MBOSE Class 11 Sample Question Paper 2026 Statistics

Meghalaya Board of School Education

SAMPLE
PAPERS

Statistics
Class XI
Full Marks:80
Time: 3 hours
The igures in the margin indicate full marks for the questions
General Instructions:
(i) Write all the answers in the Answer Script.
(ii) Attempt Part -A (Objective Questions) serially.
(iii) Attempt all parts of a question together at one place.
(PART: A- OBJECTIVE)
(Marks:32)

SECTION- I
(Marks :16)
1. Choose and write the correct answer: 1X8 = 8.
(a) The number of combinations of ‘n’ different things taken ‘r’ at a time in which two
particular things never occur is
(i) n-2Cr

(ii) n-2Cr-2

(iii) nCr-2
(iv) 2! n-2Cr-2
(b) The value of nC0+ nC1+ nC2+……………………………+ nCn=
(i) 2n
(ii) n2
(iii) 2n
(iv) n
(c) If A and B are two events such that P(A) = , P(B) = and P(AꓵB) = , then P (𝐵 𝐴) is
(i)
(ii)
(iii)
(iv)
(d) If 𝑃 𝐴 𝐵 = 𝑃(𝐴) then the events A and B are
(i) Complementary
(ii) Mutually Exclusive
(iii) Exhaustive
(iv) Independent.
(e) If 𝑓(𝑥) = 3 and h =1 then the value of ∆𝑓(𝑥) is
(i) 3x+1
(ii) 32x
(iii) 2.3x
(iv) 0
(f) In a ‘more than’ ogive, the vertical axis represents:
(i) Frequencies.
(ii) Cumulative frequencies.
(iii) Class limits.

(iv) Midpoints.
(g) For a symmetrical distribution, the relationship between mean, median and mode is
(i) Mean = Mode = Median
(ii) Mean > Mode > Median
(iii) Mean < Mode < Median
(iv) Mean =3 Median -2Mode
(h) If 𝑙𝑜𝑔 (𝑥 + 2) = 3, then the value of x is
(i) 11
(ii) 25
(iii) 12
(iv) 29.

2. Fill in the blanks: 1X4 = 4.
a) E(C)= _____________ when C is a constant.
b) The value of 10 is _______________ .
c) If A and B are independent events then P (𝐴̅ ∩ 𝐵) = __________________ .
d) The relationship between variance and standard deviation of a variable X is ___________.

3. State whether the following statements are True or False. 1X4 =4.
a) In Lagrange interpolation, the values of the arguments must be equidistance.
b) The middle term in the expansion of (𝑥 + 𝑦) if n is even is 𝑇 .

c) Two coins are tossed simultaneously. Then the probability of getting at least one head is .
d) Median is not affected by extreme values.

SECTION – II
(Marks :16)

4. Answer the following questions (any eight): 2 X8 =16.
a) Find ‘n’ if n-1P3: nP4 = 1 :9.
b) Find the middle term in the expansion of 𝑥 − .
c) If 𝑓(𝑥) = (3𝑥 + 5), ind ∆𝑓(𝑥) when the interval of differencing is unity.
d) Evaluate ∆ 𝑙𝑜𝑔𝑥.
e) The probability of solving a problem by A is and that of solving by B is . What is the
probability that the problem is solved.
f) De ine primary and secondary data.
g) De ine Standardized Death rate.
h) Write three uses of vital statistics.
i) Differentiate between a frequency polygon and an ogive.
j) Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly
one ace in each combination.
(PART: B – DESCRIPTIVE)
(Marks :48)

Answer four questions, taking atleast one from each group.
Group-A
5. a) Find 𝑎 if the 17th and 18th terms of the expansion (2 + 𝑎) are equal. 5.
b) If 1 ≤ 𝑟 ≤ 𝑛, prove that nCr+nCr-1 = n+1Cr. 5.

c) If + = , ind x. 2.
! ! !

6. a) Prove that ∆ (𝑎𝑒 ) = (𝑒 − 1) 𝑎𝑒 where 𝑎 is constant. 5.
b) Write down Newton’s forward interpolation formula, and use this to ind the value of 𝑓(14)
from the following data. 3+4=7.
x: 12 16 20 24
f(x): 1 5 10 17

Group-B

7. a) De ine with example. 3X2=6.
(i) Sample space.
(ii) Mutually exclusive events.
(iii) Independent Events.
b) Two dice are thrown. Find the probability that
(i) The sum of the numbers is 7.
(ii) The product of the numbers is at least 10. 3+3=6.

8. a) State and prove total probability theorem of two events A and B. 2+3=5.
b) A bag contains 4 white and 5 black balls. Two balls are drawn at random one by one
(without replacement). Find the probability that both the balls are white. 4.
c) Show that if 𝑃 𝐵 𝐴
𝐴 = 𝑃(𝐵) and 𝑃(𝐵) ≠ 0, then 𝑃 𝐵 = 𝑃(𝐴). 3.

Group-C

9. a) What do you understand by the term “Vital Statistics”? How are such statistics collected and
what are their defects? What suggestion can you give for their improvement? 2+2+2=6.
b) Calculate the Crude Death Rate from the following data for locality A and B, also the
standard death rate for A considering B as standard. 6.

Age Group Locality A Locality B
(in years) Population Death Population Death
(in thousand) (in thousand)
0 - 10 20 500 10 370
10 – 20 10 260 30 650
20 – 40 50 1140 62 1400
40 – 60 30 1050 15 700
Above 60 10 500 3 280

10. a) Explain the term “classi ication” and “tabulation” and point out their importance in a
statistical investigation. 5.
b) Write a short note on Age-Speci ic Death rate 3.
c) Find the median from the following data. 4.

Value: 10 – 14 15- 19 20-24 25- 29 30 – 34 35- 39
Frequency: 5 10 13 20 7 5

Answer Key
(PART: A – OBJECTIVE)
(Marks :32)

SECTION – I
(Marks:16)
1. Choose and write the correct answer: 1X8=8.
a) (i) n-2Cr

b) (iii) 2n
c) (ii)
d) (iv) Independent.
e) (iii) 2.3x
f) (ii) Cumulative frequencies.
g) (i) Mean = Mode=Median
h) (ii) 25

2. Fill in the blanks: 1X4 =4.
a) E(C)=C.
b) .
c) P (𝐴̅ ∩ 𝐵)= P (𝐴̅) . 𝑃(𝐵)
d) Variance = (Standard Deviation)2.

3. State whether the following statements are True or False. 1X4 =4.
a) False.
b) True.
c) False.
d) True.

SECTION-II
(Marks:16)

4. Answer the following questions (any eight): 2X8=16.
a) Given n-1P3 : nP4 = 1 :9.
( )!
( )!
⟹ ! =
( )!
( )! ( )!
⟹( )!
× =
!
( )!
⟹ =
!
⟹ = ∴ 𝑛 = 9.

b) Here n=12 which is even and so there is one middle term in its expansion which is given
by 𝑇 =𝑇 =𝑇

now 𝑇 = 𝑇 =12 C6 (𝑥 ) −
! . . . . .
= 𝑥 . = 𝑥 = 924𝑥
! ! . . . . .

c) Given 𝑓(𝑥) = 3𝑥 + 5, interval of differencing =1
∆𝑓(𝑥) = 𝑓(𝑥 + 1) − 𝑓(𝑥)
Now, 𝑓(𝑥 + 1) = 3(𝑥 + 1) + 5 = 3𝑥 + 8
∴ ∆𝑓(𝑥) = (3𝑥 + 8) − (3𝑥 + 5)
=3

d) ∆ 𝑙𝑜𝑔𝑥 = ∆(∆𝑙𝑜𝑔𝑥) = ∆[log(𝑥 + ℎ) − 𝑙𝑜𝑔𝑥]
= ∆𝑙𝑜𝑔
= 𝑙𝑜𝑔 − 𝑙𝑜𝑔

= 𝑙𝑜𝑔
( )
= 𝑙𝑜𝑔 × = 𝑙𝑜𝑔 ( )

e) Given P(A) = , P(B) = ∴ P(𝐴̅) = , P(𝐵) =
∵ 𝑃(𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑠𝑜𝑙𝑣𝑒𝑑) + 𝑃(𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑛𝑜𝑡 𝑠𝑜𝑙𝑣𝑒𝑑) = 1
⟹ 𝑃(𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑠𝑜𝑙𝑣𝑒𝑑) = 1 − 𝑃(𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑛𝑜𝑡 𝑠𝑜𝑙𝑣𝑒𝑑)
⟹ = 1 − P(𝐴̅) × P(𝐵 )
=1− ×
=1− ×
=1− =

f) Primary data: Collected directly by the investigator (e.g., surveys).
Secondary data: Obtained from existing sources (e.g., reports).
g) Standardized Death Rate adjusts the death rate to a standard population to eliminate the
effect of age structure differences.
h) i)Vital statistics help us in analysing demographic trends in the country.
ii) Vital statistics ill the gap in the census method and supplement it.
iii) Knowledge of vital events makes it easy to know the future population igures with the
help of population projection.
i) Frequency Polygon: Line graph connecting midpoints of class intervals to show frequency
distribution.
Ogive: Cumulative frequency curve (less than or more than type) showing cumulative
frequencies against class boundaries.
j) A pack of 52 cards contains 4 aces and 48 other cards.
Since, each combination must include exactly 1 ace
∴ 1 ace can be selected from 4 aces in 4C1 ways.
Other 4 cards can be selected from the remaining 48 cards (non-ace) in 48C4 ways.
∴ Total number of combinations = 4C1 × 48C4.

PART: B – DESCRIPTIVE)
(Marks :48)
Answer four questions, taking atleast one from each group.
Group-A

5. a) Given, in the expansion of (2 + 𝑎) 17th term and 18th terms are equal. 5.
∴ T17 = T18.

⟹ 50C16 250 -16. 𝑎 = 50C17 250 -17. 𝑎
! !
⟹ 2 𝑎 = 2 𝑎 .
!× ! !× !
! !× !
⟹ × × =𝑎
!× ! !
⟹ ×2=𝑎
⟹ 𝑎 = 1.

b) Refer textbook. Proof of a standard formula 5.

c) Given, + = 2.
! ! !

⟹ =
! !
⟹ × 8! = 𝑥
!
⟹ x = 64.

6. a) L. H.S = ∆ (𝑎𝑒 ) 5.
= ∆ 𝑎(𝑒 −𝑒 )
= 𝑎∆ [∆(𝑒 ) − ∆(𝑒 )]
= 𝑎∆[{𝑒 −𝑒 } − {𝑒 − 𝑒 }]
=𝑎[∆(𝑒 ) − 2∆(𝑒 ) + ∆𝑒 ]
= 𝑎[{𝑒 −𝑒 } − 2{𝑒 −𝑒 } + {𝑒𝑥+ℎ − 𝑒 }]
= 𝑎[𝑒 − 3𝑒 + 3𝑒 −𝑒 ]
=𝑎𝑒 [𝑒 − 3. 𝑒 + 3. 𝑒 − 1]
= (𝑒 − 1) 𝑎𝑒
=R.H.S

b) Newton’s forward interpolation formula – Refer textbook 3+4=7.
2 Part: Since the values of x are equidistant and 14 lies in the beginning of x series, Newton’s
nd

forward interpolation formula can be applied
Here 𝑢 = = 0.5
Difference table
X f(x) ∆𝑦 ∆ 𝑦 ∆ 𝑦
12 1
4
16 5 1
5 1
20 10 2
7
24 17

( ) ( )( )
∵ 𝑦 = 𝑓(𝑥) = 𝑦 + 𝑢 ∆𝑦 + ∆ 𝑦 + ∆ 𝑦
! !
. ( . ) . ( . )( . )
⟹ 𝑦 = 𝑓(14) = 1 + 0.5 × 4 + ×1+ ×1
= 1 + 2 − 0.125 + .0625 = 2.94

Group-B

7. a) (i) Sample space: Set of all possible outcomes of an experiment. 3X2=6.
Example: For a die roll,{1,2,3,4,5,6}.
(ii)Mutually exclusive events: Events that cannot occur together.
Example: Rolling an even number ({2,4,6}) and an odd number ({1,3,5}) on a die.
(iii)Independent events: Occurrence of one does not affect the other.
Example: Tossing a coin (H or T) and rolling a die (1 to 6).
b) Total out comes: 6×6=36.
(i)Sum is 7: 3.
Outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)→6 outcomes.
P(sum=7) = =
(ii)Product is at least 10: 3.
Outcomes: (2,5), (2,6), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4),
(5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6).
Total=19outcomes.
P(product≥10) =

8. a) Refer textbook. 5.
b) Let A= 1st ball drawn is white, B= 2nd ball drawn is white 4.
Hence, 𝑃(𝐴) =
Since the irst ball is not replaced, before the second ball drawn, 𝑃(𝐵) =
Hence the probability that both the balls drawn are white
is 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵) = . = .
c) Given 𝑃 𝐵 𝐴 = 𝑃(𝐵) 3.
( ∩ )
⟹ = 𝑃(𝐵)
( )
⟹ 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵)
( ∩ )
⟹ = 𝑃(𝐴) as 𝑃(𝐵) ≠ 0
( )
⟹𝑃 𝐴 𝐵 = 𝑃(𝐴)
Group-C
9. a) Refer textbook 6.
b) 6.
Age Locality A Locality B
Group Population Death Death Population Death Death 𝑚 𝑃𝐵𝑥
P (Thousand) 𝐷 Rate (Thousand) 𝐷 Rate
𝑃 𝑚 𝑃 𝑚
0 - 10 20 500 25 10 370 37 250
10 – 20 10 260 26 30 650 21.67 780
20 – 40 50 1140 22.8 62 1400 22.58 1413.6
40 – 60 30 1050 35 15 700 46.67 525
Above 60 10 500 50 3 280 93.33 150
Total 120 3450 120 3400 3118.6

∑ 𝐷𝐴𝑥
C.D.R for A=∑ × 1000 = × 1000 = 28.75 per thousand
𝑃𝐴
𝑥
∑ 𝐷𝐵𝑥
C.D.R for B= ∑ × 1000 = × 1000 = 28.33 per thousand
𝑃𝐵
𝑥
∑ 𝑚𝐴𝑥 𝑃𝐵𝑥 .
Since locality B is taken as standard, STDR for A = ∑𝑩 𝑃𝐵𝑥
× 1000 = × 1000
= 25.99 per thousand.

10. a) Refer textbook. 5.
b) Refer textbook. 3.
c) Here the class intervals are not continuous. So irst all of these are made continuous by
subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. 4.

Class Interval f Cumulative Frequency
9.5 – 14.5 5 5
14.5 - 19.5 10 15
19.5 - 24.5 13 28
24.5 – 29.5 20 48
29.5 – 34.5 7 55
34.5 – 39.5 5 60
N=60

Median =𝐿+ ×ℎ

= 24.5 + × 5 = 24.5 + 0.5 = 25

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