NCERT Solutions for Miscellaneous Exercise Chapter 6 Class 11 - Linear Inequalities

# NCERT Solutions for Miscellaneous Exercise Chapter 6 Class 11 - Linear Inequalities

Edited By Vishal kumar | Updated on Nov 16, 2023 12:38 PM IST

## NCERT Solutions for Class 11 Maths Chapter 6: Linear Inequalities Miscellaneous Exercise- Download Free PDF

NCERT solutions for Class 11 Maths Chapter 6 miscellaneous exercises are the solutions to the practice questions given at the end of the chapter. The Class 11 Maths Chapter 6 miscellaneous exercise solutions cover all the topics of the chapter linear inequalities. The questions based on all other three exercises of the chapter are discussed in the Class 11 Maths chapter 6 miscellaneous solutions. Miscellaneous exercise Chapter 6 Class 11 gives more practice questions of the concepts discussed in the chapter linear inequalities.

The class 11 chapter 6 maths miscellaneous solutions provided by subject experts at Careers360 are presented in a clear and easily understandable language. They are crafted in a step-by-step format, ensuring simplicity and clarity for students. Additionally, PDF versions of the class 11 maths ch 6 miscellaneous exercise solutions are available, allowing students to choose the format that suits their preferences and comfort. The other exercises discussed in the NCERT book chapter linear inequalities are discussed below.

**In the CBSE Syllabus for 2023-24, miscellaneous exercise class 11 chapter 6 has been renumbered as Chapter 5.

## Access Linear Inequalities Class 11 NCERT Maths- Miscellaneous Exercise

Given : $2\leq 3x-4\leq5$

$2\leq 3x-4\leq5$

$\Rightarrow\, \, 2+4\leq 3x\leq 5+4$

$\Rightarrow\, \, 6\leq 3x\leq 9$

$\Rightarrow\, \, \frac{6}{3}\leq x\leq \frac{9}{3}$

$\Rightarrow\, \, 2\leq x\leq 3$

Thus, all the real numbers greater than equal to 2 and less than equal to 3 are solutions to this inequality.

Solution set is $\left $2,3 \right$$

Given $6 \leq -3(2x - 4) < 12$

$6 \leq -3(2x - 4) < 12$

$\Rightarrow\, \ \frac{6}{3}\leq -(2x-4)< \frac{12}{3}$

$\Rightarrow\, \ -2\geq (2x-4)> -4$

$\Rightarrow\, \ -2+4\geq 2x> -4+4$

$\Rightarrow\, \ 2\geq 2x> 0$

$\Rightarrow\, \ 1\geq x> 0$

Solution set is $(01]$

Given $-3 \leq 4 - \frac{7x}{2}\leq 18$

$\Rightarrow \, \, -3 \leq 4 - \frac{7x}{2}\leq 18$

$\Rightarrow \, \, -3-4 \leq - \frac{7x}{2}\leq 18-4$

$\Rightarrow \, \, -7 \leq - \frac{7x}{2}\leq 14$

$\Rightarrow \, \, 7 \geq \frac{7x}{2} \geq -14$

$\Rightarrow \, \, 7\times 2 \geq 7x\geq -14\times 2$

$\Rightarrow \, \, 14 \geq 7x \geq -28$

$\Rightarrow \, \, \frac{14}{7} \geq x \geq \frac{-28}{7}$

$\Rightarrow \, \, 2 \geq x \geq -4$

Solution set is $[-4,2]$

Given The inequality

$-15 < \frac{3(x-2)}{5} \leq 0$

$-15 < \frac{3(x-2)}{5} \leq 0$

$\Rightarrow\, \ -15\times 5< 3(x-2)\leq 0\times 5$

$\Rightarrow\, \ -75< 3(x-2)\leq 0$

$\Rightarrow\, \ \frac{-75}{3}< (x-2)\leq \frac{0}{3}$

$\Rightarrow\, \ -25< (x-2)\leq 0$

$\Rightarrow\, \ -25+2< x\leq 0+2$

$\Rightarrow\, \ -23< x\leq 2$

The solution set is $(-23,2]$

Given the inequality

$-12<4-\frac{3x}{-5} \leq 2$

$-12<4-\frac{3x}{-5} \leq 2$

$\Rightarrow\, \, -12-4< -\frac{3x}{-5}\leq 2-4$

$\Rightarrow\, \, -16< -\frac{3x}{-5}\leq -2$

$\Rightarrow\, \, -16< \frac{3x}{5}\leq -2$

$\Rightarrow\, \, -16\times 5< 3x\leq -2\times 5$

$\Rightarrow\, \, -80< 3x\leq -10$

$\Rightarrow\, \, \frac{-80}{3}< 3x\leq \frac{-10}{3}$

Solution set is $(\frac{-80}{3}, \frac{-10}{3}]$

Given the linear inequality

$7 \leq \frac{(3x+ 11)}{2}\leq 11$

$7 \leq \frac{(3x+ 11)}{2}\leq 11$

$\Rightarrow \, \, 7\times 2 \leq (3x+ 11)\leq 11\times 2$

$\Rightarrow \, \, 14 \leq (3x+ 11)\leq 22$

$\Rightarrow \, \, 14-11 \leq (3x)\leq 22-11$

$\Rightarrow \, \, 3 \leq 3x\leq 11$

$\Rightarrow \, \, 1 \leq x\leq \frac{11}{3}$

The solution set of the given inequality is $[1,\frac{11}{3}]$

Given : $5x + 1 > -24,\ 5x - 1 <24$

$5x + 1 > -24\, \, \, \, \, \, \, and\, \, \, \, \, \, \ 5x - 1 <24$

$\Rightarrow 5x > -24-1\, \, \, \, \, \, \, and\, \, \, \, \, \, \ 5x <24+1$

$\Rightarrow 5x > -25\, \, \, \, \, \, \, and\, \, \, \, \, \, \ 5x <25$

$\Rightarrow x > \frac{-25}{5}\, \, \, \, \, \, \, and\, \, \, \, \, \, \ x <\frac{25}{5}$

$\Rightarrow x > -5\, \, \, \, \, \, \, and\, \, \, \, \, \, \ x <5$

$(-5,5)$

The solution graphically on the number line is as shown :

Given : $2(x-1) 2 -x$

$2(x-1) 2 -x$

$\Rightarrow \, \, 2x-2 2 -x$

$\Rightarrow \, \, 2x-x<2+5\, \, \, \, and\, \, \, \, \, \ 3x+x> 2 -6$

$\Rightarrow \, \, x<7\, \, \, \, and\, \, \, \, \, \ 4x> -4$

$\Rightarrow \, \, x<7\, \, \, \, and\, \, \, \, \, \ x> -1$

$(-1,7)$

The solution graphically on the number line is as shown :

Given : $3x - 7 > 2(x-6),\ 6-x > 11 - 2x$

$3x - 7 > 2(x-6)\, \, \, \, and\, \, \, \, \, \ 6-x > 11 - 2x$

$\Rightarrow \, \, 3x - 7 > 2x-12\, \, \, \, and\, \, \, \, \, \ 6-x > 11 - 2x$

$\Rightarrow \, \, 3x - 2x >7-12\, \, \, \, and\, \, \, \, \, \ 2x-x > 11 - 6$

$\Rightarrow \, \, x >-5\, \, \, \, and\, \, \, \, \, \ x > 5$

$x\in (5,\infty )$

The solution graphically on the number line is as shown :

Given : $5(2x-7)-3(2x+3)\leq 0,\quad 2x + 19 \leq 6x +47$

$5(2x-7)-3(2x+3)\leq 0\, \, \, \, \, and\, \, \, \, \, \, \, \quad 2x + 19 \leq 6x +47$

$\Rightarrow \, \, 10x-35-6x-9\leq 0\, \, \, \, \, and\, \, \, \, \, \, \, \quad 2x -6x\leq 47-19$

$\Rightarrow \, \, 4x-44\leq 0\, \, \, \, \, and\, \, \, \, \, \, \, \quad -4x\leq 28$

$\Rightarrow \, \, 4x\leq 44\, \, \, \, \, and\, \, \, \, \, \, \, \quad 4x\geq - 28$

$\Rightarrow \, \, x\leq 11\, \, \, \, \, and\, \, \, \, \, \, \, \quad x\geq - 7$

$x\in [-7,11]$

The solution graphically on the number line is as shown :

Since the solution is to be kept between 68° F and 77° F.

$68< F< 77$

Putting the value of $F = \frac{9}{5}C + 32$ , we have

$\Rightarrow \, \, \, 68< \frac{9}{5}C + 32< 77$

$\Rightarrow \, \, \, 68-32< \frac{9}{5}C < 77-32$

$\Rightarrow \, \, \, 36< \frac{9}{5}C < 45$

$\Rightarrow \, \, \, 36\times 5< 9C < 45\times 5$

$\Rightarrow \, \, \, 180< 9C < 225$

$\Rightarrow \, \, \, \frac{180}{9}< C < \frac{225}{9}$

$\Rightarrow \, \, \, 20< C < 25$

the range in temperature in degree Celsius (C) is between 20 to 25.

Let x litres of 2% boric acid solution is required to be added.

Total mixture = (x+640) litres

The resulting mixture is to be more than 4% but less than 6% boric acid.

$\therefore \, 2\%x+8\%\, of\, 640> 4\%\, of\, (640+x)$ and $2\%x+8\%\, of\, 640< 6\%\, of\, (x+640)$

$\Rightarrow \, 2\%x+8\%\, of\, 640> 4\%\, of\, (640+x)$ and $2\%x+8\%\, of\, 640< 6\%\, of\, (x+640)$

$\Rightarrow \, \frac{2}{100}x+(\frac{8}{100}) 640> \frac{4}{100} (640+x)$ $\Rightarrow \, \frac{2}{100}x+(\frac{8}{100}) 640< \frac{6}{100} (640+x)$

$\Rightarrow \, 2x+5120> 4x+2560$ $\Rightarrow \, 2x+5120< 6x+3840$

$\Rightarrow \, 5120-2560> 4x-2x$ $\Rightarrow \, 5120-3840< 6x-2x$

$\Rightarrow \, 2560> 2x$ $\Rightarrow \, 1280< 4x$

$\Rightarrow \, 1280> x$ $\Rightarrow \, 320< x$

Thus, the number of litres 2% of boric acid solution that is to be added will have to be more than 320 and less than 1280 litres.

Let x litres of water is required to be added.

Total mixture = (x+1125) litres

It is evident that amount of acid contained in the resulting mixture is 45% of 1125 litres.

The resulting mixture contain more than 25 % but less than 30% acid.

$\therefore \, 30\%\, of\, (1125+x) > 45\%\, of\, (1125)$ and $25\%\, of\, (1125+x)< 45\%\, of\, 1125$

$\Rightarrow \, 30\%\, of\, (1125+x) > 45\%\, of\, (1125)$ and $25\%\, of\, (1125+x)< 45\%\, of\, 1125$

$\Rightarrow \, \frac{30}{100}(1125+x)> \frac{45}{100} (1125)$ $\Rightarrow \, (\frac{25}{100}) (1125+x)< \frac{45}{100} (1125)$

$\Rightarrow \, 30\times 1125+30x> 45\times (1125)$ $\Rightarrow \, 25 (1125+x)< 45(1125)$

$\Rightarrow \, 30x> (45-30)\times (1125)$ $\Rightarrow \, 25 x< (45-25)1125$

$\Rightarrow \, 30x> (15)\times (1125)$ $\Rightarrow \, 25 x< (20)1125$

$\Rightarrow \, x> \frac{15\times 1125}{30}$ $\Rightarrow \, x< \frac{20\times 1125}{25}$

$\Rightarrow \, x> 562.5$ $\Rightarrow \, x< 900$

Thus, the number of litres water that is to be added will have to be more than 562.5 and less than 900 litres.

Given that group of 12 years old children.

$80\leq IQ\leq140$

For a group of 12 years old children, CA =12 years

$IQ= \frac{MA}{CA}\times 100$

Putting the value of IQ, we obtain

$80\leq IQ\leq140$

$\Rightarrow \, \, 80\leq \frac{MA}{CA}\times 100\leq140$

$\Rightarrow \, \, 80\leq \frac{MA}{12}\times 100\leq140$

$\Rightarrow \, \, 80\times 12\leq MA\times 100\leq140\times 12$

$\Rightarrow \, \, \frac{80\times 12}{100}\leq MA\leq \frac{140\times 12}{100}$

$\Rightarrow \, \, 9.6\leq MA\leq 16.8$

Thus, the range of mental age of the group of 12 years old children is $\, \, 9.6\leq MA\leq 16.8$

## More About NCERT Solutions for Class 11 Maths Chapter 6 Miscellaneous Exercises

Question number 1 to 6 of miscellaneous exercise chapter 6 Class 11 is to solve linear inequalities in one variable. Question number 7 to 10 gives graphical solutions of linear inequalities in one variable. The remaining questions are applications of linear equations and are illustrated in NCERT solutions for Class 11 Maths chapter 6 miscellaneous exercise.

Also Read| Linear Inequalities Class 11th Notes

## Benefits of NCERT Solutions for Class 11 Maths Chapter 6 Miscellaneous Exercises

• Students can practice all the questions of miscellaneous exercise chapter 6 Class 11 to revise the chapter.

• All the questions in the Class 11 Maths Chapter 6 miscellaneous exercise solutions are equally important and students should solve the first three exercises prior to NCERT solutions for Class 11 Maths Chapter 6 miscellaneous exercise.

JEE Main Highest Scoring Chapters & Topics
Just Study 40% Syllabus and Score upto 100%

## Topics Covered in Miscellaneous Exercise Class 11 Chapter 6

The Miscellaneous Exercise in NCERT Solutions for Class 11 Maths Chapter 6 - Linear Inequalities covers key topics such as:

1. Introduction to Linear Inequalities
2. Inequalities
3. Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation

The class 11 chapter 6 miscellaneous exercise solutions align with the latest CBSE guidelines, offering students a comprehensive approach to mastering the subject. Regular practice with these questions not only follows CBSE recommendations but also boosts students' confidence, making it essential for those aiming for high scores in the Maths board examination. Regular and thorough practice of NCERT Solutions for Class 11 Maths is crucial for success.

## Key Features of Class 11 Maths ch 6 Miscellaneous Exercise Solutions

1. Comprehensive Coverage: Class 11 Chapter 6 maths miscellaneous solutions address all the miscellaneous exercise problems in Chapter 6, ensuring comprehensive coverage of the topics.

2. CBSE Guidelines: The class 11 maths miscellaneous exercise chapter 6 solution are designed in accordance with the latest CBSE guidelines, ensuring relevance to the prescribed curriculum.

3. Clarity and Accuracy: Class 11 maths ch 6 miscellaneous exercise solutions are presented with clarity and accuracy to facilitate better understanding for students.

4. Step-by-Step Approach: Each problem is solved in a step-by-step manner, providing a clear and logical progression to aid in better comprehension.

5. Graphical Representation: Class 11 Chapter 6 miscellaneous exercise solutions include graphical representations where applicable, helping students visualize concepts and solutions.

6. Relevance to Board Examination: The Class 11 chapter 6 maths miscellaneous solutions are tailored to help students prepare effectively for the Class 11 Maths board examination, aligning with the exam pattern and question types.

7. Accessible PDF Format: The solutions are available in a PDF format, offering convenience for students to use them according to their preferences and ease of access.

Also see-

## Subject Wise NCERT Exampler Solutions

### Frequently Asked Question (FAQs)

1. What is the relation between degree Celsius (C) and Fahrenheit (F)?

F=9C/5+32

2. What is question number 11 of NCERT solutions for Class 11 Maths chapter 6 miscellaneous exercise?

It’s about finding the range of temperature in degrees Celsius. Given the range of Fahrenheit.

3. What is the solution of 3x-6=0?

3x-6=0

3x=6

x=2

4. Give the solutions of 3x-6≥0

The solution is  3x≥6

Or x ≥ 2. The solution is the right-hand side of the line x=2 including the points on the line. The graphical representation is given in the NCERT Class 11 Maths book chapter 6 example 10.

5. What is the difference between the solutions of x ≥ 2 and x>2?

Both inequalities have solutions to the right of the line x=2. For x ≥ 2 points on the line x=2 are included and for x>2 points on the linex=2 are excluded.

6. How can you represent x>120 and x<300 as a double inequality?

120<x<300

7. How many exercises are solved in NCERT syllabus chapter 6 Class 11?

4 exercises are discussed in NCERT Class 11 Maths chapter 6

8. How many questions are solved in the Class 11 Maths chapter 6 miscellaneous exercise solutions?

Fourteen questions are solved in Linear Inequality Class 11 NCERT Maths chapter 6 miscellaneous exercise

## Upcoming School Exams

#### National Means Cum-Merit Scholarship

Application Date:01 August,2024 - 16 September,2024

#### National Rural Talent Scholarship Examination

Application Date:05 September,2024 - 20 September,2024

#### International General Knowledge Olympiad

Exam Date:19 September,2024 - 19 September,2024

#### Unified Cyber Olympiad

Exam Date:20 September,2024 - 20 September,2024

#### National Institute of Open Schooling 12th Examination

Exam Date:20 September,2024 - 07 October,2024

Get answers from students and experts

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

 Option 1) Option 2) Option 3) Option 4)

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

 Option 1) 2.45×10−3 kg Option 2)  6.45×10−3 kg Option 3)  9.89×10−3 kg Option 4) 12.89×10−3 kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

 Option 1) Option 2) Option 3) Option 4)

A particle is projected at 600   to the horizontal with a kinetic energy . The kinetic energy at the highest point

 Option 1) Option 2) Option 3) Option 4)

In the reaction,

 Option 1)   at STP  is produced for every mole   consumed Option 2)   is consumed for ever      produced Option 3) is produced regardless of temperature and pressure for every mole Al that reacts Option 4) at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?

 Option 1) 0.02 Option 2) 3.125 × 10-2 Option 3) 1.25 × 10-2 Option 4) 2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

 Option 1) decrease twice Option 2) increase two fold Option 3) remain unchanged Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

 Option 1) Molality Option 2) Weight fraction of solute Option 3) Fraction of solute present in water Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

 Option 1) twice that in 60 g carbon Option 2) 6.023 × 1022 Option 3) half that in 8 g He Option 4) 558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

 Option 1) less than 3 Option 2) more than 3 but less than 6 Option 3) more than 6 but less than 9 Option 4) more than 9