CBSE Class 10 Arithmetic Progressions Sure Shot Questions Set B

CBSE Class 10 Arithmetic Progressions Sure Shot Questions Set B. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

1. Find the sum of first 24 terms of the AP 5, 8, 11, 14,…….

2. Find the sum: 25 + 28 + 31 +……….. + 100.

3. Find the sum of first 21 terms of the AP whose 2^nd term is 8 and 4^th term is 14.

4. If the nth term of an AP is (2n + 1), find the sum of first n terms of the AP.

5. Find the sum of first 25 terms of an AP whose nth term is given by (7 – 3n).

6. Find the sum of all two-digit odd positive numbers.

7. Find the sum of all natural number between 100 and 500 which are divisible by 8.

8. Find the sum of all three digit natural numbers which are multiples of 7.

9. How many terms of the AP 3, 5, 7, 9,… must be added to get the sum 120?

10. If the sum of first n, 2n and 3n terms of an AP be S₁, S₂ and S₃ respectively, then prove that S₃ =3(S₂ – S₁).

11. If the sum of the first m terms of an AP be n and the sum of first n terms be m then show that the sum of its first (m + n) terms is –(m + n).

12. If the sum of the first p terms of an AP is the same as the sum of first q terms (where p ≠q) then show that the sum of its first (p + q) terms is 0.

13. If the pth term of an AP is 1/q and its qth term is 1/ p, show that the sum of its first pq terms is1/2( p +q) .

14. Find the sum of all natural numbers less than 100 which are divisible by 6.

15. Find the sum of all natural number between 100 and 500 which are divisible by 7.

16. Find the sum of all multiples of 9 lying between 300 and 700.

17. Find the sum of all three digit natural numbers which are divisible by 13.

18. Find the sum of 51 terms of the AP whose second term is 2 and the 4^th term is 8.

19. The sum of n terms of an AP is (5n² – 3n). Find the AP and hence find its 10th term.

Case Based MCQs

Case I : Read the following passage and answer the questions from 39 to 43.

Discussion on A.P.
In a class the teacher asks every student to write an example of A.P. Two friends Geeta and Madhuri writes their progressions as –5, –2, 1, 4, … and 187, 184, 181, …. respectively. Now, the teacher asks various students of the class the following questions on these two progressions. Help students to find the answers of the questions.

Question. Find the \(34^{th}\) term of the progression written by Madhuri.
(a) 286
(b) 88
(c) –99
(d) 190
Answer: (b)

Question. Find the sum of common difference of the two progressions.
(a) 6
(b) –6
(c) 1
(d) 0
Answer: (d)

Question. Find the \(19^{th}\) term of the progression written by Geeta.
(a) 49
(b) 59
(c) 52
(d) 62
Answer: (a)

Question. Find the sum of first 10 terms of the progression written by Geeta.
(a) 85
(b) 95
(c) 110
(d) 200
Answer: (a)

Question. Which term of the two progressions will have the same value?
(a) 31
(b) 33
(c) 32
(d) 30
Answer: (b)

Case II : Read the following passage and answer the questions from 44 to 48.

Number Cards Game
Amit was playing a number card game. In the game, some number cards (having both +ve or –ve numbers) are arranged in a row such that they are following an arithmetic progression. On his first turn, Amit picks up \(6^{th}\) and \(14^{th}\) card and finds their sum to be –76. On the second turn he picks up \(8^{th}\) and \(16^{th}\) card and finds their sum to be –96.

Question. What is the difference between the numbers on any two consecutive cards?
(a) 7
(b) –5
(c) 11
(d) –3
Answer: (b)

Question. The number on first card is,
(a) 12
(b) 3
(c) 5
(d) 7
Answer: (d)

Question. What is the number on the \(19^{th}\) card?
(a) –88
(b) –83
(c) –92
(d) –102
Answer: (b)

Question. What is the number on the \(23^{rd}\) card?
(a) –103
(b) –122
(c) –108
(d) –117
Answer: (a)

Question. The sum of numbers on the first 15 cards is
(a) –840
(b) –945
(c) –427
(d) –420
Answer: (d)

Case III : Read the following passage and answer the questions from 49 to 53.

A sequence is an ordered list of numbers. A sequence of numbers such that the difference between the consecutive terms is constant is said to be an arithmetic progression (A.P.).

Question. Which of the following sequence is an A.P.?
(a) 10, 24, 39, 52, ….
(b) 11, 24, 39, 52, …
(c) 10, 24, 38, 52, …
(d) 10, 38, 52, 66, ….
Answer: (c)

Question. If \(x, y\) and \(z\) are in A.P., then
(a) \(x + z = y\)
(b) \(x – z = y\)
(c) \(x + z = 2y\)
(d) None of these
Answer: (c)

Question. If \(a_1, a_2, a_3, ….., a_n\) are in A.P., then which of the following is true?
(a) \(a_1 + k, a_2 + k, a_3 + k, ….., a_n + k\) are in A.P., where \(k\) is a constant.
(b) \(k – a_1, k – a_2, k – a_3, ….., k – a_n\) are in A.P., where \(k\) is a constant.
(c) \(ka_1, ka_2, ka_3, ….., ka_n\) are in A.P., where \(k\) is a constant.
(d) All of these
Answer: (d)

Question. If the \(n^{th}\) term (\(n > 1\)) of an A.P. is smaller than the first term, then nature of its common difference (\(d\)) is
(a) \(d > 0\)
(b) \(d < 0\)
(c) \(d = 0\)
(d) Can’t be determined
Answer: (b)

Question. Which of the following is incorrect about A.P.?
(a) All the terms of constant A.P. are same.
(b) Some terms of an A.P. can be negative.
(c) All the terms of an A.P. can never be negative.
(d) None of these.
Answer: (c)

Assertion & Reasoning Based MCQs

Directions (Q.54 to 60) : In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices :
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion : If \(a, b, c\) are in A.P., then \(\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}\) are also in A.P.
Reason : If a constant is added to each term of an A.P., then the resulting pattern of numbers is also an A.P.
(a) A
(b) B
(c) C
(d) D
Answer: (b)

Question. Assertion : The \(n^{th}\) term of a sequence is \(3n – 2\). It is an A.P.
Reason : A sequence is not an A.P. if its \(n^{th}\) term is not a linear expression in \(n\).
(a) A
(b) B
(c) C
(d) D
Answer: (a)

Question. Assertion : The \(10^{th}\) term from the end of the A.P. 7, 10, 13, ...., 184 is 163.
Reason : In an A.P. with first term \(a\), common difference \(d\) and last term \(l\), the \(n^{th}\) term from the end is \(l – (n – 1)d\).
(a) A
(b) B
(c) C
(d) D
Answer: (a)

Question. Assertion : The common difference of the A.P. 19, 18, 17, .... is 1.
Reason : Let \(a_1, a_2, a_3, a_4, ...\) is an A.P. Then, common difference of this A.P. will be the difference between any two consecutive terms, i.e., common difference \((d) = a_2 – a_1\) or \(a_3 – a_2\) or \(a_4 – a_3\) and so on.
(a) A
(b) B
(c) C
(d) D
Answer: (d)

Question. Assertion : The ninth term of an A.P. is equal to seven times the second term and twelfth term exceeds five times the third term by 2. Then the first term is 1.
Reason : If \(S_n\) and \(S_{n–1}\) are the sum of first \(n\) terms and \((n – 1)\) terms of an A.P., then \(n^{th}\) term, \(a_n = S_{n–1} – S_n\)
(a) A
(b) B
(c) C
(d) D
Answer: (c)

Question. Assertion : Sum of first 20 multiples of 4 is 480.
Reason : In an A.P., sum of \(n\) terms, \(S_n = \frac{n}{2}[a + l]\) where, \(n, a\) and \(l\) are number of terms, first term and last term respectively.
(a) A
(b) B
(c) C
(d) D
Answer: (d)

Question. Assertion : If the first term of an A.P. is 4, last term is 81 and the sum of the given terms is 510. Then, there are 12 terms in the given A.P.
Reason : If \(a\) is the first term, \(l\) is the last term and \(n\) is the number of terms of an A.P., then \(S_n = \frac{n}{2}(a + l)\).
(a) A
(b) B
(c) C
(d) D
Answer: (a)

SUBJECTIVE TYPE QUESTIONS

Very Short Answer Type Questions (VSA)

Question. Find the \(9^{th}\) term from the end of the A.P. 5, 9, 13, ..., 185.
Answer: \(l = 185, d = 4, n = 9\). \(a_9\) from end \(= l - (n-1)d = 185 - (8)4 = 153\).

Question. Check whether −150 is a term of the A.P. : 17, 12, 7, 2,… or not.
Answer: \(a = 17, d = -5\). \(a_n = -150 \Rightarrow 17 + (n-1)(-5) = -150 \Rightarrow -5(n-1) = -167 \Rightarrow n-1 = 167/5\). Not an integer, so not a term.

Question. What will be the \(21^{st}\) term of the A.P. whose first two terms are –3 and 4?
Answer: \(a = -3, d = 4 - (-3) = 7\). \(a_{21} = -3 + (20)7 = 137\).

Question. If the sum of first \(n\) terms of an A.P. is given by \(S_n = 5n^2 + 3n\), then find its \(n^{th}\) term.
Answer: \(a_n = S_n - S_{n-1} = (5n^2 + 3n) - [5(n-1)^2 + 3(n-1)] = 10n - 2\).

Question. Which term of the A.P. : 21, 42, 63, 84,... is 210?
Answer: \(a = 21, d = 21, a_n = 210 \Rightarrow 21 + (n-1)21 = 210 \Rightarrow 21n = 210 \Rightarrow n = 10\).

Question. Find the \(25^{th}\) term of the A.P. \(-5, \frac{-5}{2}, 0, \frac{5}{2}, .....\)
Answer: \(a = -5, d = 2.5\). \(a_{25} = -5 + (24)2.5 = -5 + 60 = 55\).

Question. Show that the sequence defined by \(a_n = 2n^2 + 1\) is not an A.P.
Answer: \(a_1 = 3, a_2 = 9, a_3 = 19\). \(a_2 - a_1 = 6\) while \(a_3 - a_2 = 10\). Differences are not constant.

Question. Check whether the following situation form an arithmetic progression or not : The cost of digging a tube well after every metre of digging when it costs ₹ 250 for the first metre and rises by ₹ 25 for each subsequent metre.
Answer: \(a = 250, d = 25\). Yes, it forms an A.P. as the cost increases by a constant amount.

Question. In an A.P., the first term is 12 and the common difference is 6. If the last term of the A.P. is 252, then find \(n\).
Answer: \(12 + (n-1)6 = 252 \Rightarrow 6(n-1) = 240 \Rightarrow n-1 = 40 \Rightarrow n = 41\).

Question. If \(\frac{2}{3}, k, \frac{5k}{8}\) are in A.P., then find the value of \(k\).
Answer: \(2k = \frac{2}{3} + \frac{5k}{8} \Rightarrow 2k - \frac{5k}{8} = \frac{2}{3} \Rightarrow \frac{11k}{8} = \frac{2}{3} \Rightarrow k = \frac{16}{33}\).

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