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NCERT Exemplar Solution for CBSE Class 10 Mathematics: Arithmetic Progressions (Part-II)

Jul 5, 2017 11:16 IST

    Class 10 Maths NCERT Exemplar, Arithmetic Progressions NCERT Exemplar Problems, NCERT Exemplar Problems, Class 10 Chapter 5 NCERT ExemplarHere you get the CBSE Class 10 Mathematics chapter 5, Arithmetic Progressions: NCERT Exemplar Problems and Solutions (Part-II). This part of the chapter includes solutions for Exercise 5.2 of NCERT Exemplar Problems for Class 10 Mathematics Chapter: Arithmetic Progressions. This exercise comprises of only the Very Short Answer Type Questions framed from various important topics in the chapter. Each question is provided with a detailed solution.

    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 – 5.2

    Very Short Answer Type Questions

    Question. 1 Which of the following form of an AP? Justify your answer.

    Arithmetic Progressions Short Questions

    Solution.

    A series of numbers will form an AP, if d1 = d2 = d3

    (i) −1, −1, −1, −1, ….

    Here, t1 = −1, t2 = −1, t3 = −1 and t4 = − 1

    Now,   d1 = t2 – t1 = − 1 + 1 = 0

                d2 = t3 − t2 = − 1 + 1 = 0

                d3 = t4 − t3 = − 1 + 1 = 0

    As d1 = d2 = d3, therefore given list of numbers forms an AP.

    (ii) 0, 2, 0, 2, …

    Here, t1 = 0, t2 = 2, t3 = 0 and t4 = 2

    Now,   d1 = t2 −  t1 = 2 − 0 = 2

                d2 = t3 − t2 = 0 − 2 = − 2

                d3 = t4 −  t3 = 2 − 0 = 2

    As d1 ≠ d2, therefore given list of numbers does not form an AP.

    (iii) 1, 1, 2, 2, 3, 3, …..

    Here, t1 = 1, t2 = 1, t3 = 2 and t4 = 2

    Now,   d1 = t2 −  t1 = 1 − 1 = 0

                d2 = t3 − t2 = 2 – 1 = 1

                d3 = t4 − t3 = 2 − 2 = 0

    As d1 ≠ d2, therefore given list of numbers does not form an AP.

     (iv) 11, 22, 33, …..

    Here, t1 = 11, t2 = 22 and t3 = 33

    Now,   d1 =  t2 − t1 = 22 − 11 = 11

                d2 = t3 − t2 = 33 − 22 = 11

                d3 = t4 − t3 = 33 − 22 = 11

    As d1 = d2 = d3, therefore given list of numbers forms an AP.

    Arithmetic Progressions Class 10

    As d1 ≠ d2, therefore given list of numbers does not form an AP.

    (vi) 2, 22, 23, 24,…. i.e., 2, 4, 8, 16, ……

    Here,   t1 = 2, t2 = 4, t3 = 8 and t4 = 16

    Now,   d1 = t2 − t1 = 4 − 2 = 2

    And     d2 = t3 − t2 = 8 − 4 = 4

    As d1 ≠ d2, therefore given list of numbers does not form an AP.

    Arithmetic Progressions Important Questions

    As d1 = d2 = d3, therefore given list of numbers forms an AP.

    Arithmetic Progressions NCERT Exemplar

    Since, here a2 – a1 = a3 – a2 but a3 – a2 ≠ a4 – a3, i.e., d2 ≠ d3 therefore given series of numbers does not form an AP.

    Question. 3 For the AP −3, −7, −11,….. can we find directly a30 – a20 without actually finding a30 and a20? Give reason for your answer.

    Solution.

    Yes.

    We know that nth term of an AP ia given as,

                            an = a + (n − 1) d

    Therefore,        a30 = a + (30 − 1) d = a + 29d

    And                 a20 = a + (20 - 1) d = a + 19d                                     

    ⟹                   a30 – a20 = (a + 29d) – (a + 19d) = 10d

    Now, form given AP common difference, d = −7 – (−3) = −7 + 3 = −4

    Therefore,        a30 – a20 = 10 (–4) = –40                                             

    Question. 4 Two AP’s have the same common difference. The first term of one AP is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference any two corresponding terms. Why?

    Solution.

    Let the same common difference of two APs be d.

    Given, first term of first AP, a1 = 2

    And, first term of second AP, a1’= 7

    Using, an = a + (n − 1)d,   

    10th term of first AP, a10 = 2 + 9d

    And 10th term of second AP, a10’ =7 + 9d

    So, a10 − a10’ = (2 + 9d) −  (7 + 9d) = −5

    Again using an = a + (n − 1)d,

    21st term of first AP, a21 = 2 + 20d 

    And 21st term of second AP, a21’ =  7 + 20d

    So, a21 − a21’ = (2 + 9d) − (7 + 20d) = −5

    Also, if the an and an’ are the nth terms of first and second AP.

    Then, an – an’ = [2 + (n – 1)d] – [7 + (n – 1)d] = –5

    Hence, the difference between any two corresponding terms of such AP’s is the same as the difference between their10th and 21st terms.

    Question. 5 Is 0 a term of the AP 31, 28, 25,…? Justify your answer.

    Solution.

    Let 0 be the nth term of given AP, i.e., an = 0.

    Now, for given AP,

    First term, a = 31

    And common difference, d = 28 – 31 = −3

    As nth terms of an AP is given as,

                an = a + (n − 1)d

    According to question,

    Since, n comes out to be fraction but not a positive integer. Therefore, 0 cannot be a term of the given AP.

    Question. 6 The taxi fare each km, when the fare is Rs. 15 for the first km and Rs. 8 for each additional km, does not form an AP as the total fare (in Rs.) After each km is 15, 8, 8, 8, …… . Is the statement true? Given Reasons.

    Solution.

    No, the statement is not ture as the total fare (in Rs.) after each km is given as:

                15, (15 + 8), (15 + 2 × 8), (15 + 3 × 8),… . = 15, 23, 31, 39,…

    Let       t1 = 15, t2 = 23, t3 = 31 and t4 = 39

    Now,   d1 = t2 – t1 = 23 – 5 = 8

                d2 = t3 – t2 = 33 – 23 = 8

    And     d3 ­= t4 – t3 = 39 – 31 = 8

    As d1 = d2 = d3, therefore given list of numbers formed by taking total fares after each Km forms an AP.

    Question. 7 In which of the following situations, do the lists of numbers involved form an AP? Give reasons for your answers.

    (i) The fee charged from a student every month by a school for the whole session, when the monthly is Rs. 400.

    (ii) The fee charged every month by a school from classes I to XII, when the monthly fee for class I is Rs. 250 and it increase by Rs. 50 for the next higher class.

    (iii) The amount of money in the account of Varun at the end of every year when Rs. 1000 is deposited at simple interest of 10% per annum.

    (iv) The number of bacteria in certain food item after each second, when they double in every second.

    Solution.

    The fee charged from a student every month by a school for the whole session is

                                                    400, 400, 400, 400,…

    As the difference between successive terms of given list of numbers is same as 0, so it forms an AP, with common difference, d = 0.

    (ii) The monthly fee from I to XII is given as:

                250, (250+50), (250 + 2 × 50), (250 + 3 × 50),….

    i.e.,      250, 300, 350, 400,….

    Here,   d1 = 300 – 250 = 50

                d2 = 350 – 300 = 50

                d3 = 400 – 350 = 50

    As, d1 = d2 = d3, therefore given list of numbers formed by monthly fees from I to XII forms an AP.

    So, the amount of money in the account of Varun at end of every year is given as:

                1000, (1000 + 100 × 1), (1000 + 100 × 2), (1000 + 100 × 3),…..

    i.e.,      1000, 1100, 1200, 1300,….

    Here,   d1 = 1100 – 1000 = 100

                d2 = 1200 – 1100 = 100

                d3 = 1300 – 1200 = 100

    As, d1 = d2 = d3, therefore given list of numbers forms an AP.

    (iv) Let the number of bacteria present in food initially = x

    Since, they double in every second.

    Therefore, number of bacteria after eavery second are given as:

                x, 2x, 2 (2x), 2 (2·2·x),…..

    i.e.,      x, 2x, 4x, 8x,……

    Here,   d1 = 2xx = x

                d2 = 4x – 2x = 2x

                d2 = 8x – 4x = 4x

    As, d1 = d2 = d3, therefore given list of numbers forms an AP.

    Question. 8 Justify whether it is true to say that the following are the nth terms of an AP.

    (i) 2n – 3

    (ii) 3n2 + 5

    (iii) 1 + n + n2

    Solution.

    Concept used:To prove for the given term to be nth term of an AP, the series formed by that term must have same common difference.

    (i) Yes.

    Here, an = 2n – 3

    ⟹       a1 = 2(1) – 3 = −1

                a2 = 2(2) – 3 = 1

                a3 = 2(3) – 3 = 3

                a4 = 2(4) – 3 = 5

    Now, d1 = a2 – a1 = 1 – (–1) = 1 + 1 = 2

                d2 = a3 – a2 = 3 – 1 = 2

                d3 = a4 – a3 = 5 – 3 = 2

    As, d1 = d2 = d3, therefore list of numbers −1, 1, 3, 5,…. forms an AP.

    Thus, 2n – 3 is the nth term of an AP.

    (ii) No.

    Here an = 3n2 + 5

    ⟹       a1 = 3(1)2 + 5 = 8

    ⟹       a2 = 3(2)2 + 5 = 3(4) + 5 = 17

    ⟹       a3 = 3(3)2 + 5 = 3(9) + 5 = 27 + 5 = 32

    ⟹       a4 = 3(4)2 + 5 = 3(16) + 5 = 48 + 5 = 53

    Now, d1 = a2 – a1 = 17 – 8 = 9

    d2 = a3 – a2 = 32 – 17 = 15

    d3 = a4 – a3 = 53 – 32 = 21

    As, d1 ≠ d2 ≠ d3, therefore list of numbers 8, 17, 32, 53,……does not form an AP.

    Thus, 3n2 + 5 is not the nth term of an AP.

     (iii) No.

    Here    an = 1 + n + n2

    ⟹       a1 = 1 + 1 + (1)2 = 3

    ⟹       a2 = 1 + 2 + (2)2 = 1 + 2 + 4 = 7

    ⟹       a3 = 1 + 3 + (3)2 = 1 + 3 + 9 = 13

    ⟹       a4 = 1 + 3 + (4)2 = 1 + 4 + 16 = 21

    Now,   d1 = a2 – a1 = 7 – 3 = 4

                d2 = a3 – a2 = 13 – 7 = 6

                d3 = a4 – a3 = 21 – 13 = 8                   

    As, d1 ≠ d2 ≠ d3, therefore list of numbers 3, 7, 13, 21,…. does not form an AP.

    Thus, 1 + n + n2 is not the nth term of an AP.

    You may also like to read:

    CBSE Class 10 Mathematics Syllabus 2017-2018

    CBSE Class 10 NCERT Textbooks & NCERT Solutions

    NCERT Solutions for CBSE Class 10 Maths

    NCERT Exemplar Problems and Solutions Class 10 Science: All Chapters

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