz are said to be parallel if they are not meet
with infinitely produce.

Note:

a Alternate angles: A pair of non-adjacent interior angles
Iying both sides of a transversal that is formed when it is
cuts two line segments is called alternate angle. If the lines
are parallel then the alternate angles are equal.

b. Corresponding angles: A pair of an exterior and another
pair of interior angles lying on the same side of a
transversal but not adjacent when the transversal cuts two
line segments are called corresponding angle. If the lines

lo are parallel then the corresponding angles are equal.

105

C. Co-interior angles: A pair of interior angles lsyeinggmneeornnttahree
same side of transversal when it cuts two line

called co-interior angles. If the lines are parallel then tthhaeir
sum is 180°.

3. Types of triangle:
According to sides:

Equilateral triangle: All three sides are equal.

Isosceles triangle: Any two sides are equal

Scalene triangle: All three sides are different.

According to angles:

Acute angle triangle: All angles are less than 90°,

Obtuse angle triangle: One of the angle more than 90°

Right angle triangle: An angle is 90°.

4. Height (altitude) & median of a triangle:

The perpendicular draw from any vertex to its opposite side is
called height of triangle. The straight line joining any vertex of
a triangle to the middle point of its opposite side is called
median. The point of intersection of all medians of a triangle is
called centroid of a triangle.

5. Properties of triangle:

a. The sum of three angles of a triangle is equal to two right
angles (180°).

b. The exterior angle of a triangle is equal to the sum of the
opposite two interior angles.

c. The sum of exterior angles of a triangle is equal to 360°.

d. The sum of two sides of a triangle is greater than the third
side.

e. The side opposite to the greater angle of a triangle is
greater than the side opposite to the smaller angle.

f The base angles of isosceles triangle are equal.

g. The perpendicular drawn from a vertex to the base of an
isosceles triangle bisects the vertical angle and the base.

h Each angles of an equilateral triangle is always 60°

i. A line segment joining the mid-points of any two sides of
a triangle is parallel to third side and is half of it.

106

A median of a triangle divides into two triangles of a equal
area.

k The centroid divide the median in the ratio 2:1 from the
vertex.

1. The point of intersection of altitudes of the triangle is
called orthocentre.

m. The point of intersection of internal angle bisector of
triangle is called in-centre.

n. The point intersection of perpendicular bisectors of sides
of the triangle is called circum centre.

6 Congruency of triangles:

Two triangles are said to be congruent if they have same shape
and same size. If two triangles are congruent then
corresponding angles and sides are equal. It is denoted by =

Condition for congruency of triangle:
a. S.A.S. [Two sides and included angle]
b. A.S.A. [Two angles and included side]

C. S.S.S. (Three sides]

d. R.H.S. (Right angle, hypotenuse and any other side]

e. A.A.S. [Two angles and corresponding side]

1. Similarity of triangles:
Two triangles are said to be similar if they have sameshape. If two

triangles are similar then their corresponding angles are equal and
corresponding sides are proportional. It is denoted by ~

[Note: Every congruent triangles are similar but converse may
not be hold]

Condition for similarity oftriangles.
A.A. Similarly.

b A.A.S. Similarly.

C. S.S.S. Similarly.

Objective questions

1 Which is the supplement angle of 45°.

459 b. 135°

C. 315° d. 235°

107

DHG-2. In the given figure AB||CD, AGH = a and DHG

Find the value of ab.

a. 2 b. 4
c. 5 d. 6

A. B

C HF D

3. In the adjoining figure AB||CD. Calculate the length of y
and x.

a. 3.75 cm, 12cm b. 10cm, 5 cm

C. 15cm, 4 cm d. 12cm, 5 cm

A 3cm Bca

4.5cm 10cm
B

4. Ifr is ananglesatisfying: 90° <I< 180º then x is called

a. Acute angle b. Obtuse angle

C. Reflex angle d. Complement angle

5. What is the complement ofž of right angle?

a 54° b. 36°
d. 126°
C. 108°

6. What is the sum of the acute angles of an isosceles right

angled triangle?

a. 25° b. 70°

C. 80° d. 90°

7. Two complementary angles are in the ratio 7:3 find them.

a. 55°, 35° b. 80°, 10º

C. 70°, 20° d. 63°, 27°

8. Find the size of an angle which is two third of it5

complement

a. 36° b. 72°

C. 82° d. 420

108

9. Find the size of an angle which is one fifth of its supplement

a 30° b. 40°

c 50° d. 40°

10. The angles of a triangle are (x + 37°), (2x + 15°) and (3x+
8°). Find each angles of triangle.

a. 52°, 45°, 83° b. s9°, 59°, 62°

C. 60°, 40°, 80° d. 57°, 55°, 68°

11. If the sum of two angles of a triangle is 114° and difference
is 48°. Find these angles.

a. 81° & 33° b. 80° & 30°

C. 19° & 95° d. 72° & 42°

12. In atriangleABC,AB = BC, B= ° and LA = 2x- 20°.
Find <B

a. 20° b. 40°

C. 44° d. 80°

13. If three angles of a triangles are in the ratio 2:3:4 then find

the angles.

a. 20°, 60°, 80° b. 60, 80°,40°

C. 40°, 80°, 60° d. 40°, 60°, 80°

14. If LXYZ = 40°, OX bisects the YXZ, OY bisects the

<XYZ. Findtheangleof XY

a. 80 b. 90°

C. 140° d 120°

7

15. In atriangleABC,2(&A + <B) =<C, find C

a. 60° b. 80°

C. 100° d. 120°

16. From the figure, find OM.

a. 2 cm b. 3cm

C. 4cm d. 12cm

109

B

6cm 4cm

3cm

M

17. What is thecomplementaryangle of (90° - 0)

a. 90° b. 80°

C. 600 d.

18. Two complementary angles are in ratio 5:4, then these

angles are

a 50°,40° b. 100°, 80°

C. 75°, 60° d. 80°, 80°

19. An angle is 10° more than 3 times its supplement. Find the
angle?

a 30° b. 70°

C 135° d. 137.5° ges

20. Find the value of x° from figure.

a. 200o b. 240°

C. 280° d 320°

A45°

21. Find the value x from the figure

a. 60° b. 120°

C. 50° d. 70°

120°

22. Find the value ofy from the figure.

P 130

10

a. 100° b. 50°
C. 70° d. 85°

110

23. Which of the following is not the axiom of congruent?

a. S.A.S b. R.H.S

C. S.S.S d. A.A.A

24. A triangle ABC right at B, then which is the true statement?

a. AB'= AC' + BC² b. BC'= AC?+ AB

c. AC- AB' = Bc² d. BC'= AB- AC?

25. In the given figure, ZPQS = LPRQ then

P

AR

a. PQ =PS.QS b. PQ° =PS.SR

C. PQ' =PS.PR d. PQ' =PS.QR

26. Find the value of x from the figure.

a. 6 cm b. 8 cm

C. 9 cm d. 1l cm

17cm

10cm

X cm
6cm

27. If the sum of two sides of a right angled triangle is 17cm

and the hypotenuse is 13 cm, then the length of sides are

. 5 cm and 2 cm b. 12 cm and 5 cm

C. 10 cm and 3 cm d. 13 and 4

28. Find the coordinate of a point which is equi-distant from
the points (7,-3) (0,-2) and (8,-2):

a. (1, 4) b. (1, 3)

c. (4, 1) d. (5, 2)

29. In a right angle triangle x < y<zwhich is true?

y+ła. x+y=2 b. x'-y'=z
d. x'=?-y²

111

30. The area of right triangle is 12 sq. cm. The ratio of its legs is
2:3. Find the hypotenuse of the triangle.

a. V13 cm b. y26 cm

c. 313 cm d. y52 cm

31. The base of an isosceles triangle, exceed each of the equal
sides by 8 em. If the perimeter is 89 cm, then what is the
length of the base?

a. 35 cm b. 27 cm

e. 29fcm d. 70 cm

32. The perimeter of triangle is 12 cm and ratio of sides are
3:4:5 then area is

a. 12 cm b. 6 cm

C. cm d. 2 cm

33. The acute angle of right angled triangle are

a. Complementary b. Supplementary.

C. Equal d. Adjacent

34. Find the length of NM from the adjoining figure if <LPM

=L LN0.

2Cmo

6cm

L N
Scm M

a. 4.75 cm b. 4.6 cm
C. 4.5 cm d. 4.25 cm

35. In figure QPR = APSQ, themeasurementof QR is

a. 2 cm 10cm
C. 6 cm
Q R As
8cm
b. 5 cm
d. 5.5 cm
112

36. The value of x in the figure is DA 3cm
b. 4cm
a. 5 cm tuti Xcm

C. 6 cm d. 8 cm

B 13cm

37. If LDBA = 39°, FBE = 79°,then DG
LGBC hasmeasure of

a 390 b. 62°

C. S1° d. 152°

FE

Answers:

I. b 2.a3.a4.b 5. b 6. d 7. d8. a 9. a 10. d
L. 3 12.e 13. d14, c |1S. d16, a|17. d18. b19, d 20.e
21.a 22. b 23. d 24. c25. c 26., c27. b 28.c29, db0. d

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

113

Chapter- 16

QUADRILATERAL AND PARALLELO0GRAMS

1. Quadrilateral:

A quadrilateral is a plane figure bounded by four straight line
segments.
Note:
1. Sum of interior angles of quadrilateral is 360
2. If one angle is reflex then it is called concave

quadrilateral.
3. Sum of Exterior angles of quadrilateral is 360°.
2. Types of quadrilateral:
a. Parallelogram: A parallelogram is a quadrilateral in

which opposite sides are parallel.
Properties of a parallelogram

Opposite sides and angles are equal.

Diagonals are not equal.
Diagonal divides it into two equal parts.

• Diagonal bisect each other. (mid points of diagonal

same)

Triangles on equal base and between the same
parallels are equal in area.
Area of parallelograms on the same base and between
the same parallel lines is equal.

The area of a triangle is equal to half the area of a
parallelogram on the same base and between the
same parallels.
Area of parallelogram = base x height
Area of parallelogram = ab sin®.( where a and b be
the length of adjacent sides and 9 is angle between
them)
b. Rectangle: A parallelogram in which one angle is 90°
then it is called rectangle.

114

Properties ofrectangle

Opposite sides are equal.
Diagonals bisect each other

Diagonals are equal.
Diagonals bisect opposite angles.
Each angle is a right angle.
C. Rhombus: A parallelogram having all sides are equal, is
called rhombus.

Properties of rhombus

Opposite sides and angles are equa.
Diagonals bisects each other at right angle.

Diagonals are not equal.
Diagonal divides it into two equal parts.
Area of rhombus: 1/2 (Product of diagorals)
Also, Area = base x height
d. Trapezium (Trapezoid): A trapezium is a quadrilateral in
which one pair of opposite sides are parallel.
Note: If non-parallel sides of a trapezium are equal then it
is called an isosceles trapezium.

Properties of Trapezium

An isosceles trapezium has equal diagonals.
Median of trapezium is parallel to the pair of parallel
sides and half in length to the sum of parallel sides.

Area =x heightx(Sumof parallelsides)

e, Square: A rectangle in which two adjacent sides are equal
then it is called square.
Properties of square
All sides are equal.
Diagonals bisect each other at right angles.
Diagonals are equal.
Diagonals bisect opposite angles.
Each angle is a right angle

115

f. Kite: A kite is a quadrilateral in which pairs of adjacent
sides are equal.

Note: The diagonals meet at 90° and one of the two
diagonals divide the kite into two congruent triangles
while the other divides into two isosceles triangles.

Objective questions

1. What is the sum of the exterior angle of quadrilateral?

a. 180 b. 360°

c. 270° d. 108°

2 In a square ABCD, What is the value of x?

a. 58° b. 103
C. 133o
d. 135°
D
3. The value of x and y from the figure is,

a. 50° & 63 b. 152°& 44°

c. 92° & 32° d. None

°PBy20 8g•

AY

4. In the given figure AB|DC, AD|BC and BC = Scm and
<DAB = 30° and the area of ABC = 1Sem, find the length
of AB.

a. Scm b. 7cm
C. 9cm d. 12cm

5. Diagonalsareequalin ....

a. Rhombus b. Rectangle

C. Trapizium d. All of the above

6. If an angle of a parallelogram is 90°, then it is called...

a. Rectarngule b. Rhombus

c. Trapezium d. Square

116

1. Find thevalue of (x + y

a. 30° b 80°

C. 110° d. 150°

8. From the adjoining figure find LXYZ

a. 45° b. 60°

75° d. 120°

60

9. Find the value of y in the given figure in which ABCE is
parallelogram, AE = AD &ÁDAE =30°

B

a. 30° b. 75° eho tn

C. 80 d. 150°

10. If , 2x°, 4x° and Sx° areanglesofquadrilateral.Find

greatest one?

a. 60° b. 120°

C. 150° d 160°

11. If any two consecutiveangles of a parallelogram are in the

ratio 2:3, find the angles of parallelogramn

a. 72°, 108° b. 60°, 90°

C. 90°, 120° d. 120°, 108°

12. In the adjoining figure. AABC ~ AAXC. AX 1 BC, AB =

3em and AC = 4cm, Find the length of CX.

2cm 4cm
C 3.2cm 3cm

X
b. 3.2cnı
d. 4 cm

117

13. In the given trapezium ABCD, E and F are mid points of
AD and BD then the value of EF is ...

a Scm b. 7cm

C. 8.5cm d. 14cm

7cm

D
10cm

14. Any two angles of a quadrilateral are 50° & 70, Ir
remaining two angles are in the ratio 2:3, find these angles?

a. 72°, 108° b. 95, 85°

C. 120°, 180º d. 96°, 144°

15. In adjoining figure, D, E, F & G are mid-points of AB, AC,
AD, & AE respectively. Also H is midpoint DE, and then
which is the true statement?

a. Areaof AFGH =;AABC

b. Area of AFGH = AABC

c. Area of AFGH = AABC

d. Area of AFGH = AABC

16. The area of rhombus is 60 cm if its one side measures
12cm, what is its height?

a. Scm b. 12cm

C. 8cm d. 10cm

17. Find area of quadrilateral ABCD?

a. 30cm? b. 48cm Scmy

C. 78cm d. 9lcm 13cm

18. In the given parallelogramPQRS. 80
Where PS = ST. Find the value of

a. 40° b. 80°
C. 100° d. 110°

118

Find the area of rhombus ABCD.

19.

a4 b. 16 cm

c. 32 cm? d. 64 cm

20. What is the measure of y in the parallelogram MNOP,
where OQ and PQ are the bisectors of <PON and OPM

respectively? M

a. 60°

b. 70° T8cm

C. 800 N

d. 90°

21. What is the area of parallelogram ABCD in which AB = 4
cm, BC =8 cm and B=30°.

a. 12 cm b. 16 cm

C. 32 cm d. 64

22. If D, E, F are mid-point of AC, BD & BC respectively.
Which following relation is true.

a. ABEF =;AABC b. ABEF-AABC

c. ABEF = AABC 1

d. ABEF=nABC

23. Find the value of a in the ABCD

a. 18° b 36°

C 60° d. 720

24. The area of Trapezoid is 3a/2A

a. 35cm? 6cm

b. 17.5cm Bcm
C. 16 cm

d. 20cm 8cm

25, The area of the following figure

a 30 cm 2cm
C
b. 35cm?
3cm
C. 45 cm 4cm B

d. 49cm

119

26. In the figure ABCD is a square and AED IS an equilateral

triangle. If AB= 2, what is the area of the shaded region?

a. y3 DC

b. 2

c. 4-3

d. 4+y3

27. In the figure p° + q +r+s+#+x°+y+equal

X

a. 720° b. 360°
C. 540° d. 240o

Answers:
. b2. b3. c 4. d 5. b 6. a 7. b 8.
|9. b 10. c

1. a 12. c 13. c 14. d |15. d16. a 17. c 18. a 19. c20.
21. b 22. b|23. d 24. a25. d 26. c 27. 1

120

Chapter– 12

CIRCLE AND ITS PROPERTIES

1. Circle: The locus of moving point in a plane from a fixed point
at a constant distance is called circle. The constant distance is
radius and fixed point is center of circle.

2. Basic terms of circle:
a. Circumference: The total length of a boundary
(perimeter) of circle is called circumference of the circle.
Chord: A line segment joins any two points of
circumference is caled chord.

b. Segment of a circle: A region enclosed by an arc and a
chord of a circle is called segment of a circle.

Major segment: The segment of a circle greater than a
half circle is called major segment

Minor segment: The segment of a circle smaller than half
of the circle is called minor segment.

C. Diameter: A line segment joining any two points of the
circumference and passing through center of circle is
called diameter.

d. Arc: A part of the circumference is called arc of a circle.
e. Sector ofa circle: A region enclosed by any two radius

and an arc of a circle is called a sector of the circle.
f. Concentric cireles: Two or more than two circles are said

to be concentric circles if they have same center but
different radius.
Inscribed circle: A ircle in which its circumference
touches all the sides of polygon is called inscribed circle.
i. Circumscribed circle: A circle is calledcircumscribed if
all the vertices of closed figure are on the circumference of
the circle. It is also known as circum-circle and its centre
is called circum-centre.
Line of centers: A linesegmentis called line ofcenters if
it joins the centers of two circle.

k. Central angle: Any angle at the centre of circle is called
central angle.

121

1. Angle at the circumference: An angle inscribed bbyy aann arc

of a circle at its circumference is an angle at the
circumference. It is also known as inscribed angle.
m. Cyclie quadrilateral: A quadrilateral lying its vertices on
the circumference of circle is called cyclic quadrilateral.
n. Tangent of circle: If a line touches a circle at exactly one
point is called tangent.
0. Intersection circles: Two circle are said to be intersectins
if they intersect at two different point.
p. Alternate Segment
A chord is drawn through the point of contact of a tangent
to a circle. The segment of the circle opposite to the angle
formed by the chord with the tangent at the point of
contact is called the alternate segment for that angle.

A

P

T
Here PT is a tangent at T and TA is a chord of contact.
LABT is called the angle in the alternate segment
Note: Angles in the alternate segment of a circle are equal.
i.e., ABT = LATP
3. Properties of circles:
a. All the radius and diameters of circle are equal.
b. The radius of the circle is perpendicular to the tangent at
the point of contact.
C. Any angle in semi circle is right angle.
d. Arc subtending equal angles at the centre of a circle are
equal.
d. Equal chords are equidistant from the centre.

f Angles in the same segment are equal.

g. The central angle is double of inscribed angle standing on
the same arc.

122

The perpendicular drawn from centre to a chord bisects the
h.

chord.

i. The opposite angles of a cyclic quadrilateral are
supplementary.

The longerchord liesnearesthecentre.

k. There is only one circle passing through three non
collinear points.

1. The angle in major segment is always acute.

m. The angle is minor segment is always obtuse.

n The length of tangents from an external point to a circle

are equal.

The angles on the conjugate segment are supplementary.

p. The tangents at the ends of diameter of a circle are
parallel.

q. There is two tangents from the external point to the circle.

4. Polygon: A bounded figure by three or more than three straight
lines is called polygon.

Regular Polygon: If all the sides and angles of a polygon are
equal then it is called regular polygon.

Note:

1. 7Numberofsidesandpolygon; 8 9 10
Name 56
Decag-
No. of Pentag- hexag- Heptag- Octag- Nonag- on
sides
on on on on
2.
n
Each interior angle of a regular polygon of n sides = n

x 180°

3. Each exterior angle of a regular polygon of n sides = n

4. If at least one angle of a polygon is more than 180° then it

is called a concave polygon.

5. The angle at the 360° Where n= no of side of
centre = n

regular polygon.

6 The sum of exterior angle of regular polygon = 360

7. The sum of interior angle of regular polygon = (n 2) x
180°.

123

8. The number of diagonals of regular polygon havine

sides= nn-3)

2

Objective questions

1. In a circle with centre 0, if <PRO = 65° Find the value of
LQPO.

a. 25°

b. 30°

c. 35°

shon d. 45°

2. Calculate the value of X°

a. 20° b. 30° 260°x

C. 40° b d. 50°

3. Find the value of y.

5la. 20°

1. b. 30°

C. 60°

d. 70°

4. Calculate the value of x°, where O be the centre

a. 70°

b. 110°

C. 140 B

d. 130°

5. Find the value of y, where O be the centre of circle

a. 30°

b. 60° 30
C. 100°

d. 120°

s6. IfO be thecentre ofcircle then AoC=a. 60°b. 90

C. 120° d. 130°

124

In a figure XOY is diameter.

1. LWYZ= 132°.Find <wxy

a. 90° b. 75°

C 48° d. 420

Eor what value of x in the given figure AB is diameter?

8.

a. 2 cm

b. 3 cm A Scm R

C. 4cm X cm

d. 8 cm

9. Caleulate the value of y from the figure, where O be the
center

a. 65°

b. 50°

C. 70°

d. 115°
10. Find CDA from the figure, where O be the center.

a 33

b 66°

C. 73.5o

d. 1470

11. Which is the largest chord of a eircle?

a. Segment b. Secant

C. Diameter d. Circumference

12 From the figure which one of holds. A
a. BAD <<BCD

b. ABAD > BCD

C. KBAD= BCD

d. <BAD =2& BCD

13. Find the area between two concentric circles having radius

7cm and 14cm.

a. 154 cm? b. 462 cm?

C. 616cm d. 2009 cm?

125

14. In the given circle with centre 0, PQ, is R
the tangent and P is the point of
contact. If OP = Scm, and PQ =12 cm, P 12cm
find the length of QR?

a. Scm b. 8 cm

C. 10 cm d. 13cm

15. In cyclicquadrilateralABCD. If AB = AD & BCD =86o
find thevalue of ACD.

a. 86° b. 48°

c. 43° d. 40°

16. Tis thecontactpointoftangentPR.If PT= AT & LAPR=

20°, find the value ofLABT.

a. 20°

b. 40°

C 60° T
d. 80°

17. In a circle, the longest chord measures 14 cm. What is the
area of semi circle.

a 70cm b. 77cm?

C. 80cm? d. 79cm

18. How many sides in heptagon?

a. 6 b. 7

C. 8 d. 9

19. What is the exterior angle of regular pentagon?

a. 90° b 108°

C. 72° d. 80°

20. The sum of exterior angles of regular octagon is

a. 90 b 180°

C. 360° d. 45°

21. The value of x + y in the figure is

a. S0° b. 30°

C 80° d. 70°

126

Inthe given figure, chord AB and CD of a circle intersect at
22 E.1 CE =8cm, DE = 4cm and AE = 16 cm. Then the length

of BE is B
a. 4 cm

b. 8 cm

C. cm

d. 10 cm

23.Thearea of the circle x +y= 49is

a. 149 sq. unit b. 152 sq.unit

C. 154 sq.unit d. 160 sq,unit

24. Findthevalueof x+y

a. 12 cm Scm
b 24 cm

C. 16 cm 13cm

d 17 cm

25. Caleulate the value of y b. 30°
a 40° d. 75°
C. 70°

26. Find the area of the shaded region. If radius of each circle is
7 em & A, B, C, D are centre of circles.
a. 42 cm'
b. 40 cm
40

C. 1

d. 44 cm

27. In the given figure, the point O is .. of the triangle
ABC

a. In centre

b. Orthocentre

C. Circumcentre P
d. Centroid

28. The value of x in the figure is 3cm5cm)

a. 3 cm b. 3.75 cm

C. 4 cm d. 5 cm

127

29. Find the value of (r°+ y") from the figure, when O be the

B )centre of the circle and BA and BC are tangents.
a. 112° b. 119°
C. 121° d. 126

30. In the given figure BAC+ LACB = 90°. What is AC
called?

a. chord b. segment

C. diameter d. sector

31. R
In the given figure, OA and OB are tangents. If <OAB =
65°,calculate ACB.

a. 65.50 b. 65°
C. 130° d. 140°

32. In the given figure, PQ and PR are the tangents to the

circle.If PQ= QR=7 cm,find QPR.

>p

a. 60° b. 65°
c. 55° d. 50°

128

13. İIsnathtaenagdejnoitn. Iinf.TgLfiQgPuSre,= O is the centre of the circle and TPS
46°, then the value of the POQ is

P

a. 46° b. 82°
C. 92° d. 90°

4. Thearea of shaded region is

Tcm

a. 66cm 14 cm
b. 68 cm?
C 77cm d. 82 cm

Answers:

L. e122..db 313., bc414., b5. d 6. c 7. d||8. c 9. d 10. c

b |15. c|16. b|7. b 18. b 19. c 20. c
21.d 22. e 23. c 24. b 25. a 26. a 27. c 28. b 29. e 30.
31.b 32. a 33. c 34,

129

hapter - IN

PROBABILITY

Some basic terms:
a. Experiment: An action which produce result (outcome)
b. Random experiment: when the result of an experiment is
not certain.

C. Out come: The result obtained in each random experiment

d. Sample space: The set of all possible outcomes of a
random experiment.

e. Event: Any subset of sample space.
f. Sample point: Each result of event of a sample space.

Elementary event: An event containing only a single
sample point (simple event).

h Equally likely event: Two or more than two events are

said to be equally likely if there is equal chance of getting
any outcome of an experiment.
i. Exhaustive cases: The total number of all possible
outcomes of a random experiment is called the exhaustive
case for the experiment.
j. Sure event and impossible event: An event A is said to be
sure if it is sure to happen. If A is sure event then P(A) =1
If not sure to happen then it is called impossible event. If
A is impossible event then P(A) = 0.

k Mutually exclusive events: Two or more than two events

are said to be mutually exclusive event if occurrences of
one event excludes the occurrence of the others.
i. e. An B= then A & B are mutually exclusive event.
1. Independent event: Two or more than two events are said
to be independent if the happening of an event does not
affect to happen the other events. If A and B are
independent event then

P(A & B) = P(A) × P(B)
Or, P(An B) =P(A) x P(B)

130

m. Odds in favour and odds against on event. Let A be an
event. Then the ratio P(A): P( A) is called the odds in
favour of happening of the event A and the ratio P( A):
P(A) is called the odds against the happening of the event
A.

i. If odds in a favourof anevent A are m:n, then

probability of happening of the event A is given by
m

P(A)m +n

ii. Similarly, if odds againstan event A are m :n then
the probability of getting an event A is given by

P(A) m+n

IMPORTANT RESULTS OF PROBABILITY

n(A)

a. P(A) =

b. P(O) = 0 [4 is impossible event]
C. P(S) = 1 [S is sure event]
d. 0s P(A) S1 [rangeof probability]
e. P(A) + P( A) = (sum of probability of occurrence and non

occurrence event)
f. P(A U B) = P(A) + P(B) (A & B are mutually exclusive

event)
P(A U B) = P(A) + P(B) - P(ANB) [A&B are not
mutually exclusive]
h. P(An B)= P(A) x P(B) [A & B areindependentevent]
i. PỊA O B)= P(A) × P(B/A) [A & B are dependent events]

Objective questions

1 A card is drawn at random from a pack of 52 playing cards.

Find the probability getting not a jack?

b.

c. d.

131

2. What do you call the set of results that come after cer

experiment?

a. sample b. sample space

c. elementary event d. all of the above

3. The value of probability is always lies?

a. between 0 & 1001 b. between-1 & 1

C. between0 &1 d. between -100& 100

4. What is the probability that a pregnant woman give birth to
a child on Sunday of February month?

a. 1/7 b. 2/7

C. 77365 d. 7/30

5. A bag contains 25 balls in which 12 are red, 7 are black and
6 are yellow. What is the probability of getting the ball is
yellow?

a 7/25 b. 12/25

C. 19/25 d. 6/25

6. A coin is tossed 3 times. What is the probability of getting at
least two heads?

a, 1/2 b. 2/3

C. 3/5 d. 3/8

7. How many possible out comes when a die is rolled?

a. 2 b. 4

C. 36 d. 6

8 A card is drawn at random from a deck of 52 cards. Find

the probability of getting king?

1a 13 b.

3 4

C. 13 d
13

9. A die is rolled two times, then the possible outcomes
becomes.

a. 6 b. 12

C. 36 d. 66

132

10. Vhen a coin and die rolled at same time what is the number
of possible outcomes?

a. 6 b. 12
C. 36 d. 7

11. A red card is removed from a pack of 52 cards. What is the
probabilityy of getting red jack?

2a. b. 51

1 d.
52
C. 51

12. A letter is chosen from the word "LANGURBURJA". What
is the probability getting not a vowel?

a. 4/11 b. 5/11

C. 6/11 d. 7/11

13. A couple has 3 children, what is the probability of having at
least 2 daughter?

a. 1/4 b. 3/8

C. 7/8 1

d.

14. If three coins are tossed simultaneously, then the

probability getting at most two head.

a. 7/8 b 3/8

C. 2/8 d. 1/8

15. A die is rolled twice, what is the probability (getting) that a

4 will show first and 3 on the second?

a. 11/36 b 1/36

C. 1/6 d. 2/6

16. A die is thrown, what is the probability that the outcomes
are a prime number or an odd number?

a. 1/2 b. 1/6

C. 2/3 d. 5/6

17. What is the probability of getting queen or diamond when a

card from 52 cards is removed?

a. 5/13 b 17/52

C. 1S/52 d. 4/13

133

A number card numbered from 1 to 30 is drawn rando
18. whatistheprobabilityofgettingacard having the u

which is divisible by4 or 5.
b. 3/5

a 2/5 d. 2/7

C. 3/7

A number drawn randomly from number 31 to 60, what:

19. theprobability ofgettingthe number multiple of 5 or 62

a. 2/30 b. 11/30
c. 1/3
d. 1/10

Coin is tossed and a die is thrown. What is the probabilithity

20. of obtaining a head on the coin and an even number onthe

die? b. 1/6
a. 1/2 d. V4
C. 4/6

21. If probability ofoccurrenceevent is 2/3 then probability of
nonoccurrence event is

a. 1/3 b. 1/6

C. S/6 d. 1

22. From a set of cards numbered 1 to 25 a card is drawn at
random. Find the probability that the number of card is

divisible by 5 greater than 10?

a. 8/25 b. 5/25
C. 3/25 d. 4/25

23. What is the probability getting both ace card when two
cards are drawn from a pack of cards without replacement?

a. 10/17 b. 1/221

C. 33/221 3

d 465

24. For any events A and B such that P (A) 1/3 and P (An B)
= 1/12. If A and B are independent events then P (B) =?

a. 1/6 b. 1/2

c. 3/4 d. 1/4

25. Find the probability that a month (according to English
calendar)selectedfrom a year at random starts from "A":

a. 1/12 b. 1/6

C. 1/4 d. 1/3

134

nrobability that a student passes an English test is 3/5
26. d the probability that hepassesboth English andscience

test is 12/40. The probability that the passes at least one test

0. What is the probability that hepassesthescience

test?

a 3/5 b. 2/5
d. 4/5
C. 1/5

27. There are five branches in a tree. The no. of birds in the
Grst branch is 2. The no. of birds is twice that of previous

hranch in each branch; give the total no. of birds?

. 70 b. 60

b. 62 d. 180

28. In a family having four children the probability of being all
of the same sex (either male or female).

a. 5/8 b. 3/8
C. 7/8 d. 1/8

29. If the odds against an event A is 4:5, then the probability of
getting this event A is

4 b
a.

C. d.

30. If the odds in favour of an event A are 3:5, then the
probability of getting this event A is

a b.

c. d.

31. Yam and Ram appear for an interview for two vacancies.
The probability of Yam's selection is 1 and that of Ram's

selectionis . Then theprobabilitythatnoneofthem will

be selected is

a. b.

C. d. 15

135

32. A leap year is selected at random. What is the probabili
that it will contain 53 Sunday?

a. 7 b.

53 52
C. 365
d. 365

33. The probability of solving a mathematical problem by thren

12

students P, Q and R are and respectively. Ir the

problem is given all of them, what is the probability that the
problem will be solved?

a. b.

C.

34. Three children were born from a married couple. Find the
probability of having one son and two daughter?

3 5
b.
a. 8

7 1
C.
d.

35. Ifa card is drawn from a pack of 52 cards and at the same
time a ball is drawn at random from a bag containing 13
red and 5 black balls. What is the probability of a red ball
and a red card?

a. b.

13 12
C. 36
d.. 52

36. From a number card numbered from 2 to 36, a card is
drawn at random. What is the probability of getting a card
which has square number or cube number?

a. b.

c. d.

136

Erom a pack of cards, three cards are drawn in succession
31. at random without replacement. What is the probability

thatthe cards so drawn are no facecards.

17 b. 38
a. 85 85

C. 52
38. Two dices are thrown at once, What is the probability of the

eum of the number on both being a multiple of 4?

a. 4 b.

d.

Answers:

. e2.b 3. c4. a5. d6. a 7. d 8. a 9. c10. b

. d12. d13. d14. a|15. b16. c 17. d18,a 19.e20. d

L. 12.c 23 b 24," d 25. b|26. b27. b 28. d39. b 30. a
B. d 32. b 33. b 34.a35. c36. d 37.b 38.a

oriba

d

137

Chapter-19

STATISTICS

1, Arithmetic mean (Average):

The arithmetic mean is defined as the total sum of observations
divided by total number of observation. It is denoted by (X).

a. x=For individuasel ries]

b. X= Zf ordiscreteseries]where N=Zf.
N

C. X=[for continuousseries]

Where, X=Observations
n=Number ofitems

f=Corresponding frequency
m= Mid-value of eachclass interval
2. Median (M):
When the data are arranged either ascending or descending
order of magnitude, the-middle most value in a distribution is
called median. It is also known as positional average.

a M=value ofr( item [individualseries]

b. M= valueof item [fordiscretseeries]

Note: Data must be arranged either ascending or
descending order.

C. For continuous series:

Find median class

-Ncf

=L+ i

Where, L = Lower limit of the medianclass

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f= Frequencyofmedianclass

c.f= Preceding cumulative frequency of medianclass

i =Clasisnterva(sl ize)
N= Totalno.ofrfequency

3. Range:
The difference between the highest and the lowest values of a
variable of any set of data is called range.

a. Range(R) =L-s

b Coefficienotrfange -s

4. Quartiles:
uartile divide the set of observations into four equal parts after
arranging the data ascending or descending order. There are
three quartiles Q1, Q, and Q,.

a. For individual and diserete data:

i. Qi = valueof item

i. Q= valueof3 item

Note: Median is same as Q2

b. For continuous data:

i. Q liesontheclass- item.

:Q-L+4Ncf.xf i

ii. Q, lies on the class = item.

Q.h=L3+Nfc.fii

Where, L = lower term of class
N= total no. offrequency
c.f= preceding c. f. of quartile class

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f=frequencyofquartile
i=classinterval
5. Mode:

The maximum repeated value in the distribution is called mode
It is denoted by M,

a. Mode (M,]= Most repeated value. [Individual (series)
data]

b. Mode [Ma] = Corresponding value of highest frequency
[discrete data]

C. Mode[MJ] =Ltff x[icontinuoudsata]

Where, L = lower limit of the modal class

f,= frequencyof the classprecedingthe modal class.
f =frequency of themodalclass.
f= frequencyof theclasssucceedingthe modal

class.

i= magnitudeof themodalclass.
[Note : Mode = 3Median -2 Mean]

Objective questions

1, The marks obtained in 7 students by a subject are 56, 54,
60, 71, 62, 59 and 72. Find the average mark.

a. 60 b. 62

C. 71 d. 72

2. The mean of five numbers is 27. If one number is excluded
their mean is 25. Find the excluded number.

a. 30 b. 34

C. 35 d. 2

3. The median of the following observations arranged in
ascending order is 25. Find x.

11, 13, 15, 19, x +2, x + 4, 30, 35, 39, 46

a. 22 b. 24

C. 25 d. 22.5 ot

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What is the positional average of 12, 22, 25, 19, 21, 25, 23,
4. and 30?

a. 19.5 b. 20

C. 22 d. 22.5

1r the mean of 37, 12, y, 18, 17, 13, 12 is 20. What is the
5. value of y?

a. 19 b. 20
C. 31 d. 22

The sum of age of some students is 360 years and the

6.

average age of is 20, find the number of students.

a. 18 b. 24

C. 26• d. 28

Find the median of 10, 12, 13, 8,9, and 15?

a. 9 b. 11
C. 12 d. 15

8 Which of the following values divides the two data arranged
in ascending or descending order into two equal parts?

a. Mean b. Median

C. Mode d. Average

9 If there is n terms of a series, which of the following falls in

place.

a. Mean C. Median
C. Mode d. Average

10. In a frequency distribution, the measure of the range is....

a. L+S b. S-L

C. L-S dd.. L-S

L+S

11. If the mode of 9, 11, 12, x, 15, 12, 11, 16 is 12. What is the
value of x?

a., 9 b. 11

C. 12 d. 15

12, Which is called "Positional average "?

Mean b. Mode

C. Median d. Quartile

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