Here you get the CBSE Class 10 Mathematics chapter 5, Arithmetic Progressions: NCERT Exemplar Problems and Solutions (Part-IVB). This part of the chapter includes solutions of Question Number 6 to 10 from Exercise 5.4 of NCERT Exemplar Problems for Class 10 Mathematics Chapter: Arithmetic Progressions. This exercise comprises of only the Long Answer Type Questions framed from various important topics in the chapter. Each question is provided with a detailed solution.
CBSE Class 10 Mathematics Syllabus 2017-2018
NCERT Exemplar problems are a very good resource for preparing the critical questions like Higher Order Thinking Skill (HOTS) questions. All these questions are very important to prepare for CBSE Class 10 Mathematics Board Examination 2017-2018 as well as other competitive exams.
Find below the NCERT Exemplar problems and their solutions for Class 10 Mathematics Chapter, Arithmetic Progressions:
Exercise 5.4
Long Answer Type Questions (Q. No. 6-10):
Question. 6 The ratio of the 11^{th} term to the 18^{th} term of an AP is 2 : 3. Find the ratio of the 5^{th} term to the 21^{st} term and also the ratio of the sum of the first five terms to the sum of the first 21 terms.
Solution.
Let a and d be the first term and common difference of an AP.
Now nth term of an AP is given as:
an = a + (n – 1)d
∴ 11^{th} term of AP, a_{11} = a + 10d
And, 18^{th} term of AP, a_{18} = a + 17d
According to question,
So, ratio of the sum of the first five terms of the AP to the sum of the first 21 terms is given as
S_{5} : S_{21} = 30d : 294d = 5 : 49
Question. 7 Show that the sum of an AP whose first term is a, the second term b and the last
Hence proved.
Question. 8 Solve the equation – 4 + (–1) + 2 + ... + x = 437.
Solution.
Given equation is, – 4 – 1 + 2 + .... + x = 437 .... (i)
LHS of equation (i) forms an AP with
First term, a = – 4,
Common difference, d = 3,
And last term, l = a_{n} = x
Here, x cannot be negative as for x = −53, n will be negative which is not possible
Hence, the required value of x is 50.
Question. 9 Jaspal singh repays his total loan of Rs. 118000 by paying every month starting with the first installment of Rs. 1000. If he increases the installment by Rs. 100 every month, what amount will be paid by him in the 30^{th} installment? What amount of loan does he still have to pay after the 30^{th} installment?
Solution.
Jaspal singh takes total loan = Rs. 118000
Monthly instalments paid by Jaspal are Rs. 1000, Rs. 1100, Rs. 1200, …. Which form an AP with:
First term, a = 1000
Common difference, d = 100
30^{th} term of this AP, a_{30} = 1000 + (30 − 1)100 [Using, a_{n} = a + (n−1)d]
= 1000 + 29 × 100
= 1000 + 2900 = 3900
So, Rs. 3900 will be paid by Jaspal in the 30^{th} installment.
Now, Total amount paid in 30 installments is = 1000 + 1100 + 1200 + ..... + 3900
Here, First term, a = 1000,
Last term, l = 3900
And, number of terms, n = 30
Total amount that Jaspal still have to pay after the 30^{th} installment
= (Amount of loan) – (Sum of 30 installments)
= 118000 – 73500 = Rs. 44500
Question. 10 The students of a school decided to beautify the school on the annual day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags.
Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance she did cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?
Solution.
Thus Ruchi have to place 13 flags on the left of 14th flag and 13 flags on its right.
Similarly, for placing third flag and return his initial position, distance travelled = 4+ 4= 8m
For placing fourth flag and return his initial position, distance travelled = 6+ 6 = 12 m.
For placing fourteenth flag and return his initial position, distance travelled
= 26 + 26 = 52m
Proceed same manner into her right position from middle most flag i.e., 14th flag.
Distance between two adjacent flags = 2m
Therefore distance travelled to place first flag = 4m
Distance travelled to place second flag = 8m
Distance travelled to place 13^{th} flag = 52 m
Also, when Ruchi placed the last flag she returns his middle most position and collects her books. This distance also included in placed the last flag.
So, these distances form a series.
4 + 8+ 12 + 16 + .... + 52 [For left side]
And 4 + 8 + 12 + 16 + ..... + 52 [For right side]
Here, first term, a = 4
Common difference, d = 8 – 4 = 4
And, number of terms, n = 13
Now, the maximum distance she travelled carrying a flag = Distance travelled by Ruchi during placing the 14th flag in her left position or 27th flag in her right position
= (2 + 2 + 2 + ... + 13 times)
= 2 × 13 = 26 m
Hence, the maximum distance she travelled carrying a flag is 26 m.
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