Ncert Solutions Maths class 11th

68. Given, f (x) = (x + cos x) (x tan x)

So, f?(x) = (x + cos x) $\frac{d}{dx}-tanx)+\frac{d}{dx}(x+cosx)·(x-tanx)$

$=(x+cosx)\left(\frac{dx}{dx}-\frac{d}{dx}tanx\right)+\left(\frac{d}{dx}+\frac{dcosx}{dx}\right)(x-tanx)$

Let g (x) = tan x.

So, g?(x) =

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{sin(x+h)}{cos(x+h)}-\frac{sinx}{cosx}\right]$

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{sin(x+h)cosx-sinxcos(x+h)}{cos(x+h)·cosx}\right]$

$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{sin(x+h-x)}{cosx·cos(x+h)}\right]$

$=\underset{h\to 0}{lim}\frac{1}{cosx·co}$$\frac{}{}$$\frac{s(x+h)}{}*\underset{h\to 0}{lim}\frac{sinh}{h}$

$=\frac{1}{co{s}^{2}x}*1$

= sec²x ______ (2)

     Put (2) in (1) we get,

     f?(x) = (x + cos x) (1 - sec²x) +(1 - sin x) (x- tan x)

     We know that,

     1 + tan²x = sec²x

     Þ 1 - sec²x = - tan²x

     So, f?(x) = - tan²x(x + cos x) +(x- tan x) (1 - sin x).

5. Given,

4 x + 3 < 5 x+ 7.

3 – 7 < 5 x 4 x.

–4 < x.

So, x ( –4, ∞)

4. Given, 3x + 8 > 2

3x + 8 – 8 > 2 – 8

3x > – 6

$\Rightarrow \frac{3x}{3}>-\frac{6}{3}$

x > – 2

(i) when x is an integer, x ∈ z all integer greater than – 2 are the soln. so, x = { – 1, 0, 1, 2, 3, …}.

(ii) when x is a real number, all real number greater than – 2 are the soln. so, x ( –2, ∞  )

3. Given, 5x – 3 < 7.

5x – 3 + 3 < 7 + 3

5x< 10

$\frac{5x}{5}<\frac{10}{5}$

x< 2.

(i) when x ∈ z, i. e, x is an integer all integers less than 2 are the solⁿ so, x = {…., – 3, – 2, – 1, 0, 1}

(ii) when x ∈ R i, e x is a real number all t real number less than 2. are the solⁿ so, x∈ (– ∞, 2)

2. – 12 x> 30.

Dividing both sides by 12 we get,

$-x>\frac{30}{12}$

$-x>\frac{5}{2}$

Multiplying both side by ( – 1) the inequality will change.

i e, x< $\frac{5}{2}$ = – 2.5

(i) As x is a natural number, the soln of the given inequality does not exist in natural numbers.

(ii) As x is an integer, the soln of the given inequality will be all the integer less than – 5/2

i. e, – 3, – 4, – 5, ….

1. 24x < 100.

Dividing both sides by 24 we get,

$\frac{24x}{24}<\frac{150}{24.}$

x< $\frac{25}{6}$ = 4.166

(i) As x is a natural no the solⁿ of the given inequality are 1, 2, 3, 4.

(ii) As x is an integer the solⁿ of the given inequality are 4, 3, 2, 1, 0, 1, 2, 3, ….

34. Given, n = 100.

incorrect mean ( $\overline{x}$ ) = 20.

incorrect standard deviation (σ) = 3

We know that,

$\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{n}_i}$

$\Rightarrow 20=\frac{1}{100}{\displaystyle \sum _{i=1}^n{n}_i}$

$\Rightarrow {\displaystyle \sum _{i=1}^n{n}_i}=2000.$

So, incorrect sum of observation = 2000

Correct sum of observation $=2000-21-21-18$

= 1940

33. C.V in mathematics = $\frac{12}{42}*100=28.57.$

C.V in Physics = $\frac{15}{32}*100=46.87.$

C.V in chemistry = $\frac{20}{40.9}*100=48.89$

$\therefore$  Chemistry has the highest variability and mathematics has the lowest variability.

32.  (i) Given, n = 20.

Incorrect mean $(\overline{x})=10$

Incorrect standard deviation $(\sigma )=2$

We know that,

$\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}$

$\Rightarrow 10=\frac{1}{20}{\displaystyle \sum _{i=1}^{20}{x}_i}$

$\Rightarrow {\displaystyle \sum _{i=1}^{20}{x}_i}=200$

So, incorrect sum of observation = 200.

correct sum of observation =200 – 8 = 192

And correct mean $=\frac{192}{19}=10.1$

31. For n observations x₁, x₂,……., x_n .

We have mean = $\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\left(1\right)$

and variance = ${\sigma }^2=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2\left(2\right)}$

Let y_i  be the new observations with same n.

So,  y_i = ax_i      (3)

Now mean, $\overline{y}=\frac{{\displaystyle \sum _{i=1}^n{y}_i}}{n}=\frac{{\displaystyle \sum _{i=1}^{n}a{x}_i}}{n}=\frac{a{\displaystyle \sum _{i=1}^n{y}_i}}{n}=a\overline{x}\left[\left(1\right)\right]$

So  $\overline{y}=a\overline{x}\left(4\right)$

And, putting (3) and (4) in (2) we get,

${\sigma }^2=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(\frac{y_i}{a}-\frac{\overline{y}}{a}\right)}^2}$

$\Rightarrow {\sigma }^2=\frac{1}{a^2}\left[\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}\right]$

$\Rightarrow {\left({\sigma }^{\text{'}}\right)}^2={\sigma }^2{\alpha }^2.$

Hence, the mean and variance of ax₁, ax₂, ……, ax_n  are $a\overline{x}$ and a² σ² .