ICSE Class 10

General instructions: Answer all questions. Marks are shown in brackets [ ].

Objective

1. 1.
   If $a, b, c$ are in A.P., which of the following identities must hold true?

   1. A. $(a + b + c)^2 = 3(ab + bc + ca)$
   2. B. $(a + b + c)^2 = ab + bc + ca$
   3. C. $(a + b + c)^2 = 2(ab + bc + ca)$
   4. D. $(a + b + c)^2 = 4(ab + bc + ca)$
   [1]
2. 2.
   Find the number of all natural numbers between 20 and 80, which are divisible by 3.

   1. A. 19
   2. B. 20
   3. C. 21
   4. D. 26
   [1]
3. 3.
   The sum of first 16 terms of the AP 10, 6, 2, ... is

   1. A. -320
   2. B. 320
   3. C. -352
   4. D. -400
   [1]
4. 4.
   Find the 30th term of the sequence: 1/2, 1, 3/2, ...

   1. A. 14
   2. B. 15
   3. C. 29/2
   4. D. 30
   [1]
5. 5.
   The $n^{\text{th}}$ term of an arithmetic progression is given by $15 - 7n$. What is its common difference?

   1. A. 15
   2. B. -7
   3. C. 7
   4. D. 8
   [1]
6. 6.
   If the first term of an AP is $-5$ and the common difference is 2, then the sum of the first 6 terms is

   1. A. 0
   2. B. 5
   3. C. 6
   4. D. 15
   [1]
7. 7.
   Solve the equation: $(x + 1) + (x + 4) + (x + 7) + \dots + (x + 28) = 165$. What is the value of $x$?

   1. A. 2
   2. B. 3
   3. C. -1
   4. D. 5
   [1]
8. 8.
   A sum of ₹ 8,000 is invested at 10% simple interest per annum. Calculate the interest at the end of each year. Does the sequence of interests at the end of consecutive years form an arithmetic progression (A.P.)? If it does, what are its first term and the common difference?

   1. A. The sequence forms an A.P. with first term ₹ 800 and common difference ₹ 0.
   2. B. The sequence forms an A.P. with first term ₹ 800 and common difference ₹ 800.
   3. C. The sequence does not form an A.P. because the interest varies each year.
   4. D. The sequence forms an A.P. with first term ₹ 1,600 and common difference ₹ 800.
   [1]
9. 9.
   The first term of an A.P. is 5, the last term is 45 and the sum of its terms is 1000. What are the number of terms and the common difference of the A.P.?

   1. A. Number of terms = 40, common difference = 1
   2. B. Number of terms = 50, common difference = 0.8
   3. C. Number of terms = 20, common difference = 2
   4. D. Number of terms = 30, common difference = 1.5
   [1]
10. 10.
    Determine the value of $k$ for which the expressions $k^2 + 4k + 8$, $2k^2 + 3k + 6$, and $3k^2 + 4k + 4$ are in arithmetic progression (A.P.).

    1. A. $k = 0$
    2. B. $k = 2$
    3. C. $k = -1$
    4. D. $k = 1$
    [1]
11. 11.
    For what value of $n$, the $n^{\text{th}}$ term of the arithmetic progression 63, 65, 67, ... and the $n^{\text{th}}$ term of the arithmetic progression 3, 10, 17, ... are equal to each other?

    1. A. 9
    2. B. 12
    3. C. 13
    4. D. 15
    [1]
12. 12.
    Find the common difference and the 99th term of the arithmetic progression: 7 3/4, 9 1/2, 11 1/4, ...

    1. A. Common difference = 1 3/4, 99th term = 179 1/4
    2. B. Common difference = 1 1/2, 99th term = 183 1/2
    3. C. Common difference = 2, 99th term = 197
    4. D. Common difference = 1 3/4, 99th term = 183 1/2
    [1]
13. 13.
    Houses in a row are numbered consecutively from 1 to 49. There exists a value of $X$ such that the sum of the numbers of houses preceding $X$ equals the sum of the numbers of houses following $X$. What is the value of $X$?

    1. A. 25
    2. B. 35
    3. C. 42
    4. D. 49
    [1]
14. 14.
    How many terms of the arithmetic progression 24, 21, 18, ... must be taken so that their sum is 78?

    1. A. 4 terms
    2. B. 5 terms
    3. C. 6 terms
    4. D. 7 terms
    [1]
15. 15.
    Which term of the A.P. 3, 10, 17, ... will be 84 more than its 13ᵗʰ term?

    1. A. 20th term
    2. B. 24th term
    3. C. 25th term
    4. D. 26th term
    [1]
16. 16.
    If the pth term of an arithmetic progression (A.P.) is given by (2p + 3), which of the following represents the A.P.?

    1. A. 3, 5, 7, 9, ...
    2. B. 5, 7, 9, 11, ...
    3. C. 2, 5, 8, 11, ...
    4. D. 5, 9, 13, 17, ...
    [1]
17. 17.
    The sum of three terms in an arithmetic progression (A.P.) is 33 and their product is 1155. Find the terms.

    1. A. 5, 11, 17
    2. B. 7, 11, 15
    3. C. 3, 11, 19
    4. D. 9, 11, 13
    [1]
18. 18.
    The digits of a positive three-digit number are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. What is the original number?

    1. A. 753
    2. B. 852
    3. C. 963
    4. D. 642
    [1]
19. 19.
    Divide 56 into four parts in AP such that the ratio of the product of their extremes (1st and 4th) to the product of means (2nd and 3rd) is 5:6. What are the four parts?

    1. A. 6, 12, 18, 20
    2. B. 8, 12, 16, 20
    3. C. 7, 14, 21, 14
    4. D. 10, 12, 14, 20
    [1]
20. 20.
    The sum of three numbers in arithmetic progression (A.P.) is 15 and the sum of the squares of the extreme terms is 58. Find the numbers.

    1. A. 2, 5, 8
    2. B. 3, 5, 7
    3. C. 1, 5, 9
    4. D. 4, 5, 6
    [1]