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NCERT Solutions for Class 11 Maths Chapter 9 Sequences And Series Miscellaneous Exercise- In this article, you will get NCERT solutions for Class 11 Maths Chapter 9 miscellaneous exercise. The miscellaneous exercise consists of mixed kinds of questions from all the topics covered in this chapter. You must have solved the previous exercise of this chapter. Now you can try to solve problems from the miscellaneous exercise chapter 9 Class 11.
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As you have already familiar with the NCERT syllabus Class 11 Maths chapter 9 concepts, you can try to solve problems from miscellaneous exercise chapter 9 Class 11. These problems are a bit lengthy and complex as compared to the other exercise of this chapter but these problems will check your understanding of this chapter. You don't need to worry if you are not able to solve these problems by yourself at first. Class 11 Maths Chapter 9 miscellaneous exercise solutions are here to help you where you will get detailed miscellaneous solutions. The class 11 maths ch 9 miscellaneous exercise solutions Maths Miscellaneous Exercises are meticulously crafted by subject experts from Careers360. Each step is explained in detail, providing comprehensive and well-explained solutions. Furthermore, the availability of a PDF version of class 11 chapter 9 maths miscellaneous solutions allows students to access the solutions offline, providing flexibility and convenience to study according to their preferences. You can check NCERT Solutions link if you are looking for NCERT solutions for all the Classes at one place.
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** As per the CBSE Syllabus for the academic year 2023-24, it has been noted that miscellaneous exercise class 11 chapter 9 has been renumbered and is now recognized as Chapter 8.
Answer:
Let a be first term and d be common difference of AP.
Kth term of a AP is given by,
Hence, the sum of and terms of an A.P. is equal to twice the term.
Question:2 If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Answer:
Let three numbers of AP are a-d, a, a+d.
According to given information ,
When d=3, AP= 5,8,11 also if d=-3 ,AP =11,8,5.
Thus, three numbers are 5,8,11.
Question:3 Let the sum of n, 2n, 3n terms of an A.P. be , respectively, show that
Answer:
Let a be first term and d be common difference of AP.
Subtract equation 1 from 2,
Hence, the result is proved.
Question:4 Find the sum of all numbers between 200 and 400 which are divisible by 7.
Answer:
Numbers divisible by 7 from 200 to 400 are
This sequence is an A.P.
Here , first term =a =203
common difference = 7.
We know ,
The sum of numbers divisible by 7from 200 to 400 is 8729.
Question:5 Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
Answer:
Numbers divisible by 2 from 1 to 100 are
This sequence is an A.P.
Here , first term =a =2
common difference = 2.
We know ,
Numbers divisible by 5 from 1 to 100 are
This sequence is an A.P.
Here , first term =a =5
common difference = 5.
We know ,
Numbers divisible by both 2 and 5 from 1 to 100 are
This sequence is an A.P.
Here , first term =a =10
common difference = 10
We know ,
Thus, the sum of integers from 1 to 100 that are divisible by 2 or 5 is 3050.
Question:6 Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Answer:
Numbers divisible by 4, yield remainder as 1 from 10 to 100 are
This sequence is an A.P.
Here , first term =a =13
common difference = 4.
We know ,
The sum of numbers divisible by 4 yield 1 as remainder from 10 to 100 is 1210.
Question:7 If f is a function satisfying f (x +y) = f(x) f(y) for all x, y N such that f(1) = 3 and
, find the value of n.
Answer:
Given : f (x +y) = f(x) f(y) for all x, y N such that f(1) = 3
Taking , we have
is forms a GP with first term=3 and common ratio = 3.
Therefore,
Thus, value of n is 4.
Answer:
Let the sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2
Therefore,
Thus, the value of n is 6.
Last term of GP=6th term
The last term of GP =160
Answer:
Given: The first term of a G.P. is 1. The sum of the third term and fifth term is 90.
Thus, the common ratio of GP is .
Answer:
Let three terms of GP be
Then, we have
...............................................1
from an AP.
....................................................................2
From equation 1 and 2, we get
If r=2, GP = 8,16,32
If r=0.2, GP= 32,16,8.
Thus, the numbers required are 8,16,32.
Answer:
Let GP be
Number of terms = 2n
According to the given condition,
Let the be GP as
Thus, the common ratio is 4.
Answer:
Given : first term =a=11
Let AP be
Given: The sum of the first four terms of an A.P. is 56.
Also, The sum of the last four terms is 112.
Thus, the number of terms of AP is 11.
Question:13 If then show that a, b, c and d are in G.P.
Answer:
Given :
Taking ,
Taking,
From equation 1 and 2 , we have
Thus, a,b,c,d are in GP.
Question:14 Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that
Answer:
Ler there be a GP
According to given information,
To prove :
LHS :
Hence proved
Question:15 The pth, qth and rth terms of an A.P. are a, b, c, respectively. Show that
Answer:
Given: The pth, qth and rth terms of an A.P. are a, b, c, respectively.
To prove :
Let the first term of AP be 't' and common difference be d
Subtracting equation 2 from 1, we get
Subtracting equation 3 from 2, we get
Equating values of d, from equation 4 and 5, we have
Hence proved.
Question:16 If are in A.P., prove that a, b, c are in A.P.
Answer:
Given: are in A.P.
Thus, a,b,c are in AP.
Question:17 If a, b, c, d are in G.P, prove that are in G.P.
Answer:
Given: a, b, c, d are in G.P.
To prove: are in G.P.
Then we can write,
Let be in GP
LHS:
Hence proved
Thus, are in GP
Answer:
Given: a and b are the roots of
Then,
Also, c, d are roots of
Given: a, b, c, d form a G.P
Let,
From 1 and 2, we get
and
On dividing them,
When , r=2 ,
When , r=-2,
CASE (1) when r=2 and x=1,
i.e. (q + p) : (q – p) = 17:15.
CASE (2) when r=-2 and x=-3,
i.e. (q + p) : (q – p) = 17:15.
Question:19 The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that
Answer:
Let two numbers be a and b.
According to the given condition,
...................................................................1
We get,
.....................................................2
From 1 and 2, we get
Putting the value of a in equation 1, we have
Question:20 If a, b, c are in A.P.; b, c, d are in G.P. and 1/c , 1/d , 1/e are in A.P. prove that a, c, e are in G.P.
Answer:
Given: a, b, c are in A.P
Also, b, c, d are in G.P.
Also, 1/c, 1/d, 1/e are in A.P
To prove: a, c, e are in G.P. i.e.
From 1, we get
From 2, we get
Putting values of b and d, we get
Thus, a, c, e are in G.P.
Question:21(i) Find the sum of the following series up to n terms:
Answer:
is not a GP.
It can be changed in GP by writing terms as
to n terms
Thus, the sum is
Question:21(ii) Find the sum of the following series up to n terms: .6 +. 66 +. 666+…
Answer:
Sum of 0.6 +0. 66 + 0. 666+….................
It can be written as
to n terms
Question:22 Find the 20th term of the series terms.
Answer:
the series =
Thus, the 20th term of series is 1680
Question:23 Find the sum of the first n terms of the series: 3+ 7 +13 +21 +31 +…
Answer:
The series: 3+ 7 +13 +21 +31 +…..............
n th term =
Question:24 If are the sum of first n natural numbers, their squares and their cubes, respectively, show that
Answer:
To prove :
From given information,
Here ,
Also,
From equation 1 and 2 , we have
Hence proved .
Question:25 Find the sum of the following series up to n terms:
Answer:
n term of series :
Here, are in AP with first term =a=1 , last term = 2n-1, number of terms =n
Question:26 Show that
Answer:
To prove :
the nth term of numerator
nth term of the denominator
Numerator :
Denominator :
From equation 1,2,3,we have
Hence, the above expression is proved.
Answer:
Given : Farmer pays Rs 6000 cash.
Therefore , unpaid amount = 12000-6000=Rs. 6000
According to given condition, interest paid annually is
12% of 6000,12% of 5500,12% of 5000,......................12% of 500.
Thus, total interest to be paid
Here, is a AP with first term =a=500 and common difference =d = 500
We know that
Sum of AP:
Thus, interest to be paid :
Thus, cost of tractor = Rs. 12000+ Rs. 4680 = Rs. 16680
Answer:
Given: Shamshad Ali buys a scooter for Rs 22000.
Therefore , unpaid amount = 22000-4000=Rs. 18000
According to the given condition, interest paid annually is
10% of 18000,10% of 17000,10% of 16000,......................10% of 1000.
Thus, total interest to be paid
Here, is a AP with first term =a=1000 and common difference =d = 1000
We know that
Sum of AP:
Thus, interest to be paid :
Thus, cost of tractor = Rs. 22000+ Rs. 17100 = Rs. 39100
Answer:
The numbers of letters mailed forms a GP :
first term = a=4
common ratio=r=4
number of terms = 8
We know that the sum of GP is
costs to mail one letter are 50 paise.
Cost of mailing 87380 letters
Thus, the amount spent when the 8th set of the letter is mailed is Rs. 43690.
Answer:
Given : A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually.
Interest in fifteen year 10000+ 14 times Rs. 500
Amount in 15 th year
Amount in 20 th year
Answer:
Cost of machine = Rs. 15625
Machine depreciate each year by 20%.
Therefore, its value every year is 80% of the original cost i.e. of the original cost.
Value at the end of 5 years
Thus, the value of the machine at the end of 5 years is Rs. 5120
Answer:
Let x be the number of days in which 150 workers finish the work.
According to the given information, we have
Series is a AP
first term=a=150
common difference= -4
number of terms = x+8
Since x cannot be negative so x=17.
Thus, in 17 days 150 workers finish the work.
Thus, the required number of days = 17+8=25 days.
Class 11 Maths chapter 9 miscellaneous solutions consist of questions related to finding the nth term of the arithmetic progression, the sum of the terms of arithmetic progression, arithmetic mean, nth term of the geometric progression, the sum of the terms of geometric progression, geometric mean, etc. There are few solved examples given before the miscellaneous exercise chapter 9 Class 11 that you can try to solve.
The Miscellaneous Exercise in NCERT Solutions for Class 11 Maths Chapter 9 - Sequences and Series covers the following topics:
By engaging with the concepts presented in NCERT Solutions for Class 11 Maths, students can address any doubts related to these topics, establishing a strong foundation for their understanding of Class 12 Maths.
Also Read| Sequences And Series Class 11 Notes
Also see-
Happy learning!!!
Sequence means an arrangement of numbers in a definite order according to some rule.
The finite sequence is s sequence containing a finite number of terms.
The infinite sequence is s sequence containing an infinite number of terms.
If the terms of a sequence are expressed as the sum of terms then it is called a series.
Sum of A.P. = n(a+l)/2
Arithmetic mean (A.M.) = (5+9)/2 = 7
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A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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