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CBSE 11th Standard CBSE all question papers, important notes , study materials , Previuous Year questions, Syllabus and exam patterns. Free 11th Standard CBSE all books and syllabus online. Practice Online test for free in QB365 Study Material. Important keywords, Case Study Questions and Solutions. Updates about latest education news and Scholorships in one place.

## Latest 11th Standard CBSE Case study Questions Update

Cbse 11th physics motion in a straight line chapter case study questions with answers, cbse 11th physics units and measurements chapter case study questions with answers, cbse 11th chemistry structure of atom chapter case study question with answers, cbse 11th chemistry some basic concept of chemistry chapter case study questions with answers, 11th biology biological classification chapter case study question with answers cbse, 11th biology the living world chapter case study question with answers cbse, 11th standard cbse study materials.

## 11th Standard CBSE Subjects

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## Class VI to XII

Tn state board / cbse, 3000+ q&a's per subject, score high marks.

## Latest CBSE 11th Standard CBSE Study Material Updates

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## CBSE Class 11 Term-1 Maths 2021-22: Topic-Wise MCQs, Case-Based & Assertion/Reason

CBSE Class 11 Term-1 Maths 2021-22: Topic-Wise MCQs, Case-Based & Assertion/Reason

In this Post I Have Provided Mathematics Class 11 Chapter-Wise MCQs, Case-Based Question And Assertion/Reason For Term-1 Session 2021-22 With Solutions. CBSE has Recently Included These Types Of MCQs And Assertion/Reason For Term-1 Exam 2020. Every Student Knows these types of Questions Are Very Important For Their Term-1 Examination.

Any student who want to download these MCQs, then they have to click on given respective download links and it will be automatically download in your Google Drive.

## CBSE Class 11th Physics Term-1 2021-22: Topic-Wise MCQs, Case-Based & Assertion/Reason

For Downlod Click Here

Given Below Are The Class 11 Mathematics Chapter Name with Respective Download Links Containing Study Material:

1. Sets | |

2. Relations and Functions | |

3. Complex Numbers | |

4. Sequence and Series | |

5. Straight Lines | |

6. Limits | |

7. Statistics |

CBSE Sample Papers For Class 11 Mathematics

CBSE sample paper for class 11th Maths is an important tools to analyse itself. Before going to final examinations every student try to solve different types of sample question paper. It will enhance your knowledge and also provide to increase mental ability. Sample question paper are the best resources for the student to prepare for their Board examinations.

These sample question papers are very helpful to the student to get an entire experience before attempting the Board examinations papers. If students solve different types of sample question papers then they get the good confidence about the appropriate answer and they make good score in their Board examinations.

As every student know CBSE always change their exam pattern and also the model paper for their Board examinations that is why it is very important to all the student to understand the Model paper pattern before going to the final examinations.

In this articles I have provided different types of question papers on the basis of CBSE latest exam pattern which will definitely help to the student for the good preparation for their examinations.

In every year before the Board examination, CBSE provide model test paper to the student, on the basis of these model test papers I have prepared sample question papers for class 11th Maths which will help to the student to understand the concept of model test paper and also it will help to the student for good preparation for their board examination.

CBSE Sample Paper:

Maths And Physics With Pandey Sir is a website which provide free study material and notes to every students who are preparing for their Board examinations. By going in Level Section you can select CBSE sample papers. In this section, I have provided CBSE sample paper for class 9th to 12th.

Also if you are preparing for any competitive exams, they can find out study material and important notes in Level Section. In this website I have also provided most important MCQs questions, Assertion/Reason based questions and case study based questions for Board examinations or any other competitive examinations.

Benefits Of Solving CBSE Sample Papers for Examination:

Time Management:-

With the help of CBSE sample question papers you can make a proper strategy for their examinations also by practicing sample question papers you can manage your time during the Board examinations.

Examination Strategy:-

After practicing different types of sample question papers you can understand on which section you have spend more time and which section is easy and which section is hard. In this way you can make a proper examination strategy.

Self Evaluation:-

Self evaluation is very important to every student because with the help of self evaluation, student can understand their weak and strong point, which is very important for any Board examinations. After practicing more and more simple paper you can evaluate himself.

To Identify Silly Mistakes:-

As every student know there are many questions in their question paper which are very easy but student make mistakes in this types of question. By solving different types of sample question papers you can reduce their silly mistakes.

Advantages Of CBSE Sample Question Papers:

* To get a good exam ideas about your examination paper patterns

* Student can easily access to the question papers

* It is very helpful to know about the mark distribution system.

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* CBSE sample paper 2021-22 class 12 Solutions

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## Class 11 Maths Sample Papers, NCERT Solutions | Mock Tests

Designed for the toppers, the course content of CBSE Class 11 Maths like Sample Papers, NCERT Solutions, Notes, and Important Questions can be availed as PDF on myCBSEguide.

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## CBSE Sample Papers

Cbse last year papers, user submitted papers, case study questions, mock tests, relations and functions, trigonometric functions, complex numbers and quadratic equations, linear inequalities, permutations and combinations, binomial theorem, sequences and series, straight lines, conic sections, introduction to 3d geometry, limits and derivatives, probability.

- CBSE Test Papers
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## CBSE Important Questions

Other useful resourses, online tests, learning videos.

Regarded as a scoring subject, Mathematics is equally important, especially when it comes to the senior secondary level. One can master this subject with little effort. All they need to do is to apply the flow of reasons while proving a result or solving a problem. To attain this level of competence you need good practice. For having the good practice you need to have quality course material. You can get the best resources for CBSE class 11 Mathematics on myCBSEguide as we gave a huge question bank with a variety of questions. Download the latest Sample Papers for CBSE Class 11 Mathematics, chapter-related Revision Notes, NCERT solution, and test papers all in PDF files.

## CBSE Class 11 Mathematics Resources

Trusted by millions of teachers and students alike, myCBSEguide is a one-stop solution for quality course content for CBSE subjects. Students must put some extra effort to excel in Mathematics at the senior secondary level, as it will be working as the foundation of their future careers and profession. We provide a wide range of products with the help of which you can prepare for senior secondary Mathematics on myCBSEguide beginning from the syllabus to the model test papers. There are online tests, mock tests, learning videos, revision notes, NCERT solutions, and some important questions for class 11 Mathematics.

## Class 11 Mathematics Syllabus

The syllabus is regularly updated based on the latest development. Students of senior secondary level, i.e., Class 11 Mathematics are on the threshold of higher academics. The marks they score here will play a decisive role in choosing their career options. Therefore, the syllabus of class 11 Maths includes every such topic that helps the students to develop a positive attitude to think, analyze and articulate logically. Given below are the name and the weightage of chapters of Class 11 Mathematics.

23 25 12 08 12 | |

The prescribed books for class 11 Mathematics are:

- Mathematics Textbook for Class XI, NCERT Publications
- Mathematics Lab Manual class XI, published by NCERT

To know the chapter and topic-related details check out our detailed blog on the Class 11 Mathematics syllabus .

## Class 11 Mathematics Notes and NCERT Solutions

All CBSE students should practice NCERT textbook solutions because most of the time the questions asked in the final board examination are based on them. As we know that the CBSE encourages following NCERT textbooks because they cover almost all possible topics of any central examination. Some of the topics of class 11 Mathematics are:

- Binomial Theorem Miscellaneous
- Complex Numbers and Quadratic Equations Miscellaneous
- Conic Sections Miscellaneous
- Limits and Derivatives Miscellaneous

To get the complete notes based on the exercise check out Class 11 Mathematics NCERT Solutions .

Although Mathematics is not a theory subject, an explanation of each chapter is needed. One cannot understand the chapter if they are not aware of the subject-specific terms and jargon along with the explanation of theorems. You can find a step-by-step explanation of each example sum of a particular topic of class 11 Mathematics. Do check our Revision Notes for Class 11 Mathematics and download them in PDF.

## CBSE Class 11 Mathematics Case Study Questions

Case study questions are the latest type of questions included in the CBSE question paper. This change is to promote competency-based learning and motivate topics from real-life and other subject areas, greater emphasis has been laid on the application of various concepts. The class 11 Mathematics case study questions look forward to acquainting students with different aspects of Mathematics used in daily life. Being the newest question format, it is difficult to get an assured quality of CBQs in the market or over the internet. Check our Class 11 Maths case study questions to know the actual standard of the questions. These case-based questions have sub-questions which can be objective or subjective type depending on the CBSE sample question paper. To get these questions get a subscription to our student dashboard.

## CBSE Class 11 Mathematics Sample Papers

A student who aims to get good grades must solve sample papers religiously. Based on the official sample papers of the CBSE, myCBSEguide prepares around ten sample papers in each subject. Knowing the blueprint and MS can help the students to write their papers confidently. Even the school-level examinations follow the CBSE pattern, therefore it is advised to solve full-length class 11 Mathematic model test papers. Get the app to download the latest Class 11 Maths Sample Papers 2023 .

Sample papers are also helpful for periodic assessment too. This is because a sample paper has an assortment of questions. A standard sample paper has all types of questions such as very short answer (VSA), short answer (SA) and, long answer (LA) to effectively assess the knowledge, understanding, application, skills, analysis, evaluation and, synthesis. We can say that ideally, a sample paper matches the modalities of a proper pen-paper test.

## CBSE Class 11 Maths Test Papers & Important Questions

Memorizing the rules and formulas is one thing and applying them is another. Good coordination between the two can make wonders. This can be achieved by enthusiastic practice. You can get the Test paper set for Class 11 Mathematics, which you can practice according to the level of your preparation. This helps in segmented learning. Test papers are good ways to learn and test simultaneously. The class 11 mathematics test papers have all types of questions. Besides, we make efforts to chalk out the most important questions from every chapter so that our students do not miss out on crucial information. Class 11 Mathematics important questions are available on myCBSEguide.

You can find such valuable study material for other CBSE subjects in our CBSE module .

## Class 11 Mathematics Case Study Questions

If you’re seeking a comprehensive and dependable study resource with Class 11 mathematics case study questions for CBSE, myCBSEguide is the place to be. It has a wide range of study notes, case study questions, previous year question papers, and practice questions to help you ace your examinations. Furthermore, it …

## CBSE Class 11 Mathematics Syllabus 2022-23

CBSE Class 11 Mathematics Syllabus 2022-23 includes Sets and Functions, Coordinate Geometry, Statistics and Probability, Calculus etc for the session 2022 – 2023. Here is the detailed syllabus. To download class 11 Mathematics CBSE latest sample question papers for the 2023 exams, please install the myCBSEguide App which is the …

## CBSE Term-1 MCQ Sample Papers 2021-22

CBSE will ask only MCQs in Term-1 Examination this year. The board will hold this first term exam in Nov-Dec 2021. As CBSE has already cleared that 1st term exam will carry 90 minutes, now it’s time to start preparation. Students can download CBSE Term-1 MCQ Sample Papers from myCBSEguide …

## CBSE to Conduct Two Term Exams in 2021-22

CBSE has issued a circular on 5th July 2021 regarding the Special Scheme of Assessment for Board Examination Classes X and XII for the Session 2021-22. Here is the complete text of the circular. COVID 19 pandemic caused almost all CBSE schools to function in a virtual mode for the …

## CBSE Class-11 New Stream Selection Rules

The New Education Policy-2020 has suggested eliminating rigid stream selection rules. We know that there are three streams in class 11 and the choices to select subjects are limited. Students have to select subjects within the stream only. Thus, if you are taking Science Stream, you have no option to …

## CBSE Reduced Syllabus by 30% for Session 2020-21

CBSE, New Delhi has reduced the syllabus for classes 9 to 12 by 30%. The CBSE Revised Syllabus 2020-21 is available in CBSE official website and myCBSEguide mobile app. Here is the complete circular regarding CBSE Revised Syllabus 2020-21: CBSE Reduced the Syllabus The prevailing health emergency in the country …

## Multiple Choice Questions for CBSE Class 10 and 12

The new curriculum issued by CBSE, New Delhi includes objective type questions along with Multiple Choice Questions (MCQ). Now CBSE will ask around 20% questions based on objective type questions. myCBSEguide App has more than one lakh MCQ from class 3 to 12 in almost all major subjects. MCQ questions …

## CBSE Online MCQ Quiz Maker

The ultimate quiz builder helps CBSE schools and teachers, create MCQ quiz online and share the same with their students. Now schools and coaching institutes can create and share MCQ quiz online with their own Name and Logo. The students will attempt the quiz and teachers will get the result …

## Create CBSE Question Papers Online

Creating question papers online is easy. You can create question papers online with your name and logo in minutes. So, why to waste time consulting many books and finding suitable questions. Just log in to myCBSEguide Test Generator here and create question papers online. Follow 3 simple steps: Select your …

## CBSE Syllabus for Class 11 Mathematics 2019-20

CBSE Syllabus for Class 11 Mathematics 2019-20 contains all the topics of this session. myCBSEguide provides you latest Syllabus for Class 11 Mathematics. Mathematics is the study of numbers, shape, and patterns. Mathematics is not only important for success in life, but it is all around us. Mathematics is the …

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## CBSE Important Questions for Class 11 Maths

CBSE Important Questions for Class 11 Maths is crucial for the students as there is sudden progress in the level of challenge in the subject. This article will help you get an idea of the different types of questions that can be framed in the examination. Students will find chapter-wise sets of CBSE Important Questions in our question bank that will enhance their level of understanding of the subject. Also, some of these questions are expected to be asked in the exam. So, practise daily to excel in the examination.

## CBSE Class 11 Maths Important Questions

There are a total of 16 chapters in the Class 11 Maths textbook. Students can get the Maths chapter-wise CBSE Important Questions of Class 11 by clicking on the links below.

Solving these questions will boost the student’s exam preparation. They will feel more confident while writing the Maths CBSE Class 11 exam. So, students must keep practising and continue with their studies.

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## CBSE Sample Papers for Class 11 Maths Set 4 with Solutions

January 6, 2024 by Bhagya

Students must start practicing the questions from CBSE Sample Papers for Class 11 Maths with Solutions Set 4 are designed as per the revised syllabus.

Time Allowed: 3 Hours Maximum Marks: 80

General Instructions:

- This Question paper contains five sections A, B, C, D, and E. Each section is compulsory. However, there are internal choices in some questions.
- Section A has 18 MCQs and 2 Assertion-Reason-based questions of 1 mark each.
- Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
- Section C has 6 Short Answer (SA)-type questions of 3 marks each.
- Section D has 4 Long Answer (LA)-type questions of 5 marks each.
- Section E has 3 source-based/case-based/passage-based/integrated units of assessment of 4 marks each with sub-parts.

Section – A

(Multiple Choice Questions) Each Question Carries 1 Mark.

Question 1. Which of the following is an empty set? (A) {x | x is a prime number less than 5} (B) {x | x is an even number greater than 10} (C) {x | x is a multiple of 7} (D) {x | x is a number in the word “set”} Answer: (D) {x | x is a number in the word “set”} Explanation: An empty set is a set that has no elements. Option A is not an empty set as there are two prime numbers less than 5 (namely 2 and 3). Option B is also not an empty set as there are many even numbers greater than 10. Option C is not an empty set as there are many multiples of 7. Option D is an empty set as there are no numbers in the word “set”.

Question 2. The domain of the function f defined by f(x) = \(\sqrt{4-x}+\frac{1}{\sqrt{x^2-1}}\) is equal to (A) (-∞, -1) ∪ (1, 4] (B) (-∞, -1] ∪ (1, 4] (C) (-∞, -1) ∪ [1, 4] (D) (-∞, -1] ∪ [1, 4) Answer: (A) (-∞, -1) ∪ (1, 4] Explanation: For domain of f(x), f(x) ≥ 0 i.e., \(\sqrt{4-x}+\frac{1}{\sqrt{x^2-1}}\) ≥ 0 ⇒ \(\sqrt{4-x}\) ≥ 0 or \(\sqrt{x^2-1}\) ≥ 0 ⇒ 4 – x ≥ 0 or x 2 – 1 ≥ 0 ⇒ -x ≥ -4 or (x – 1)(x + 1) ≥ 0 ⇒ x ≥ 4 or x < -1 and x > 1 Thus, Domain of f(x) = (-∞, -1) ∪ (1, 4]

Question 3. \(\lim _{x \rightarrow 0} \frac{|\sin x|}{x}\) is equal to (A) 1 (B) -1 (C) 0 (D) does not exist Answer: (D) does not exist Explanation: Right hand side limit = \(\lim _{x \rightarrow 0^{+}} \frac{|\sin x|}{x}\) = \(\lim _{x \rightarrow 0^{+}} \frac{\sin x}{x}\) = 1 Left hand side limit = \(\lim _{x \rightarrow 0^{-}} \frac{|\sin x|}{x}\) = \(\lim _{x \rightarrow 0^{-}} \frac{-\sin x}{x}\) = -1 RHS ≠ LHS Therefore, the solution does not exist.

Question 7. The coefficient of y in the expansion of \(\left(y^2+\frac{c}{y}\right)^5\) is (A) 10c (B) 10c 2 (C) 10c 3 (D) none of these Answer: (C) 10c 3 Explanation: Given binomial expression is \(\left(y^2+\frac{c}{y}\right)^5\) Here, \(T_{r+1}={ }^5 C_r \times\left(y^2\right)^{5-r} \times\left(\frac{c}{y}\right)^r\) = \({ }^5 C_r \times y^{10-3 r} \times c^r\) Now, 10 – 3r =1 ⇒ 3r = 9 ⇒ r = 3 So, coefficient of y = 5 C 3 × c 3 = 10c 3

Question 8. If x is a real number and |x| < 3, then (A) x ≥ 3 (B) -3 < x < 3 (C) x ≤ -3 (D) -3 ≤ x ≤ 3 Answer: (B) -3 < x < 3 Explanation: We have, |x| < 3 ∴ -3 < x < 3 [Since, |x| < a then -a < x < a]

Question 9. Let R be the relation in the set N given by R = {(a, b): a = b – 2, b > 6}. Choose the correct answer. (A) (2, 4) ∈ R (B) (3, 8) ∈ R (C) (6, 8) ∈ R (D) (8, 7) ∈ R Answer: (C) (6, 8) ∈ R Explanation: R = {(a, b): a = b – 2, b > 6} ⇒ (a, b) ∈ R ⇒ a = b – 2 where b > 6 ⇒ (6, 8) ∈ R as 6 = 8 – 2 where b = 8 > 6

Question 10. The sum of A.P. 2, 5, 8,… upto 50 terms is (A) 3775 (B) 3557 (C) 3757 (D) 3575 Answer: (A) 3775 Explanation: S n = \(\frac{n}{2}\) (2a + (n – 1)d) S 50 = \(\frac{50}{2}\) (2 × 2 + (50 – 1)3) = 25(4 + 49 × 3) = 25(4 + 147) = 25 × 151 = 3775

Question 11. School A has eight players. A five-member team will be selected out of these eight players. How many different selections can be made? (A) 66 (B) 56 (C) 46 (D) 36 Answer: (B) 56 Explanation: In this case, n = 8 and r = 5, Thus n C r = 8 C 5 = \(\frac{8 !}{3 ! \times 5 !}\) = \(\frac{8 \times 7 \times 6 \times 5 \times 4}{5 \times 4 \times 3 \times 2 \times 1}\) = 56

Question 13. If a, b, c, d, and e are the observations of the mean m and standard deviation s, then find the standard deviation of the observations a + k, b + k, c + k, d + k, and e + k is (A) s (B) ks (C) s + k (D) \(\frac{s}{k}\) Answer: (A) s Explanation: On adding a constant to each observation, the mean, median, and quartiles will be changed, but the range, interquartile range, standard deviation, and variance will remain the same.

Question 15. The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then (A) breadth > 20 cm (B) length < 20 cm (C) breadth ≥ 20 cm (D) length ≤ 20 cm Answer: (C) breadth ≥ 20 cm Explanation: If x cm is the breadth. ⇒ Length = 3x Given, the perimeter is greater than equal to 160, i.e., 2(3x + x) ≥ 160 ⇒ 2(4x) ≥ 160 ⇒ x ≥ 20 ⇒ breadth ≥ 20 cm

Question 16. The perpendicular distance of the point P(6, 7, 8) from the XY-plane is (A) 8 (B) 7 (C) 6 (D) None of these Answer: (A) 8 Explanation: Let Q be the foot of the perpendicular drawn from the point P (6, 7, 8) to the XY plane. Thus, the distance of this foot Q from P is the z-coordinate of P, i.e., 8 units.

Assertion-Reason Based Questions In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices.

(A) Both (A) and (R) are true and (R) is the correct explanation of (A). (B) Both (A) and (R) are true but (R) is not the correct explanation of (A). (C) (A) is true but (R) is false. (D) (A) is false but (R) is true.

Question 19. Assertion (A): A five-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4, and 5 with repetition. The total number formed is 216. Reason (R): If the sum of digits of any number is divisible by 3 then the number must be divisible by 3. Answer: (D) (A) is false but (R) is true. Explanation: Since a five-digit number is formed by using digits 0, 1, 2, 3, 4, and 5, divisible by 3 i.e., only possible when the sum of digits is multiple of three which gives two cases. Case I: Using digits 0, 1, 2, 4, 5 the number of ways = 4 × 4 × 3 × 2 × 1 = 96 Case II: Using digits 1, 2, 3, 4, 5 the number of ways = 5 × 4 × 3 × 2 × 1 = 120 Therefore, total numbers formed = 120 + 96 = 216 (taking numbers without repetition)

Question 20. Assertion (A): The measure of rotation of a given ray about its initial point is called an angle. Reason (R): The point of rotation is called a vertex. Answer: (B) Both (A) and (R) are true but (R) is not the correct explanation of (A).

[This section comprises very short answer-type questions (VSA) of 2 marks each]

Question 22. The reporting time of an employee is given below:

Mon | Tue | Wed | Thu | Fri | Sat | |

10:35 | 10:20 | 10:22 | 10:27 | 10:25 | 10:40 |

If the reporting time is 10:30 a.m., then find the probability of his coming late. Answer: Let ‘S’ be the sample space and ‘E’ be the event that the employee is coming late. ∴ n(S) = 6 E = {10 : 35, 10 : 40} ∴ n(E) = 2 ∴ P(E) = \(\frac{n(F)}{n(S)}=\frac{2}{6}=\frac{1}{3}\) Thus, the probability of the employee coming late is \(\frac{1}{3}\).

Question 24. In how many ways can the letters of the word “ABACUS” be arranged such that the vowels always appear together? Answer: In the word ABACUS, there are 3 vowels – 2A’s and U. Number of letters in ABACUS are 6. Let us take the vowels as one, so the number of letters now = 4 ∴ The Number of ways in which vowels occur together = 4! But the 3 vowels can rearrange amongst themselves in \(\frac{3 !}{2 !}\) ways as ‘A’ appears twice. Hence, the total number of ways in which vowels occur together = 4! × \(\frac{3 !}{2 !}\) = 72

Question 25. Are sets A = {1, 2, 3, 4}, B = {x : x ∈ N and 5 ≤ x ≤ 7} disjoint? Why? Answer: Yes, sets A and B are disjoint, because A ∩ B = φ. ∵ A = {1, 2, 3, 4} and B = {5, 6, 7} ∴ A ∩ B = {1, 2, 3, 4} ∩ {5, 6, 7} = φ

[This section comprises short answer type questions (5A) of 3 marks each]

Question 26. Prove that the line lx + my + n = 0 will touch the parabola y 2 = 4ax if ln = am 2 . Answer: Given equation of the parabola is y 2 = 4ax …….(i) and equation of the line is lx + my + n = 0 ⇒ y = \(\frac{-n-l x}{m}\) …….(ii) From (ii) and (i), \(\left(-\frac{(n+l x)}{m}\right)^2\) = 4ax ⇒ n 2 + 2lnx + l 2 x 2 = 4am 2 x ⇒ l 2 x 2 + (2ln – 4am 2 )x + n 2 = 0 ……(iii) Since, line (ii) touches the parabola (i), so roots of (i) must be equal. i.e., (2ln – 4am 2 ) 2 – 4l 2 n 2 = 0 ⇒ 4l 2 n 2 – 16lnam 2 + 16a 2 m 4 – 4l 2 n 2 = 0 ⇒ a 2 m 4 = lnam 2 ⇒ am 2 = ln Hence proved.

Question 28. Find the coefficient of x 15 in the expansion of (x – x 2 ) 10 . OR If the continued product of three numbers in G.P. is 216 and the sum of their products in pair is 156, find numbers. Answer: Given expansion is (x – x 2 ) 10 Let the term T r+1 be the general term ∴ \(T_{r+1}={ }^{10} C_r x^{10-r}\left(-x^2\right)^r\) = \((-1)^r \cdot{ }^{10} C_r \cdot x^{10-r} \cdot x^{2 r}\) = \((-1)^r{ }^{10} C_r x^{10+r}\) For the coefficient of x 15 , 10 + r = 15 ⇒ r = 5 \(T_{5+1}=(-1)^5{ }^{10} C_5 x^{15}\) ∴ coefficient of x 15 = \(-1 \times \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 !}{5 \times 4 \times 3 \times 2 \times 1 \times 5 !}\) = -3 × 2 × 7 × 6 = -252 OR Let three numbers in G.P. be \(\frac{a}{r}\), a, ar. Given, \(\frac{1}{2}\) . a . ar = 216 ⇒ a 3 = 216 = 6 3 ⇒ a = 6 Therefore sum of their product in pairs are: \(\frac{a}{r}\) × a + a × ar + ar × \(\frac{a}{r}\) = 156 ⇒ \(\frac{36}{r}\) + 36r + 36 = 156 (∵ a = 6) ⇒ \(\frac{36}{r}\) + 36r = 120 ⇒ \(\frac{3}{r}\) + 3r = 10 ⇒ 3r2 – 10r + 3 = 0 ⇒ (3r – 1)(r – 3) = 0 ⇒ r = \(\frac{1}{3}\), 3 Substituting the value of a and r in equation (i), we get 18, 6, 2 when r = \(\frac{1}{3}\) and 2, 6, 18 when r = 3

[This section comprises long answer type questions (LA) of 5 marks each]

Question 35. There are 15 points in a plane out of which only 6 are in a straight line, then (a) how many different straight lines can be made? (b) how many triangles can be made? OR If a, b, c are positive and not all equal, then prove that: (a + b + c) (bc + ca + ab) > 9abc. Answer: (a) A straight line can be formed by joining 2 points, so several ways in which we select 2 points from 15 points is 15 C 2 . But it also includes that in which 6 points are in a straight line. From those 6 points, only one line can be formed. So, we have to subtract 6 C 2 – 1 from the above result. So, Total lines are 15 C 2 – 6 C 2 + 1 = \(\frac{15 !}{2 ! 13 !}-\frac{6 !}{2 ! 4 !}+1\) = \(\frac{15 \times 14}{2 \times 1}-\frac{6 \times 5}{2 \times 1}+1\) = 105 – 15 + 1 = 91 (b) Non-collinear point 9 and collinear point 6 Case I: We choose 1 non-collinear and 2 collinear points to make a triangle. 9 C 1 × 6 C 2 = \(\frac{9 !}{8 ! \cdot 1 !} \times \frac{6 !}{4 ! \cdot 2 !}\) = \(\frac{9 \times 6 \times 5}{2}\) = 135 Case II: We choose 2 non-collinear points and 1 collinear point to make a triangle. 9 C 2 × 6 C 1 = \(\frac{9 !}{2 ! \cdot 7 !} \times \frac{6 !}{1 ! \cdot 5 !}\) = \(\frac{9 \times 8 \times 6}{2}\) = 216 Case III: We choose 3 non-collinear points to make a triangle. 9 C 3 = \(\frac{9 !}{3 ! 6 !}\) = \(\frac{9 \cdot 8 \cdot 7}{3 \cdot 2}\) = 84 Total number of triangles = 135 + 216 + 84 = 435 OR We have, (a + b + c) (bc + ca + ab) – 9abc = abc + a 2 c + a 2 b + b 2 c + abc + b 2 a + c 2 b + c 2 a + abc – 9abc = a(b 2 + c 2 ) + b(c 2 + a 2 ) + c(a 2 + b 2 ) – 6abc = a(b 2 + c 2 – 2bc) + b(c 2 + a 2 – 2ca) + c(a 2 + b 2 – 2ab) = a(b – c) 2 + b(c – a) 2 + c(a – b) 2 which is positive because each term of R.H.S. is positive. Thus, (a + b + c) (bc + ca + ab) – 9abc > 0 (a + b + c) (bc + ca + ab) > 9abc

[This section comprises three case-study/passage-based questions of 4 marks each with sub-parts.]

The first two case study questions have three sub-parts (i), (ii), and (iii) of marks 1, 1, and 2 respectively. The third case study question has two sub-parts of 2 marks each.

Question 36. Read the following passage and answer the questions given below: Ordered Pairs: The ordered pair of two elements a and b is denoted by (a, b): a is the first element (or first component) and b is the second element (or second component). Two ordered pairs are equal if their corresponding elements are equal, i.e., (a, b) = (c, d) ⇒ a = c and b = d Ordered Pair (x, y) x = first coordinate y = second coordinate Cartesian Product of Two Sets For two non-empty sets A and B, the cartesian product A × B is the set of all ordered pairs of elements from sets A and B. In symbolic form, it can be written as A × B = {(a, b) : a ∈ A, b ∈ B} (i) If (a – 3, b + 7) = (3, 7), find the value of a and b. (ii) Let A and B be two sets such that Ax B consists of 6 elements. If three elements of A × B are (1, 4), (2, 6), and (3, 6), then show that (A × B) ≠ (B × A). (iii) If (x + 2, 4) = (5, 2x + y), find the value of x and y. OR If (x + 6, y – 2x) = (0, 6), then find the value of x and y. Answer: (i) We know that two ordered pairs are equal if their corresponding elements are equal. (a – 3, b + 7) = (3, 7) ⇒ a – 3 = 3 and b + 7 = 7 [equating corresponding elements] ⇒ a = 3 + 3 and b = 7 – 7 ⇒ a = 6 and b = 0

(ii) Since, (1, 4), (2, 6), and (3, 6) are elements of A × B, it follows that 1, 2, 3 are elements of A and 4, 6 are elements of B. It is given that A × B has 6 elements. So, A = {1, 2, 3} and B = {4, 6} Hence, A × B = {1, 2, 3} × {4, 6} = {(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6)} and B × A = {4, 6} × {1, 2, 3} = {(4, 1), (4, 2), (4, 3), (6, 1), (6, 2), (6, 3)} Thus A × B ≠ B × A

(iii) (x + 2, 4) = (5, 2x + y) ⇒ x + 2 = 5 ⇒ x = 5 – 2 = 3 Also 4 = 2x + y ⇒ 4 = 2 × 3 + y ⇒ y = 4 – 6 = -2 OR (x + 6, y – 2x) = (0, 6) x + 6 = 0 ⇒ x = -6 y – 2x = 6 ⇒ y = 6 + 2(-6) = -6

(ii) x 2 = -4ay Equation of directrix is y = a ⇒ y = 4 ⇒ y – 4 = 0

## Free Resources

## NCERT Solutions

## Quick Resources

## Syllabus for the Session 2023-24

CBSE Syllabus

## Case Study Questions

Case Study on Sets CS-2 CS-3 CS-4 CS-5

Case Study on Relations & Functions CS-2 CS-3

Case Study on Trigonometric Functions

Case Study on Complex Numbers

Case Study on Linear Inequalities

Case Study on Permutations and Combinations

Case Study on Sequences & Series

Case Study on Straight Lines

Case Study on Conic Sections

Case Study on Statistics

Case Study on Probability

Pdf of Case Studies

## MCQs for Practice

Chapter 1 - Sets

Chapter 2 - Relations & Functions

Chapter 3 - Trigonometric Functions

Chapter 4 - Complex Numbers & Quadratic Equations

Chapter 5 - Linear Inequalities

Chapter 6 - Permutations & Combinations

Chapter 7 - Binomial Theorem

Chapter 8 - Sequences & Series

Chapter 9 - Straight Line

Chapter 1 0 - Conic Sections

Chapter 1 1 - Introduction to Three-dimensional Geometry

Chapter 12 - Limits & D erivatives

Chapter 13 - Statistics

Chapter 14 - Probability

Answers of MCQs

## Assertion & Reasoning Questions

Relations & Functions

Trigonometric Functions

Complex Numbers

Linear Inequalities

Permutations & Combinations

Binomial Theorem

Sequences & Series

Straight Lines

Conic Sections

Introduction to Three-Dimensional Geometry

Limits & derivatives

Probability

## Topic Wise Assignments of Previous Year Questions

Relations and Functions

Straight Line

Limits & Derivatives

## Question Papers - DoE, Delhi

Session 2015-2016

SA2(QP) SA2(MS) COMP(QP) COMP(MS)

Session 201 6 -20 17

SA1(QP) SA2(QP) SA2(MS) COMP(QP) COMP(MS)

Session 201 7 -201 8

SA2(QP) SA2(MS) COMP(QP) COMP(MS)

Session 201 8 -201 9

SA2(QP) SA2(MS) COMP(QP) COMP(MS)

Session 201 9 -20 20

SA2(QP) SA2(MS)

Session 2020-21

Video Explanation: Part A Part B Section III Part B Section IV Part B Section V

Session 2021-22

Question Paper

Video Explanation: Section A Section B Section C

Session 2022-23

Sample Question Paper for Mid-Term Exam

Video Explanation: Section A Section B Section C Section D Section E

Mid-Term Exam: Question Paper

Video Explanation: Section A Section B Section C Section D Section E

Practice Paper for Final Exam

Video Explanation: Section A Section B Section C Section D Section E

Final Exam: Question Paper Marking Scheme

Video Explanation: Section A Section B Section C Section D Section E

Support Material Issued by DoE, Delhi

Session 202 3 -24

Mid-Term Exam: Question Paper

Video Explanation: Section A Section B Section C Section D Section E

Practice Paper 1 for Final Exam: Question Paper

Practice Paper 2 for Final Exam: Question Paper

## Short Capsules (Notes to Revise Concepts)

Chapter 1 - Sets

Chapter 4 - Mathematical Induction

Chapter 5 - Complex Numbers & Quadratic Equations

Chapter 6 - Linear Inequalities

Chapter 7 - Permutations & Combinations

Chapter 8 - Binomial Theorem

Chapter 9 - Sequences & Series

Chapter 10 - Straight Line

Chapter 11 - Conic Sections

Chapter 12 - Introduction to Three-dimensional Geometry

Chapter 13 - Limits & derivatives

Chapter 14 - Mathematical Reasoning

Chapter 15 - Statistics

Chapter 16 - Probability

## Some Basic Concepts to Revise:

Number System

Mensuration Results

Quadratic Equations

Solution of Polynomial Inequality - Wavy Curve Method

Probability__Revision of the Topics studied in Earlier Classes

## Result Sheets

Trigonometric Identities

## CBSE Case Study Questions for Class 11 Maths Complex Numbers Free PDF

Mere Bacchon, you must practice the CBSE Case Study Questions Class 11 Maths Complex Numbers in order to fully complete your preparation . They are very very important from exam point of view. These tricky Case Study Based Questions can act as a villain in your heroic exams!

I have made sure the questions (along with the solutions) prepare you fully for the upcoming exams. To download the latest CBSE Case Study Questions , just click ‘ Download PDF ’.

## CBSE Case Study Questions for Class 11 Maths Complex Numbers PDF

Checkout our case study questions for other chapters.

- Chapter 3 Trigonometric Functions Case Study Questions
- Chapter 4 Principle of Mathematical Induction Case Study Questions
- Chapter 6 Linear Inequalities Case Study Questions
- Chapter 7 Permutations and Combinations Case Study Questions

## How should I study for my upcoming exams?

First, learn to sit for at least 2 hours at a stretch

Solve every question of NCERT by hand, without looking at the solution.

Solve NCERT Exemplar (if available)

Sit through chapter wise FULLY INVIGILATED TESTS

Practice MCQ Questions (Very Important)

Practice Assertion Reason & Case Study Based Questions

Sit through FULLY INVIGILATED TESTS involving MCQs. Assertion reason & Case Study Based Questions

After Completing everything mentioned above, Sit for atleast 6 full syllabus TESTS.

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- Integers Class 7 Case Study Questions Maths Chapter 1

Last Updated on August 16, 2024 by XAM CONTENT

Hello students, we are providing case study questions for class 7 maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class 7 maths. In this article, you will find case study questions for CBSE Class 7 Maths Chapter 1 Integers. It is a part of Case Study Questions for CBSE Class 7 Maths Series.

Integers | |

Case Study Questions | |

Competency Based Questions | |

CBSE | |

7 | |

Maths | |

Class 7 Studying Students | |

Yes | |

Mentioned | |

Table of Contents

## Case Study Questions on Integers

In a test $(+5)$ marks are given for every correct answer and (-3) Marks are given for every incorrect answers and no marks for not attempting any question. Daksh scored 16 marks, while Riya scores $(-8)$ marks Manya scored 10 marks.

Answer the following questions:

Q.1. Total marks scored by Daksh, Riya and Manya: (a) 34 (b) 18 (c) -18 (d) 20

Difficulty Level: Easy

Ans. Option (b) is correct. Explanation: Since, Total marks scored by Darsh, $\begin{aligned} \text { Riya and Mayan } & =16+(-8)+10 \\ & =18\end{aligned}$

Q. 2. Product of Marks scored by all of then is: (a) -2440 (b) -1280 (c) 1440 (d) 20

Ans. Option (b) is correct. Explanation: Since, Product of marks scored by all of them $$ \begin{aligned} & =(16) \times(-8) \times 10 \\ & =16 \times(-80) \\ & =-1280 \end{aligned} $$ Q.3. If Daksh get 5 correct answers, then number of questions be attempt incorrectly is: (a) 7 (b) 5 (c) 3 (d) 6

Difficulty Level: Medium

Ans. Option (c) is correct. Explanation: Total Marks scored by Daksh $=16$ but No. of correct question by Daksh $=5$ Marks scored for correct question $=5 \times 5=25$ Deducting marks of wrong question $=$ Total marks – correct question scored $$ \begin{aligned} & =26-25 \\ & =-9 \end{aligned} $$

Hence, No of incorrect questions $$ \begin{aligned} & =\frac{\text { total dedecting marks }}{\text { one mark no. of dedection }} \\ & =\frac{-9}{-3}=3 \end{aligned} $$ Q. 4. If Riya has attempted 6 questions incorrectly, then find the number of questions she has attempted correctly?

Sol. Total marks scored by Riya =(-8) No. of incorrect question =6 Marks scored by incorrect question =6×(-3)=-18 Marks scored by correct question = total scored by Riya – incorrect question numbers $$ =(-8)-(+8) $$

Hence, No. of correct question $$ \begin{aligned} & =\frac{\text { total mark by correct question }}{\text { mark of } 1 \text { corr.question }} \\ & =\frac{10}{5}=2 \end{aligned} $$

- Triangle and its Properties Class 7 Case Study Questions Maths Chapter 6
- Lines and Angles Class 7 Case Study Questions Maths Chapter 5
- Simple Equations Class 7 Case Study Questions Maths Chapter 4

## Data Handling Class 7 Case Study Questions Maths Chapter 3

Fractions and decimals class 7 case study questions maths chapter 2, topics from which case study questions may be asked.

- Properties of Addition and Subtraction of Integers
- Multiplication of Integers
- Properties of Multiplication of Integers
- Division of Integers
- Properties of Division of Integers

Integers are the set of whole numbers along with the negative numbers.

Case study questions from the above given topic may be asked.

## Operations on Integers

- To add two integers of the same sign, add their absolute values and put the common sign.
- To add two integers of the opposite signs, find the difference in their absolute values and put the sign of the number with greater absolute value.
- The product of two positive integers is always a positive integer.
- The product of a positive and a negative integer is always a negative integer.
- The product of two negative integers is always a positive integer.
- When a positive integer is divided by a negative integer, the quotient is a negative integer and vice versa.
- When a negative integer is divided by a negative integer, the quotient is a positive integer.

Zero is the additive identity and 1 is the multiplicative identity for integers. Therefore, for any integer ‘a’, a + 0 = 0 + a = a and a × 1 = 1 × a = a.

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## Frequently Asked Questions (FAQs) on Integers Case Study

Q1: what are integers.

A1: Integers are a set of numbers that include all the whole numbers (positive), their opposites (negative), and zero. In mathematical terms, integers are the set of numbers {…, -3, -2, -1, 0, 1, 2, 3, …}.

## Q2: How are integers different from whole numbers?

A2: Whole numbers are non-negative numbers including zero, such as 0, 1, 2, 3, and so on. Integers, on the other hand, include both positive and negative numbers along with zero.

## Q3: How do you add integers?

A3: To add two integers: If both integers have the same sign, add their absolute values, and give the result the same sign. If the integers have different signs, subtract the smaller absolute value from the larger absolute value and give the result the sign of the integer with the larger absolute value.

## Q4: What is the rule for subtracting integers?

A4: Subtracting an integer is the same as adding its opposite. For example, to subtract -3 from 5, you add the opposite of -3, which is +3, so the result is 5 + 3 = 8.

## Q5: Can you explain the multiplication rule for integers?

A5: The multiplication rule for integers is: When multiplying two integers with the same sign, the product is positive. When multiplying two integers with different signs, the product is negative.

## Q6: What are the properties of integers?

A6: Some important properties of integers include: Closure: The sum, difference, and product of any two integers will always be an integer. Commutative Property: The order of addition or multiplication does not affect the result. Associative Property: The grouping of numbers in addition or multiplication does not affect the result. Distributive Property: Multiplication distributes over addition or subtraction.

## Q7: Why is zero an important integer?

A7: Zero is a unique integer that is neither positive nor negative. It is the additive identity, meaning any integer added to zero remains unchanged. Zero also serves as the reference point on the number line, separating positive integers from negative integers.

## Q8: How can we represent integers on a number line?

A8: Integers can be represented on a number line with zero at the center, positive integers to the right of zero, and negative integers to the left. The distance of an integer from zero on the number line represents its absolute value.

## Q9: Are there any online resources or tools available for practicing comparing quantities case study questions?

A9: We provide case study questions for CBSE Class 8 Maths on our website . Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes.

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## Case Study Questions for Class 11 Maths Chapter 8 Binomial Theorem

- Last modified on: 11 months ago
- Reading Time: 8 Minutes

Table of Contents

## [PDF] Download Case Study Questions for Class 11 Maths Chapter 8 Binomial Theorem

Here we are providing case study questions for class 11 maths. In this article, we are sharing Class 11 Maths Chapter 8 Binomial Theorem . All case study questions of class 11 maths are solved so that students can check their solutions after attempting questions.

## Related Posts

What is meant by case study question.

In the context of CBSE (Central Board of Secondary Education), a case study question is a type of question that requires students to analyze a given scenario or situation and apply their knowledge and skills to solve a problem or answer a question related to the case study.

Case study questions typically involve a real-world situation that requires students to identify the problem or issue, analyze the relevant information, and apply their understanding of the relevant concepts to propose a solution or answer a question. These questions may involve multiple steps and require students to think critically, apply their problem-solving skills, and communicate their reasoning effectively.

## Importance of Solving Case Study Questions for Class 11 Maths

Case study questions are an important aspect of mathematics education at the Class 11 level. These questions require students to apply their knowledge and skills to real-world scenarios, helping them develop critical thinking, problem-solving, and analytical skills. Here are some reasons why case study questions are important in Class 11 maths education:

- Real-world application: Case study questions allow students to see how the concepts they are learning in mathematics can be applied in real-life situations. This helps students understand the relevance and importance of mathematics in their daily lives.
- Higher-order thinking: Case study questions require students to think critically, analyze data, and make connections between different concepts. This helps develop higher-order thinking skills, which are essential for success in both academics and real-life situations.
- Collaborative learning: Case study questions often require students to work in groups, which promotes collaborative learning and helps students develop communication and teamwork skills.
- Problem-solving skills: Case study questions require students to apply their knowledge and skills to solve complex problems. This helps develop problem-solving skills, which are essential in many careers and in everyday life.
- Exam preparation: Case study questions are included in exams and tests, so practicing them can help students prepare for these assessments.

Overall, case study questions are an important component of Class 11 mathematics education, as they help students develop critical thinking, problem-solving, and analytical skills, which are essential for success in both academics and real-life situations.

## Feature of Case Study Questions on This Website

Here are some features of a Class 11 Maths Case Study Questions Booklet:

Many Case Study Questions: This website contains many case study questions, each with a unique scenario and problem statement.

Different types of problems: The booklet includes different types of problems, such as optimization problems, application problems, and interpretation problems, to test students’ understanding of various mathematical concepts and their ability to apply them to real-world situations.

Multiple-choice questions: Questions contains multiple-choice questions to assess students’ knowledge, understanding, and critical thinking skills.

Focus on problem-solving skills: The questions are designed to test students’ problem-solving skills, requiring them to identify the problem, select appropriate mathematical tools, and analyze and interpret the results.

Emphasis on practical applications: The case studies in the booklet focus on practical applications of mathematical concepts, allowing students to develop an understanding of how mathematics is used in real-life situations.

Comprehensive answer key: The booklet includes a comprehensive answer key that provides detailed explanations and step-by-step solutions for all the questions, helping students to understand the concepts and methods used to solve each problem.

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## Biden Leaves the Stage

On president biden’s private pain since stepping aside, and his public message at the democratic national convention..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

I’m Sabrina Tavernise, and this is “The Daily.” On the first night of the Democratic National Convention, the stage belonged to the man who chose to give it up. Today, I spoke with my colleagues, Katie Rogers and Peter Baker, about Joe Biden’s private pain since stepping aside and his public message here in Chicago. It’s Tuesday, August 20th.

Is this the line we need to get on?

This is the line. Oh, gosh. I’d say we’re going to be here 45 minutes or so. Woof, my goodness. These nice birds and flowers here —

little butterfly.

Oh, there it is. I can see the United Center now. See, it’s through there.

Yeah, but can you see to the front of the line?

No, line, line, line.

I feel like the season is passing as we’re waiting in it. There’s a cicada coming up.

Yeah, I see some leaves turning, yeah.

Welcome. You got it. Have your credentials ready.

Oh, is this the end?

[MUSIC PLAYING]

OK, so we’re through the line. It was, indeed, 45 minutes, as I predicted. And we are standing outside a giant building with a big blue sign on it that says Democratic National Convention. And it is Kamala Harris’s convention. It is her week. But it is Joe Biden’s night. Tonight, he’s going to take the stage, give a speech to an entire party who forced him out.

And many of us know the broad story of how he decided to step aside. But the question always has been, what was the thing in the end that fundamentally made him drop out? But my colleague, Katie Rogers, has gotten inside the final few days for Biden in making that decision. So we’re going to go talk to her about what ultimately drove Biden to make this decision to leave the race.

OK, big “New York Times” area, may be seeing a lot of other people. I know. Oh, here she is. Katie?

How are you?

Are you ready?

Welcome. We’re so glad to have you. So, Katie, just to start, I wonder if you could reflect for a moment about what this convention was supposed to be up until a month ago.

Right, I mean, this is something that President Biden had been looking forward to. He had been workshopping a big convention speech. This is something that Biden didn’t get in 2020. If you remember, we were in the middle of a pandemic. This was all done virtually for him.

This would have been the very first time in his career that he was the presidential nominee at a huge 50,000-person strong convention. You get the balloons. You get the big banners with your Bidenisms broadcast everywhere. It was a moment this president doesn’t really ever get to have.

I mean, it’s been a moment he’s wanted for decades, right? So many people cheering for him and choosing him in a way in person.

Yeah, it’s the validation. He has wanted the validation of being that choice and will not get that now. Instead of this moment where he gets to lay the groundwork for a second term, he is ending his political career in Chicago on stage.

Katie, you and our colleagues have been reporting on why he decided to give up something he wanted so badly. Tell us about what you found.

So it starts almost exactly a month ago. The president is campaigning and comes down with COVID and is rushed home on Air Force One to his beach home in Rehoboth in Delaware. There, he is joined by his wife, Jill Biden, the First Lady. And there are two other aides who are very close to him. Another key advisor was in Rehoboth at the time sort of on standby named Steve Ricchetti. He’s the keeper of the president’s political relationships. So a lot of the party leaders were funneling their alarm and their worry and their unhappiness through Steve Ricchetti.

So Friday, the president was sick and didn’t really get out of bed much. He read the papers by the pool. The next day, though, he gets on the phone, calls Steve Ricchetti, and says, I need you and Mike at the House. And Mike is Mike Donilon, the president’s most senior strategist, the writer of his speeches, really, the keeper of the Biden flame, if you will.

Biden, Ricchetti, and Donilon go over the polling, go over what people in the party are saying, and essentially say to the president, look, if you want to do this, we can get you there. We have been around, all of us, long enough to know that the polling can be wrong. We can get you there.

It’s sort of a cultural belief in the Biden circle to not discount polls, but to look at them and say, well, we have been here before, but it is going to be a climb, and you are going to do it alone without the support of Democratic party leaders. He will be ripping apart his party if he wants to do this.

Finally, Biden, who is still sick and weakened with COVID, says to them, well, what would we tell the American people?

Meaning if he were to drop out?

Exactly, what would we say? And that’s really the first moment that Biden wants to see something on paper that is an announcement that he would be leaving the race. The president says to them that he wants to sleep on it. And they let him go upstairs. He’s sleeping in a spare bedroom because he has COVID.

So he’s alone.

So he’s alone on one of the most important nights of his life. And people close enough to him to know his thinking in this moment have told me that it was not decided. But everybody sort of knew where he was going. The next morning, he gets up. The letter gets finalized, and it is published on X exactly one minute after he calls into his senior staff meeting and is reading the letter to them, telling them that he is leaving the race.

So it sounds like the calculations for him are not about winning or losing, but really, about the party and party unity and not wanting to, as you say, rip it apart.

What do we know about how he’s been feeling about that decision since then? That was a month ago, as we said.

So in the moment, it’s been described to me as peace, a sense of understanding, that he was not willing to do what would have been required of him politically to keep going. And I think in the weeks since, I think he has experienced a mixture of emotions.

The president, it should be noted, I think, is a pretty proud person. There was a really big day a few weeks ago where three American hostages were released from Russian custody as part of a really complicated seven-nation prisoner swap. That is as Jake Sullivan, the president’s national security advisor, was fighting back tears talking to us that day, saying, this is vintage Joe Biden. This is what he’s best at, complicated foreign policy.

That was a moment for somebody like Biden to say, hey, I’m still the president. And I was on the tarmac at midnight when those hostages came off and was talking about the importance of allies, the importance of diplomatic relationships, essentially saying the importance of my experience is this. And if you know how he approaches politics, you know that he is saying in that moment, I have value. This is what I bring. This is what I am good at. And he wants the chances to be able to say that.

It’s like he’s sending a message to the people who pushed him out.

Totally, yes, yes. And speaking of the people who pushed him out, I think there are people close to the president who have told me as much, saying this party will have to reconcile with what it did to a sitting president who the voters had chosen. And I think that there is definite resentment, frustration, distrust.

Former President Obama was somebody who was making a lot of calls to party leaders, including he was in touch with the vice president in the days before the president decided to drop out. There is definitely a sense of resentment toward Nancy Pelosi, the former House Speaker, for saying things like, well, if he does drop out, I think we should have an open convention basically before he’s even made the decision to say, well, here’s how it could look and what I think.

He just sort of is pushed out, right?

Right. So one of the things he did is saying, I fully support the vice president as the person to carry on my legacy and to win this election and prevent another term for Trump. And it really shut down talk of other people jumping into this race. She locked it up. There’s got to be a land speed record for that.

There were these conversations about we don’t want it to be a coronation. But it was pretty close. And that is because Biden made the strongest case he could for her at a moment where the Democratic Party had leaders saying, well, let’s do this, or I would like this, or maybe this person. And Biden all but said, well, no, we’re going with my person. We’re going with the vice president I picked.

So, again, he doesn’t want to see this party infighting. He endorses her so that there’s not this nasty public battle, right?

Again, unity.

It was a unity play. And it was, again, a fear of his that this would happen should he not really make a strong case for her. And it’s interesting because the reporting we had leading up was also that he privately did not necessarily believe that Harris could win and got to this point by the time he decided to leave the race that he clearly believed that she was the one who could.

Yeah. So in a way, one way of looking at this convention is that it’s not his convention, but he did have a big hand in making it her convention.

I think that’s right. It’s definitely not his big event to headline anymore. He’s coming in as not the change candidate that so many Democrats are seeing in Harris now but as this throwback figure almost that is passing the baton to her. And I think he sees the vice president as somebody who is best positioned to carry forward his legacy. She’s taking the baton from me. I saved democracy. She will preserve it.

And inside the United Center where we are, delegates who are arriving are getting little coffee bean packets that say “Cup of Joe” on them. And the digital banners will say, “spread the faith” and a couple other Bidenisms. And then they go dark when he’s done, and they flip over to Harris walls and the messages associated with her campaign. So there’s a lot of talk of gratitude for him.

The thing is, though, is that Joe Biden has been around long enough to understand that the second beat to thank you so much is thank you so much for dropping a campaign that Democratic voters did not believe in. Thank you so much for getting off the stage. And for somebody like him, that has got to be very bittersweet and a tough moment to get through.

Katie, thank you.

Thank you for having me.

So right now, we’re at the arena, and we are about to go inside to talk to delegates. These are people who will formalize Kamala Harris’s nomination. And we are going to talk to them about Joe Biden. How are people seeing him in this moment?

Oh, my God, hi. Are you a delegate?

I am a delegate.

Where are you from?

The great state of Michigan.

OK, my name is Sabrina, and this is Rob.

Hi, Sabrina. Hi, Bob.

Nice to meet you.

Well, his name is Rob, actually.

I’ll take Bob. That’s fine.

Yeah, whatever.

So question for you, one word to describe Joe Biden. If you had one word, what would it be?

^ Honorable.

Courageous.

Delawarean.

He is an American Patriot. He has been such an incredible leader for our nation for years.

He’s been an amazing public servant.

I’m a little misty. I have such affection for him.

I was sad, very sad to see him make the decision he made. But he made the right decision for the country.

In some ways, does part of you wish he was the nominee still?

No, because the election has turned on a dime. It’s going in the right direction now. It was going in a very bad direction before.

I mean, in less than a month, she’s turned over this campaign. She’s brought people together in just incredible ways. Now, that’s not to take away from Joe Biden or to say — I mean, I love Joe Biden.

How is it going to feel tonight watching him speak?

I don’t know, but I brought Kleenex.

I think we’re all here to give him his flowers, to say thank you. We’re going to be proud for him. It’s a valedictory, and he’s deserved it and earned it.

I think that there’s going to be an incredible outpouring of love, respect, and honor for him. I’m not sad for him. I’m proud of him. I’m just, I’m in awe of him, really. And I think we’re living history right now. I mean, we’re living history tonight. I think the history is going to show, wow, what an incredible gift that Joe Biden has given us.

We’ll be right back.

So we are walking to the arena from the Press Center. I’m walking with my friend and admired colleague, Peter Baker.

I’m an admirer of you. And we are first going to have a conversation. And then, we’re going to go in to watch Biden’s speech together. OK, so we’re going to talk to Peter about what he expects from the speech tonight. What do you expect, Peter?

Well, look, it’s a really fascinating moment because Joe Biden has been coming to conventions like this since 1972. He’s given a lot of speeches. But this is the first one where he has to basically give a speech saying I’m done. And it’s something he’s always resisted. His whole thesis of his political career, his whole life really, is if you’re knocked down, you get back up. Well, now, he’s going to say, OK, this is the end for me after more than 50 years on the public stage. So he’s speaking to a couple of different audiences. One, of course, is the audience here in the Hall and the national audience watching on television. And he’s going to try to frame his accomplishments and pass the torch to Kamala Harris. And then, the second audience he’s talking to really is history. He’s trying to say, who is Joe Biden? What did it matter that he was president for four years? Where does he fit in our national story? And it’s going to be a really fascinating speech to watch.

And what do you make of the fact that it’s happening on Monday, the first day of the convention? But this is the day they’ve chosen for him.

Yeah, Monday is, of course, a demotion. He expected to give his speech on Thursday when he was going to accept the nomination. That’s the big night. That’s the night you want to give a speech. He’s been demoted to Monday, which is usually the night for the formers.

And you can already see in the last few weeks, if you go to the White House every day or you watch him give his public presentations, what few he’s done, you can feel the energy and the power and the attention drifting away from him. And it’s a palpable thing when a president becomes a lame duck. And you really have seen it these last few weeks.

And now tonight, he’ll get up there. He’ll give his speech. And then he’s going to get on a plane later tonight. And he’s going to head off to California. And he won’t be here for the rest of the convention. He’s going to literally cede the stage to Kamala Harris. And it’s the symbolic way the Democratic Party is going to say, we’re moving on. All right, the Joe Biden era is essentially over now. And we’re moving on to the Kamala Harris era. And he will be on vacation. You won’t see him in public really again until Labor Day.

I guess, Peter it is really a fundamentally human moment that we’re witnessing here, right? I mean, this is a man who had all of these years in public service. And the one thing he wanted most in the world was this that he’s now giving up. And that feels really just in the sweep of history pretty remarkable and a very human moment.

Yeah, it really is. In some ways, it’s the most human story. It’s actually all of our story because all of us eventually are going to run out of time. He has been in public life for a half century, more than any other president before him, by the way. He has been running for president since a lot of the people in this arena were children. His first campaign was in 1987. His second one, of course, was in 2008. He didn’t win either time. Finally, he wins in 2020.

And so to say, OK, I’m going to give it up after spending a lifetime of trying to get there, it goes against the grain. It just is completely against everything that he sees in himself. His life has been full of tragedy and setback followed by comeback and resilience. And here is the first time he is not bouncing back.

So he’s done the one thing that he prided himself on never doing. But at some point, you cannot outrace time. You cannot outrace the impact of age. And as much as he wanted to stay on the stage, his time has come. And he’s had to accept that. And it’s not an easy thing to do. But tonight brings that to the fore in a very public way. And he’s, I think, going to use this opportunity to frame that exit.

OK, let’s watch that speech. Let’s watch.

We’re going to go over. So we’re sitting at the very top row in the arena, and you can really see everything from here. It’s pretty amazing — thousands and thousands of Delegates and this huge TV screen.

And now, I would like to introduce my father, your 46th President of the United States, Joe Biden.

So he’s smiling. He’s looking up into the crowd. He’s putting his hand over his eyes.

They’re chanting. Thank you, Joe. Thank you, Joe.

Look, his eyes are tearing up. He’s always been an emotional politician, not afraid to show emotion in public. This has got to be one of the most emotional moments of his political career.

My fellow Americans, nearly four years ago, in winter on the steps of the Capitol on a cold January day, I raised my right hand, and I swore an oath to you and to God to preserve, protect, and defend the Constitution and to faithfully. execute the Office of the President of the United States.

In front of me — in front of me was a city surrounded by the National Guard. Behind me, a capital that two weeks before had been overrun by a violent mob. But I knew then from the bottom of my heart that I knew now, there is no place in America for political violence, none.

Biden’s presidency has been rooted in that January 6th attack and the aftermath basically for four years. And in a way, Kamala Harris is going to run a different campaign because rather than talk about the past, she’s trying to say that she’s about the future. As important as January 6th is, a lot of Americans want to move on. And the difference between his approach to it and hers is rather striking.

I stand before you now on this August night to report that democracy has prevailed.

“Democracy has prevailed” is the line he used in his inauguration address in January 2021.

And now, democracy must be preserved.

This was the reason he gave to run, right?

This is the reason he gave to run in 2020. It’s the same narrative, and it’s the same evaluation of where we are as a country, in effect. He’s trying to say this is the battle that we have to wage. And even though he’s not going to be the one waging it now going forward, he’s telling these Democrats that they have to continue the fight.

I also ran to rebuild the backbone of America, the middle class.

So here, he is moving on to legacy.

Yeah, yeah, he’s moving on to legacy. He wants to talk about what he has done these last three years on legislation, on foreign policy, presumably, on domestic policy.

We have done that. And many of those jobs in the so-called fabs, the buildings that make the chips that are being constructed now. And guess what? The average salary in those fabs the size of a football field will be over $100,000 a year. And you don’t need a college degree.

That’s been a big thing for him, a big priority in the last four years, talking about jobs for people who don’t have a college degree, trying to move Democrats away from the idea that they’re elites and that they only care about people who are highly educated. And it’s coming at a time when the demographics have changed. The Republicans have become the party of the less educated voters, and the Democrats have been the party of the more educated voters. And he’s trying to recapture that old Democratic appeal to working class Americans.

Make sure they thought —

Those are his roots, in fact.

Those are his roots, right, “Scranton Joe,” as he conceives himself.

Meanwhile, we made the largest investment commonly in public safety ever. Now, the murder rate is —

So he just mentioned Kamala and I. But here we are about half an hour into this speech, and he really hasn’t gotten around to talking about her yet. He’s made a couple passing references to her, but he hasn’t yet given us the torch-passing part of the speech.

Kamala and I are committed to strengthening legal immigration, including protecting Dreamers and more.

Again, in reference to Kamala, but kind of —

But again, in passing.

— sort of subsidiary.

Yeah, the subsidiary, Kamala and I and then moves on. There’s not — we haven’t yet heard him talk about who she is, what she’s done as vice president.

He’s not giving us a pitch.

No, he presumably will before it’s over. But this is not a speech he had about Kamala.

The brave service members who gave their lives of this country, he called them “suckers and losers.” Who in the hell does he think he is? Who does he think he is?

There’s no words for a person. They’re not the words of a person, not worthy of being commander in chief, period. Not then, not now, and not ever. And I mean that. I mean that from the bottom of my heart.

He’s talked more about Trump tonight so far than he has about Kamala Harris because I think he expected to give a speech tonight making the case against Trump, that he wanted to make that case himself for his own candidacy. And in that sense, a lot of the speeches, the speech he might have given a month ago, except that one key line, “I accept your nomination.”

Folks, I’ve got five months left in my presidency. I’ve got a lot to do. I intend to get it done.

It’s been the honor of my lifetime to serve as your president. I love the job, but I love my country more.

I love my country more.

And all this talk about how I’m angry with all those people who said I should step down, that’s not true.

[CROWD CHANTING]

I love my country more, and we need to preserve our democracy. In 2024, we need you to vote. We need you to keep the Senate. We need you to win back the House of Representatives.

And above all, we need you to beat Donald Trump.

This was his case, right? This was the case he made for himself.

Elect Kamala and Tim.

Elect Kamala and Tim, he’s endorsing them. He’s supporting them, but he hasn’t really talked about them.

We’ve made incredible progress. But we have more work to do. And Kamala and Tim will continue to take on corporate greed and bring down cost of food.

Peter, is he actually passing the torch?

Well, he has changed the subject of the sentence from “I will” to “they will.” And the rest of it is a lot of what he would have said otherwise.

They will continue my policy.

Exactly, exactly. He’s outlining what he would have done in effect and saying Kamala and Tim will be the ones to carry it out.

Selecting Kamala was the very first decision I made when I became our nominee.

All right, we’re talking a little bit about her.

It was the best decision I made my whole career.

He said it was the best decision he ever made in picking Kamala Harris as his vice president. Now, you can say he adds a line, but obviously, he understands right now that in some ways, she’s the most important thing to his legacy because if she wins, he will be remembered for what he accomplished and for sacrificing and stepping aside. And if she loses, people will say he hung on too long through hubris or stubbornness or pride, and he didn’t set the Democrats up for success.

In other words, his legacy is entirely dependent on her and what becomes of her in the next couple of months.

Absolutely right. 10 weeks will decide it.

America, America, I gave my best to you.

So this is the final moment.

The book ends of an extraordinary political career. Whether you support him or not, it’s an American story.

Remember who we are. We’re the United States of America.

This is the line he always uses to finish his speeches.

And there’s nothing we cannot do when we do it together.

God bless you all, and may God protect our troops.

The crowd is roaring.

He turns around, looks behind him.

So about 48 minutes, long speech. It’s now well after midnight in the east. But the crowd here was appreciative, even if some of them began to drift away. Behind him and behind Jill are graphics saying, “thank you, Joe” and “we love Joe.”

Here’s Kamala Harris.

Kamala and her husband, Doug, are joining them. They’re embracing.

I love you so much, he just said. Kamala Harris, you could see her lips. I love you so much, he said, and gave her a hug. And then she just said, I love you, again. This is effectively the torch being passed, in effect, sending the signal, this call to the country and the world that he sees her as his successor. And it’s her party now. And if she can win in November, it’ll be her White House.

So he’s clearing the stage. The next three nights of this convention, all in prime time, will all be Kamala Harris and the people who are promoting Kamala Harris. Joe Biden won’t be here anymore, and it won’t be his convention.

And he’s leaving the stage, Peter.

Now, he’s leaving the stage, literally, and you can argue, politically.

That’s it. He’s gone.

Here’s what else you should know today.

In a very constructive meeting with Prime Minister Netanyahu today, he confirmed to me that Israel accepts the bridging proposal, that he supports it.

Israel accepted an important proposal put forward by international mediators, including the US, who were trying to broker a last ditch plan for a ceasefire between Israel and Hamas. The proposal was described as a bridge toward a larger peace agreement. And Israel’s support prompted American Secretary of State, Antony Blinken, to call on Hamas to back the proposal as well.

The next important step is for Hamas to say yes, and then, in the coming days, for all of the expert negotiators to get together to work on clear understandings on implementing the agreement.

But it’s unclear whether Hamas will back the bridge proposal. Hamas has repeatedly claimed that the ceasefire negotiations have been slanted toward Israel. And in a sign of its skepticism, the group’s military wing carried out a provocative suicide bombing in Tel Aviv on Sunday night.

Today’s episode was produced by Rob Szypko, Lynsea Garrison, Carlos Prieto, Stella Tan, and Jessica Cheung. It was edited by Rachel Quester and Paige Cowett, contains original music by Marion Lozano, Dan Powell, and Rowan Niemisto, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

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Hosted by Sabrina Tavernise

Featuring Katie Rogers and Peter Baker

Produced by Rob Szypko Lynsea Garrison Carlos Prieto Stella Tan and Jessica Cheung

Edited by Rachel Quester and Paige Cowett

Original music by Marion Lozano Dan Powell Rowan Niemisto and Corey Schreppel

Engineered by Chris Wood

## Listen and follow ‘The Daily’ Apple Podcasts | Spotify | Amazon Music | YouTube | iHeartRadio

On the first night of the Democratic National Convention, the stage belonged to the man who chose to give it up.

Katie Rogers and Peter Baker, White House correspondents for The Times, discuss President Biden’s private pain since stepping aside, and his public message in Chicago.

## On today’s episode

Katie Rogers , a White House correspondent for The New York Times.

Peter Baker , the chief White House correspondent for The New York Times.

## Background reading

Biden defended his record and endorsed Kamala Harris : “America, I gave my best to you.”

Analysis: The speech Biden never wanted to give .

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Michael Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Katie Rogers is a White House correspondent. For much of the past decade, she has focused on features about the presidency, the first family, and life in Washington, in addition to covering a range of domestic and foreign policy issues. She is the author of a book on first ladies. More about Katie Rogers

Peter Baker is the chief White House correspondent for The Times. He has covered the last five presidents and sometimes writes analytical pieces that place presidents and their administrations in a larger context and historical framework. More about Peter Baker

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Importance of Solving Case Study Questions for Class 11 Maths. Case study questions are an important aspect of mathematics education at the Class 11 level. These questions require students to apply their knowledge and skills to real-world scenarios, helping them develop critical thinking, problem-solving, and analytical skills.

Hello students, we are providing case study questions for class 7 maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class 7 maths.

"Little is certain except that the fighting and human suffering are not likely to end soon." Oct. 11, 2023 The New York Police Department said it was investigating an alleged assault against an Israeli School of General Studies student—a case that garnered widespread media attention and first thrust Columbia into the national spotlight.

Case Study Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations. February 25, 2023 Physics Gurukul Leave a Comment.

Hello students, we are providing case study questions for class 7 maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class 7 maths.

Importance of Solving Case Study Questions for Class 11 Maths. Case study questions are an important aspect of mathematics education at the Class 11 level. These questions require students to apply their knowledge and skills to real-world scenarios, helping them develop critical thinking, problem-solving, and analytical skills.

Biden Leaves the Stage On President Biden's private pain since stepping aside, and his public message at the Democratic National Convention.