SAMPLE QUESTION PAPER

MATHEMATICS (211)
Time: 2½ hrs Maximum Marks: 85
Note:
i. This question paper consists of 44 questions in all.
ii. All questions are compulsory.
iii. Marks are given against each question.
iv. Section A consists of
a. Q.No. 1 to 17 – Multiple Choice type questions (MCQs) carrying 1 mark each. Select
and write the most appropriate option out of the four options given in each of these
questions. An internal choice has been provided in some of these questions. You have to
attempt only one of the given choices in such questions.
b. Q.No. 18 to 28 – Objective type questions. Q.No. 18 to 27 carry 02 marks each (with 2
sub-parts of 1 mark each) and Q.No. 28 carries 05 marks (with 5 sub-parts of 1 mark
each). Attempt these questions as per the instructions given for each of the questions 18 –
28.
v. Section B consists of
a. Q.No. 29 to 37 – Very Short questions carrying 02 marks each.
b. Q.No. 38 to 42 – Short Answer type questions carrying 03 marks each.
c. Q.No. 43 to 44 – Long Answer type questions carrying 05 marks each.

SECTION A
S.NO. Questions Marks
Q.No. 1 to 17 are MCQs (1 mark each):
An internal choice has been provided in some of these questions. You have to
attempt only one of the given choices in such questions.
1. (i) If a +b= 12 and ab = 22 then a2 + b2= ? 1
(a) 188
(b) 144
(c) 34
(d) 100
OR
(ii) Which of the following is the factored form of the expression 5x 2- 13x-6
(a) (x-3)(5x+ 2)
(b) (x+3)(5x+ 2)
(c) (x-3)(x +2)
(d) (x+3)(x+2)
2. (i) Which of the following is not a solution of the equation: 3x + 6y = 12. 1
(a) (-4, 4)
(b) (0,2)
(c) (8, -2)
(d) (3,1)
OR
(ii) The pairs of equations x+2y-5 = 0 and -4x-8y+20=0 have:
(a) Unique solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No solution
1
3. (i) simple interest on ₹ 1632 for 5 years at per annum: 1

(a) ₹ 649
(b) ₹ 510
(c) ₹ 580
(d) ₹ 630
OR
(ii) What sum of money lent for two years at compound interest will amount to ₹
968at the rate of 10% per annum, interest compounded annually?
(a) ₹ 845
(b) ₹ 827
(c) ₹ 889
(d) ₹ 800
4. (i) If P is 40% less than Q, then Q is what % more than P? 1
(a) 40 %
(b) 66.66%
(c) 60 %
(d) 33.3%
OR
(ii) The price of cooking oil has increased by 25%. By what present should a
family reduce the consumption of cooking oil so as not to increase the expenditure
in this account?
(a) 20%
(b) 25%
(c) 18%
(d) 16%
5. 1

(i) What is the measure of the angle x, when angle measure of arcs AB and AC are
840 and 1400 respectively?
(a) 1340
(b) 1350
(c) 1360
(d) 1370
OR

2
 (ii) What is the value of angle x?
(a) 500
(b) 1200
(c) 600
0
(d) 70
6. 1

What is the length of AC?

(a) 19
(b) 18
(c) 17
(d) 16
7. (i) The degree measure of the angle subtended by the diameter of a semi-circle at 1
its centre is:
(a) 90
(b) 45
(c) 180
(d) 60
OR
(ii) The radius of a circle drawn from the point of contact of a tangent to the circle
is always _____ to the tangent.
(a) equal
(b) perpendicular
(c) twice
(d) parallel

3
8. 1

If a line intersects two concentric circles with centre O at A, B, C and D, as shown

below,
Then
(a) AB = CD
(b) AB > CD
(c) AB < CD
(d) None of the above
9. Area of shaded portion in the following figure is: 1

(a) a2 + b2
(b) 2ab
(c) (a + b)
(d) a +ab2
10. 1

In the following figure Area of parallelogram ABCD is 40 cm2 . What is the area of
Rectangle BEDF?
(a) 20 cm2
(b) 24 cm2
(c) 28 cm2
(d) 32cm2
11. (i) Area of a circle whose circumference is equal to the perimeter of a square of 1
side 11 cm is:
(a) 231 cm2
(b) 140 cm2
(c) 77 cm2

4
 (d) 154 cm2
OR
2
(ii) Area of a rhombus is 156 cm and one of its diagonal is 13 cm. Its other
diagonal is:
(a) 12cm
(b) 24 cm
(c) 36 cm
(d) 48 cm
12. If sin A + sin2A be equal to 1, then what is the value of cos2A + cos4A? 1
(a) 1
(b) ½
(c) 2
(d) 3
13. (i) Value of (sin A + cos A)2 – 2 sin A cos A is equal to 1
2 2
(a) 0 (b) 1 (c) 2 (d) sin A - cos A
OR
(ii) If cos X = ⅔ then tan X is equal to:
(a) 5/2
(b) √(5/2)
(c) √5/2
(d) 2/√5
14. In ∆ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. The value of tan C is: 1
(a) 12/7
(b) 24/7
(c) 20/7
(d) 7/24
15. A card is drawn from a well shuffled deck of 52 playing cards. Find the probability that it 1
is of red colour
(a) 0.1
(b) 0.3
(c) 0.5
(d) 1.0
16. If P( E ) + P(𝐸 ) = y , value of y3 – 4 is 1
(a) 4
(b) 3
(c) – 3
(d) 0
17. Two different coins are tossed at the same time. How many outcomes are possible? 1
(a) 2
(b) 4
(c) 6
(d) 8

5
 Q.No. 18 to 27 are Objective Questions (2 marks each):
Some of these questions have 4 sub-parts. You have to do any 2 sub-parts out of 4 sub-
parts in such questions.
18. Fill in the blanks: (Attempt any two parts from following questions (i) to (iv)) 1x2
(i) Two factors (x+3) and ______ are obtained on factorising x2 + 8x + 15 .
(ii) The value of the polynomial 2x + 3x2 – 4 at x = 0 is _______
(iii) If p(x) = 0 is a quadratic equation, then p(x) is a polynomial of degree _____
(iv) The common difference of the A.P. 3, 1, –1, –3 ... is ___________
19. Match column –I statement with the right option of column - II 1x2
Column –I Column - II
2
(i) roots of 4x + 4√3x + 3 = 0 are P. real and distinct
2 Q. not real
(ii) roots of 2x + 5x + 5 = 0 are
R. real and equal
20. Write TRUE for correct statement and FALSE for incorrect statements: 1x2
(i) Graphically, the pair of equations 6x – 3y + 10 = 0 and 2x – 9y + 10 = 0 represents two
lines which are intersecting at exactly one point
2 2
(ii) One of the factors of (25x – 1) + (1 + 5x) is (5x+1).
21. Write the factorised form of following algebraic expression. (Attempt any two parts 1x2
from following questions(i) to (iv))
(i) x2 + 2xy + y2 = ……………………..
(ii) x2 - 2xy + y2 = ……………………..
(iii) x3 – y3 – 3x2y + 3xy2 = ……………………..
(iv) x3 – y3 = ……………………..
22. Read the passage and answer the questions that follow it. (i to ii) 1x2
Seema has in her kitchen 16 spoons, 4 serving spoons, 19 quarter plates, 22
full plates, 11 forks, 2 lighters and 36 boxes. Some of these boxes contain
spices, 7 of the boxes contain pulses and 6 boxes contain rice. Two boxes
have ghee and three boxes have oil in them. In this way full variety of things
are there in Seema’s Kitchen.
(i) What percentage of boxes of rice are there in the kitchen?
(a) 7
(b) 50/3
(c) 100/3
(d) 100/7
(ii) What percentage of boxes of spices are there in the kitchen?
(a) 7
(b) 14
(c) 50
(d) 100/3
23. Fill in the blanks: (Attempt any two parts from following questions (i) to (iv)) 1x2
(i) Angle in the same segment of a circle are __________.
(ii) If the sum of a pair of opposite angles of a quadrilateral is 180º, the
quadrilateral is _________.
(iii) Only _______ tangents can be drawn to a circle from an external point.
(iv) If angle between two tangents drawn from a point P to a circle of radius a
and centre O is 90°, then OP = ______

6
24. Write TRUE for correct statement and FALSE for incorrect statements: 1x2
(i) To draw a pair of tangents to a circle which are inclined to each other at an
angle of 30°, it is required to draw tangents at the end points of those two
radii of the circle, the angle between which is 140°.
(ii) Using ruler and compass it is possible to construct an angle of 25°.
25. Fill in the blanks: (Attempt any two parts from following questions (i) to (iv)) 1x2
(i) Ratio of area of a circle of radius ‘r’ to its circumference is _______
(ii) Ratio of area of a square of side ‘a’ to its perimeter is _______
(iii) A sphere of radius 3 cm is inscribed in a cylinder. The height of the cylinder
= ______
(iv) A room is in the shape of cube of side ‘a’. The area of four walls of the
room which needs to be painted is ______
26. Read the passage and answer the questions that follow it. (i to iv) 1x2
Two towers of equal heights are standing opposite each other on either side
of the road which is 100 m wide. From a point between them on the road the
angles of elevation of the top of towers are 30 o and 60o, respectively.
Attempt any two parts from following questions(i to iv):
(i) Distance of the point from the tower whose top has the angle of elevation of 30 o is:
(a) 20 m
(b) 25 m
(c) 50 m
(d) 75 m
(ii) Distance of the point from the tower whose top has the angle of elevation of 60 o is:
(a) 20 m
(b) 25 m
(c) 50 m
(d) 75 m
(iii) The height of the tower is:
(a) 20√3 m
(b) 25√3 m
(c) 50/√3 m
(d) 75/√3 m
(iv) What will be the height of the tower if point is exactly at the midpoint of the line
segment joining the foot of the towers and from the point, the angle of elevation
for the top of both the towers is 45o?
(a) 20 m
(b) 25 m
(c) 50 m
(d) 75 m
27. Read the passage and answer the questions that follow it. (i to iv) 1x2

Probability is that branch of mathematics which deals with the measure of

uncertainty in various phenomenons that gives several results/outcomes instead of
a particular one.
Sample space is the collection of all possible outcomes of a random experiment

7
 whereas event is some specific or a set of specific outcomes.
Probability of an event is denoted by P(E).
Let us consider a random experiment is in which two dice are thrown
simultaneously and the sum of the numbers appearing on them is noted.
Attempt any two parts from following questions(i to iv):
(i) The number of all possible outcomes in the sample space corresponding to this
experiment are _____
(ii) The number of outcomes related to the event (E) that sum of the numbers
appearing on the two dice is 7 are_____
(iii) The value of P(E) is _____
(iv) The value of P(𝐸 ) is ______
28. Read the passage and answer the questions that follow it. (i to vii) 1x5
As a part of a campaign a huge balloon with message of awareness on “Say no to
Drugs” was displayed from the terrace of a tall building. It was held by strings of
length 12 m each and inclined at an angle of 60o at the point where it was tied as
shown in figure. A sparrow bird sits at a point S on the balloon.
S
Say No
O
to
A
Drugs B

60o

P
Attempt any five parts from following questions(i to vii):
(i) ΔABP is
(a) Equilateral triangle
(b) Isosceles triangle
(c) Scalene triangle
(d) can be Isosceles or scalene triangle
(ii) What is the length of AB?
(a) 9 m
(b) 12 m
(c) 8 m
(d) 18 m
(iii) Find measure of reflex ∠𝐴𝑂𝐵
(a) 60o
(b) 120o
(c) 80o
(d) 240o
(iv) What is the measure of ∠𝐴𝑆𝐵
(a) 60o
(b) 120o
(c) 80o
(d) 240o

8
(v) Find the radius of balloon
(a) 3√3 m
(b) 4√3 m
(c) 6√3 m
(d) 8√3 m
(vi) Find the distance between O and P
(a) 3√3 m
(b) 4√3 m
(c) 6√3 m
(d) 8√3 m
(vii) What is the measure of ∠𝑂𝐴𝐵
(a) 15o
(b) 30o
(c) 45o
(d) 60o

SECTION B
Q. Questions Marks
No.
29. Construct a tangent to a circle at any point on it when radius of the circle is 3cm. 2
30. The surface area of a cube is 294 cm2. Find its volume. 2
Or
From a circular disc of diameter 8 cm, a square of side 1.5 cm is removed. Find
the area of the remaining portion of the disc. (Use 𝜋 = 3.14)
31. Find the value of k so that the quadratic equation 2x²+ kx +3=0 has equal roots. 2
32. Find the sum of all natural numbers upto 125 which are divisible by 5. 2
OR
How many terms of the AP 25,28,31,34, ……. are needed to give the sum 1070?
33. A refrigerator is sold for Rs 22000 cash or Rs. 10000 cash down payment and Rs 2
12600 after six months. Find the rate of simple interest charged under the
instalment plan.
34. A second hand car is sold for Rs 50000 cash down payment along with two equal 2
monthly instalment of Rs 102010 each. If the dealer charges interest at the rate of
12% p.a. compounded monthly under the instalment plan, find the cash price of
the car.
35. If point C (-2,-1) divides the line segment joining points A(1,5) and B in the ratio 2
3: 4, then find the coordinates of B.
OR
Find the centroid of the triangle whose vertices are (5,-1), (- 3,-2) and (-1,8).

9
36. A circus artist climbs a 16 m long rope whose one end is tied to the ground and 2
the other end to the top of a vertical pole. If the angle of elevation made by the
rope with the ground level is 30°, then find the height of the pole.
OR
A balloon is connected to a meterological ground station by a cable of length 100
m inclined at 60o to the horizontal. Find the height of the balloon from the ground
assuming that there is no slack in the cable.
37. By what number the median will increase if 25 is removed from the data 20, 24, 2
25, 28, 30, 31, 33, 38?
38. If the 7th term of an AP is 27 and the 11th term is 43, then find its 20th term. 3
39. Sum of two natural numbers is 12 and sum of their squares is 74. Find the greater 3
number.
Or
The product of digits of a two digit number is 12. When 9 is added to the number,
the digits interchange their places. Determine the number.
40. In a ΔABC with vertices A (6,4), B (5,-2) and C (7,-2), find the length of median 3
through A.
41. A solid is composed of a cylinder with hemispherical ends. If the whole length of 3
the solid is 90cm and the diameter of the hemispherical ends is 28cm, then find
the surface area of the solid. (Use π = )
Or
A cone, a cylinder and a hemisphere are of the same base and same height. Find
the ratio of their volumes.
42. 3
Find the value of
43. If the mean of the following distribution is 30, their find the value of p 5
Class interval 0-10 10-20 20-30 30-40 40-50
Frequency 4 8 10 p 13
Or
(a) If 𝑥̅ represents the mean of n observations x 1, x2,…… xn, then show that
∑ (𝑥 − 𝑥̅ ) = 0.
(b) If each observation of a data is increased by 'a', then show that its mean also
increases by ‘a’.
44. Construct a ΔABC in which BC= 6cm, AB= 6cm and median AD = 4cm. 5
OR
Construct a triangle whose perimeter is 9.5 cm and base angles are 60 o and 45o

10
 SAMPLE QUESTION PAPER
Mathematics (211)
Making Scheme
SECTION A
Question Correct Explanation Marks
Number option
Q.No. 1 to 17 are MCQs
1. (i) (d) (i) a+ b = 12 1
(a + b )2 = 144
a2 + b2 + 2ab = 144
a2 + b2 + 2X 22 = 144
a2 + b2 = 100
Option (d)
OR OR
(ii) (a) (ii) 5x2- 13x- 6 = 5x2 + 2x- 15x – 6
= x( 5x + 2) -3 (5x +2)
= (5x + 2) ( x – 3)
2. (i) (d) (i) Verify by putting the values of x and y in the equation. 1
OR OR
(ii) (c) (ii) a1/a2= b1/b2 = c1/c2= -1/4.
Therefore equations have infinitely many solutions
Option ( c )
3. (i) (b) (i) S.I = (1632 X 25 X 5 X100)/ (100 X 4) = ₹510 1
Option (b)
OR OR
(ii) (d) (ii) 968 = P ( 1 + 10/100)2
∴ P = 800
Option (d)
4. (i) (b) (i) Let P = 100y. 1
Then Q = 60 y
= 40y / 60 y = 2/3
Q – P = (2/3)P = 66.66 P%
Therefore Q is 66.66 % more than P
Option (b)

OR OR
(ii) (a) (ii) The percentage of reduction is calculated with the new price of
the oil.
Let the price of cooking oil = Rs. 100

Increase in price =25% of 100 = 25

∴ Increased price = 100 +25=125
Required percentage of Reduction
25
× 100% = 20%
125
Option (a)
5 (i) (c) (i) Angle measure of minor arc BC = 360 –(84 + 140) = 136 1
Option (c)
OR OR
(ii) (d) (ii) Central angle of minor arc LB = 120

11
  ∠LAB = 60 (As the angle subtended by an arc at centre is double
the angle subtended by it any point on the remaining part of the
circle)
∠ ALB = 180 -(50 +60) = 70
x = ∠ ALB= 70 (Angles in alternate segment)
Option (d)
6 (a) 𝐶𝑃 = 𝐶𝑅 = 8 1
𝐴𝑃 = AQ = AB – BQ = 15 - BR = 15 – 4 = 11
AC = AP + CP = 11 + 8 = 19
7. (i) (a) 1
OR
(ii) (b)
8 (a) OM ⊥AB. 1
Therefore AM = DM and BM = CM
Or AB = CD
Option (a)
9 (b) 1
10 (c) 1
11 (i) (d) (i) Perimeter of square of side 11 cm = 44 cm 1
Therefore circumference of circle = 44 cm
r = 7 cm
( ) 2
Or = 154 cm
Option (d)
OR
OR
(ii) Area = (13d)/2
(ii) (b)
(13d)/2 = 156
d = 24 cm
Option (b)
12 (a) sinA + sin2A = 1 1
sinA = 1 – sin2A = cos2A
sin2A = cos4A
1- cos2A= cos4A
1 = cos2A + cos4A
Option (a)
13 (i) (b) (i) (sin A + cos A)2 – 2 sin A cos A 1
= sin2 A + cos2 A+ 2 sin A cos A – 2 sin A cos A
=1
OR OR
(ii) (c)

(ii)

12
 PM2 = (3K )2 = (2k)2
PM = 5k
PM 5k
tan X  
OM 2k
5

2
Option (c)
14 ( b) 1
15 (c) 1
16 (c) Since P( E ) + P(𝐸 ) = 1 1
∴y=1
⟹. y3 – 4 = 13 – 4 = - 3
17 (b) 4 possible outcomes (H,H), (H,T), (T,H), (T,T)
Q.No. 18 to 27 are Objective type Questions of 2 marks
18 (i) (x+5) 1x2
(ii) –4
(iii) 2
(iv) –2
19 (i) – R, (ii) – Q 1x2
20 (i) F 1x2
(ii) T
21 (i) x + 2xy + y2 = (x+y)2
2
1x2
(ii) x2 - 2xy + y2 = (x-y)2
(iii) x3 – y3 – 3x2y + 3xy2 = (x-y)3
(iv) x3 – y3 = (x+y)(x2 - xy + y2)
22 (i) (b) 50/3 1x2
(ii) (c) 50
23. (i) equal 1x2
(ii) cyclic
(iii) two
(iv) √2𝑎
24. (i) F 1x2
(ii) F
25. (i) r:2 1x2
(ii) a:4
(iii) 6 cm
(iv) 4a2
26. (i) (d) 75 m 1x2
(ii) (b) 25 m
(iii) (b) 25√3 m
(iv) (c) 50 m
27. (i) 36 1x2
(ii) 6
(iii)
(iv)
28. (i) (a) equilateral triangle 1x5
(ii) (b) 12 m
o
(iii) (d) 240

13
 (iv) (a) 60o
(v) (b) 4√3 m
(vi) (b) 8√3 m
o
(vii) (b) 30

SECTION B
Marks
29 Steps of construction 1 2
1. Draw a circle with radius 3cm
2. Draw ∠OAB = 90°
AB is the required tangent

30 6a2= 294 ⇒ a = 7 1 2
Volume = 73 = 343 cm3 1
OR OR
Remaining area = 𝜋𝑟 − 𝑎 1
= 3.14  4 4 – (1.5)2 = 47.74 cm2 1
31 2𝑥 + 𝑘𝑥 + 3 = 0 2
Discriminate = 𝑘 − 4 × 2 × 3 = 𝑘 − 24 ½
Roots are equal ⇒ 𝑘 − 24 = 0 ½
𝑘=± ±√24 = ±2√6 1
32 5, 10, 15, --------- 125 ½ 2
a=5, d=5 ½
tn =125 = 5+(n-1)5⇒ n=25 1
25
𝑆 = (5 + 125) = 1625
2
OR OR
a = 25, d = 3, Sn = 1070
𝑛
𝑆 = ((2𝑎 + (𝑛 − 1)(3)) ½
2
𝑛
1070 = ((2(25) + (𝑛 − 1)(3))
2
3n2 + 47n – 2140 = 0 ½
n = 20 1

14
33 Cash price = Rs. 22000 2
Cash down payment = Rs. 10000
Balance payment = Rs. 12000 1
12000 × 6 × 𝑟 1
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 600 =
100 × 12
𝑟 = 10
34 12 2
102010 = 𝑃 1 + 𝑃 = 𝑅𝑠. 100000 ½
1200
12 ½
102010 = 𝑃 1 + 𝑃 = 𝑅𝑠. 101000
100 ½
Cash price of the car = 50000 +100000+101000 = Rs. 251000 ½

35 4 × 1 + 3𝑥 2
= −2 𝑥 = −6
7
20 + 3𝑦 1½
= −1 𝑦 = −9
7
Point B is ( -6, -9) ½
OR Or
centroid of the triangle whose vertices are (5,-1), (- 3,-2) and (-
1,8).
𝑥 +𝑥 +𝑥 5 + (−3) + (−1) 1
𝑥= = = 1
3 3 3
𝑦 +𝑦 +𝑦 −1 + (−2) + (8) 5
𝑦= = = 1
3 3 3
1 5
,
3 3

15
36
1 2

𝐴𝐵 1 1
= 𝑠𝑖𝑛 30° =
16 2
⇒ AB = 8m
OR OR

𝐴𝐵 √3
= 𝑠𝑖𝑛 60° =
100 2
⇒ AB = 50
50√3m 1

37 20, 24, 25, 28, 30, 31, 33,38 2

Median = = 29 1
When 25 is removed median = 30 ½
Median increases by 1 ½

38. t7 = a+6d = 27 1 3
t11 = a + 10d=43 1
⇒ a=3, d= 4
t20 = a+ 19d = 3+19x4=79 1
39. 𝑥
𝑥, 12 − 𝑥 ½ 3
𝑥² + (12
12 − 𝑥) = 74 1
x²- 12x+35=0 1
⇒ x=7, 5 ½
greater number is 7
40. 𝐷= , = (6, −2)) 1 3
1
𝐴𝐷 = (6 − 6
6) + (4 + 2) = 6 𝑐𝑚 1

16
41. 3
1

h = (90-2x14)
2x14) = 62 cm
Surface area
= 2πrh +4πr2
= 2πr(h+2r) 1
=2 × × 14 (62 + 28)
7920 cm 2 1
OR
Let height of cone = h Or
Height of cone = height of cylinder = height of hemisphere
Height of cone = height of cylinder = diameter of hemisphere
h = h = 2r ½
Ratio of volumes = V1 : V2 : V3
= 𝜋𝑟 ℎ ∶ 𝜋𝑟 ℎ ∶ 𝜋𝑟
1 4 1½
= 𝑟 (2 2𝑟) ∶ 𝑟 (2𝑟) ∶ 𝑟
3 3
= 2:3:4
1
42. 4 𝑐𝑜𝑠 30 + 𝑠𝑖𝑛 45 − 3𝑡𝑎𝑛 60 3
2𝑐𝑜𝑠 60 𝑠𝑖𝑛 60 + 𝑐𝑜𝑡45
√
(√ ) 1
√
=
√

= 1

= = -4 1

43. Class x −𝑎 5.
fi xi 𝑢 = f i ui
Interval ℎ
0-10 4 5 -2 -8
10-20 8 15 -1 -8 1
20-30 10 25 0 0 1
30-40 p 35 1 p 1
40-50 13 45 2 26
35+p p+10

𝑚𝑒𝑎𝑛 = 25 + × 10 = 30

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 𝑝 + 10 1
× 10 = 5𝑝 = 15
35 + 𝑝
Or 1
(a) As 𝑥̅ is the mean of n observations x1, x2,…… xn
𝑥 + 𝑥 + 𝑥 + ⋯……𝑥 Or
𝑥̅ =
𝑛
𝑥 + 𝑥 + 𝑥 + ⋯ … … 𝑥 = 𝑛𝑥̅ 3
∑ 𝑥 = 𝑛𝑥̅
Now to show: ∑ (𝑥 − 𝑥̅ ) = 0
L.H.S. = ∑ (𝑥 − 𝑥̅ ) = ∑ 𝑥 − ∑ 𝑥̅
= 𝑛𝑥̅ − 𝑥̅ × 𝑛 = 0 = R.H.S.

(b) As 𝑥̅ is the mean of n observations x1, x2,…… xn

𝑥 + 𝑥 + 𝑥 + ⋯……𝑥
𝑥̅ =
𝑛
let 𝑋 is the mean of n observations (x1+a), (x2+a), ……
(xn+a)
(𝑥 + 𝑎) + (𝑥 + 𝑎) + ⋯ … … + (𝑥 + 𝑎)
𝑋= 2
𝑛
⋯…… ⋯….
𝑋= +
=𝑥̅ + 𝑎

44.

3 5

Steps of Construction
1. Draw a line segment BC= 6cm ½
2. Bisect BC at D ½
3. Draw ∆ ABD such that AB=6cm AD=4cm ½
4. Draw AC ½
ΔABC is the required triangle
OR
To construct a triangle whose perimeter is 9.5 cm and base
angles are 60o and 45o
we go through the following steps: 2
Step 1: Draw XY = 9.5 cm

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 YXP = 30o [which is 1/2 × 60o]
Step 2: At X, construct ∠YXP
XYQ = 22½o [which is 1/2 × 45o]
Step 3: At Y, construct ∠XYQ
Let XP and YQ intersect A.
Step 4: Draw right bisector of XA intersecting XY at B.
Step 5: Draw right bisector of YA intersecting XY at C.
Step 6: Join AB and AC.

ΔABC is the required triangle.

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