NCERT Solutions Class 11th Mathematics. FREE Flip-BOOK by Study Innovations. 614 Pages

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Q2 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers): (x + a)
Answer :
Let f(x) = x + a. Accordingly,
By first principle,

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Q3 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :
By Leibnitz product rule,

Q4 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers): (ax + b) (cx + d)2

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Answer :

Let
By Leibnitz product rule,

Q5 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :

Let
By quotient rule,

Q6 :

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Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):

Answer :

By quotient rule,

Q7 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :
Let
By quotient rule,

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Q8 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :
By quotient rule,

Q9 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):

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Answer :
By quotient rule,

Q10 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :

Q11 :

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Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :

Q12 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers): (ax + b)n
Answer :
By first principle,

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Q13 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers): (ax + b)n (cx + d)m
Answer :
Let
By Leibnitz product rule,

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Therefore, from (1), (2), and (3), we obtain

Q14 :

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Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers): sin (x + a)
Answer :
Let

By first principle,

Q15 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers): cosec x cot x

Answer :

Let
By Leibnitz product rule,

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By first principle,

Now, let f2(x) = cosec x. Accordingly,
By first principle,

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From (1), (2), and (3), we obtain

Q16 :

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Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :
Let
By quotient rule,

Q17 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :
Let
By quotient rule,

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Q18 :
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-
zero constants and m and n are integers):
Answer :
Let
By quotient rule,

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a— —i

It is not evident from the sentence as to whom ‘he’ is referre
This is a question, which also contains ‘she’, and it is not evident as to who ‘she’ is. Hence, it is not a statement.

x
x

x

x
(i) The negation of the first statement is “the number x is a rational numberâ€.

(ii) The negation of the first statement is “the number x ational numberâ€. This means that the
x

p r
q

p
q

p
q
r

pq

xx xx

(i) Here, the connecting word is ‘and’.

p
q
(ii) Here, the connecting word is ‘or’.

p
q
(iii) Here, the connecting word is ‘and’.

p
q
(iv) Here, the connecting word is ‘and’.

px x x
qx x x

xx x
(i) The quantifier is “There existsâ€.

(ii) The quantifier is “For everyâ€. x

xx
(iii) The quantifier is “There existsâ€.

xyyx xy
xyyx
xy

xy x y ≠y x

State whether the “Or†used in the following statements is “exclusive “or†inclusive. Give

(i) Here, “or†is exclusive because it is not possible for the Sun to rise and
(ii) Here, “or†is inclusive since a person can have both a ration card and a passport to apply for a driving

(iii) Here, “or†is exclusive because all integers cannot be both positive and negative.

Rewrite the following statement with “if then†in five different ways conveying the same meaning.
If a natural number is odd, then its square is also odd.

xx

x x
x
x
x

xx
xx
xx

Write each of the following statement in the form “if thenâ€.

p: “If x xx x is 0†is true by

p: “If x xx x is 0â€.
q
qx x x

rx

p r
x
q

∴ x x

xx x
⇒ x

x

r

p p
xx
x x
x x
x p
xx
xx
xx

x

p r q
r
r x
∠r x x

x â‰

x x
q
∴ x x + 4) â‰
⇒ x x â‰

∠r ⇒ ∠q

p

Show that the statement “For any real numbers a b a b a b†is not true by giving a

The given statement can be written in the form of “if then†as follows.

ab ab ab

pa b ab

qa b

p, then ∠q

a b ab a ≠b

ab

ab

∴ a b
a ≠b

p If x is an integer and x2is even, then x is also even.

px p x x r q
qx q x x x

rx x
q
x
p
x

p
qx

(i) The given statement is of the form “if q râ€.
q
r q, then ∠r

p

qx p
x
x
x
x

x

p y x y, then –x < –y
q
r
sx

t

p
q

ab

xy

r
xy
⇒ –x < –y

s

t

p xx
q
r x xx
s xx

p x
x
xx
q x

r
x

s
x

p
q
r

p

r
r

Write each of the statements in the form “if p qâ€.
p
q
r

p
q
r

Re write each of the following statements in the form “p qâ€.
p
q
r

p 25 is a multiple of 5.
q: 25 is a multiple of 8.
Write the compound statements connecting these two statements with “And†and “Orâ€. In both

The compound statement with ‘And’ is “25 is a multiple of 5 and 8â€.
The compound statement with ‘Or’ is “25 is a multiple of 5 or 8â€.

p nn
qn
p
p bcde

n q n n
n n
n n
n
n

n
⇒n n

n nn

p: If triangle is equiangular, then it is an obtuse angled triangle.

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NCERT Solutions for Class 11 Maths Chapter 15

Statistics Class 11

Chapter 15 Statistics Exercise 15.1, 15.2, 15.3, miscellaneous Solutions

Exercise 15.1 : Solutions of Questions on Page Number : 360
Q1 :
Find the mean deviation about the mean for the data
4, 7, 8, 9, 10, 12, 13, 17

Answer :
The given data is
4, 7, 8, 9, 10, 12, 13, 17

Mean of the data, are

The deviations of the respective observations from the mean
–6, – 3, –2, –1, 0, 2, 3, 7

The absolute values of the deviations, i.e. , are

6, 3, 2, 1, 0, 2, 3, 7

The required mean deviation about the mean is

Q2 :
Find the mean deviation about the mean for the data
38, 70, 48, 40, 42, 55, 63, 46, 54, 44

Answer :
The given data is
38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Mean of the given data,

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The deviations of the respective observations from the mean are
–12, 20, –2, –10, –8, 5, 13, –4, 4, –6

The absolute values of the deviations, i.e. , are
12, 20, 2, 10, 8, 5, 13, 4, 4, 6
The required mean deviation about the mean is

Q3 :
Find the mean deviation about the median for the data.
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Answer :
The given data is
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Here, the numbers of observations are 12, which is even.
Arranging the data in ascending order, we obtain
10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

The deviations of the respective observations from the median, i.e. are

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–3.5, –2.5, –2.5, –1.5, –0.5, –0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

The absolute values of the deviations, , are

3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

The required mean deviation about the median is

Q4 :
Find the mean deviation about the median for the data
36, 72, 46, 42, 60, 45, 53, 46, 51, 49

Answer :
The given data is
36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Here, the number of observations is 10, which is even.
Arranging the data in ascending order, we obtain
36, 42, 45, 46, 46, 49, 51, 53, 60, 72

The deviations of the respective observations from the median, i.e. are
–11.5, –5.5, –2.5, –1.5, –1.5, 1.5, 3.5, 5.5, 12.5, 24.5

The absolute values of the deviations, , are

11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5

Thus, the required mean deviation about the median is

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Q5 :

Find the mean deviation about the mean for the data.

xi 5 10 15 20 25
6 3 5
fi 7 4
63
Answer : 16
6
xi fi fi xi 18
55
5 7 35 9 158
10 4 40 4
15 6 90 1
20 3 60 6
25 5 125 11
25 350

Q6 :
Find the mean deviation about the mean for the data

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xi 10 www.ncrtsolutions.in 70 90
fi 4 16 8
30 50
24 28 160
480
Answer :
0
xi fi fi xi 320
320
10 4 40 40 1280
30 24 720 20
50 28 1400 0
70 16 1120 20
90 8 720 40
80 4000

Q7 :

Find the mean deviation about the median for the data.

xi 5 7 9 10 12 15

fi 8 6 2 2 2 6

Answer :

The given observations are already in ascending order.

Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table.

xi fi c.f.

58 8

76 14

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92 16

10 2 18

12 2 20

15 6 26

Here, N = 26, which is even.

Median is the mean of 13th and 14th observations. Both of these observations lie in the cumulative frequency 14, for
which the corresponding observation is 7.

The absolute values of the deviations from median, i.e. are 023 5 8
622 2 6
|xi – M| 2 046 10 48
fi 8
16
fi |xi – M|

and

Q8 :
Find the mean deviation about the median for the data

xi 15 21 27 30 35
fi 3 5 6 7 8

Answer :

The given observations are already in ascending order.

Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table.

xi fi c.f.

15 3 3

21 5 8

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27 www.ncrtsolutions.in 14
30 21
35 6 29
7
Here, N = 29, which is odd. 8

observation = 15th observation
This observation lies in the cumulative frequency 21, for which the corresponding observation is 30.
∴ Median = 30

The absolute values of the deviations from median, i.e. are 9 305

|xi – M| 15

fi 3 5 678
45 45 18 0 40
fi |xi – M|

∴

Q9 : Number of persons
Find the mean deviation about the mean for the data. 4
8
Income per day 9
0-100 10
7
100-200 5
200-300 4
300-400 3
400-500
500-600
600-700
700-800

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Answer : Number of persons fi Mid-point xi fi xi 308 1232
The following table is formed. 208 1664
4 50 200 108 972
Income per day 8 150 1200
9 250 2250 8 80
0 – 100 10 350 3500 92 644
100 – 200 7 450 3150 192 960
200 – 300 5 550 2750 292 1168
300 – 400 4 650 2600 392 1176
400 – 500 3 750 2250 7896
500 – 600 50
600 – 700 17900
700 – 800

Here,

Q10 : Number of boys
Find the mean deviation about the mean for the data 9
13
Height in cms 26
95-105 30
105-115 12
115-125
125-135
135-145

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