case study questions for class 11 maths

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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.

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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.

* Helpful in self evaluation

* To know about the pattern of CBSE

* Help to adjust a proper time management

* Available in PDF format

* Chapter wise CBSE sample papers

* CBSE sample papers for class 10

* CBSE sample paper class 12

* CBSE sample paper 2021-22 class 12 Solutions

* CBSE sample paper class 11

* CBSE Sample paper class 9

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Mathematics

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 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.

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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.

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 Maths Important Questions
Class 11 Maths

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.

For more study material and the latest updates on CBSE/ICSE/State Board/Competitive exams, keep visiting BYJU’S. Also, download the BYJU’S App, to watch interactive study videos.

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NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

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

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q12

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.

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q14

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.

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q17

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 $\frac{1}{3}$.

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q23

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.

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q27

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

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q29

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

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q32

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

CBSE Sample Papers for Class 11 Maths Set 4 with Solutions Q37

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

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Case Study on Complex Numbers

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Chapter 8 - Sequences & Series

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Chapter 1 1 - Introduction to Three-dimensional Geometry

Chapter 12 - Limits & D erivatives

Chapter 13 - Statistics

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