ICSE Class 10

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

Objective

1. 1.
   Find the sum of the G.P.: $1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dots$ to 9 terms.

   1. A. $\frac{171}{256}$
   2. B. $\frac{170}{256}$
   3. C. $\frac{341}{512}$
   4. D. $-\frac{171}{256}$
   [1]
2. 2.
   What is the $6^{\text{th}}$ term from the end of the GP 3, $-6, 12, -24, \ldots, 12288$?

   1. A. 284
   2. B. 384
   3. C. $-384$
   4. D. $-284$
   [1]
3. 3.
   Which term of the geometric progression 1, $\frac{1}{3}, \frac{1}{9}, \ldots$ is equal to $\frac{1}{243}$?

   1. A. 4
   2. B. 5
   3. C. 6
   4. D. 7
   [1]
4. 4.
   Find the next three terms of the sequence: $36, 12, 4, \dots$

   1. A. $\frac{4}{3}, \frac{4}{9}, \frac{4}{27}$
   2. B. $4, 2, 1$
   3. C. $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}$
   4. D. $12, 4, \frac{4}{3}$
   [1]
5. 5.
   Find the next three terms of the sequence: $\sqrt{5}, 5, 5\sqrt{5}, \dots$

   1. A. $25, 25\sqrt{5}, 125$
   2. B. $5\sqrt{5}, 25, 25\sqrt{5}$
   3. C. $10, 10\sqrt{5}, 50$
   4. D. $5, 5\sqrt{5}, 5$
   [1]
6. 6.
   What is the next term of the series $\frac{1}{6}, \frac{1}{3}, \frac{2}{3}, \ldots$

   1. A. $\frac{4}{3}$
   2. B. $\frac{2}{3}$
   3. C. $\frac{1}{3}$
   4. D. $\frac{5}{3}$
   [1]
7. 7.
   Find the fifth term from the end of the geometric series 243, 81, 27, 9, ..., $ \frac{1}{729} $.

   1. A. $ \frac{1}{9} $
   2. B. $ \frac{1}{27} $
   3. C. $ \frac{1}{81} $
   4. D. $ 9 $
   [1]
8. 8.
   Which term of the G.P.: $-10, \frac{5}{\sqrt{3}}, -\frac{5}{6}, \dots$ is $\frac{5}{72}$?

   1. A. 4
   2. B. 5
   3. C. 6
   4. D. 7
   [1]
9. 9.
   Insert three numbers between 1 and 256 so that the resulting sequence of five numbers is a geometric progression. Which of the following sets of numbers correctly completes the sequence?

   1. A. 2, 4, 32
   2. B. 4, 16, 64
   3. C. 16, 64, 128
   4. D. 1, 16, 81
   [1]
10. 10.
    Assertion (A): The sum of first 10 terms of the GP 1, 2, 4, 8, 16, ... will make a sum of 1024. Reason (R): Sum of first $n$ terms of a GP is given by $S_n = \frac{a(r^n - 1)}{r - 1}$. Which of the following is correct?

    1. A. Both assertion (A) and reason (R) are true, and (R) is the correct explanation of (A).
    2. B. Both assertion (A) and reason (R) are true, but (R) is not the correct explanation of (A).
    3. C. Assertion (A) is true but reason (R) is false.
    4. D. Assertion (A) is false but reason (R) is true.
    [1]
11. 11.
    Find the number of terms in a geometric progression whose first term is $\frac{3}{4}$, common ratio is 2, and the last term is 384.

    1. A. 10
    2. B. 9
    3. C. 11
    4. D. 8
    [1]
12. 12.
    A geometric progression has common ratio = 3 and last term = 486. If the sum of its terms is 728, find its first term.

    1. A. 2
    2. B. 4
    3. C. 6
    4. D. 1
    [1]
13. 13.
    Find the geometric progression whose 4th term is 54 and the 7th term is 1458.

    1. A. 3, 9, 27, 54, ...
    2. B. 2, 6, 18, 54, ...
    3. C. 1, 3, 9, 27, ...
    4. D. 6, 12, 24, 48, ...
    [1]
14. 14.
    Find the $4^{\text{th}}$ term from the end of the series 8, 4, 2, ..., $ \frac{1}{128} $.

    1. A. $ \frac{1}{16} $
    2. B. $ \frac{1}{32} $
    3. C. $ \frac{1}{8} $
    4. D. $ \frac{1}{64} $
    [1]
15. 15.
    What is the sum $n$ terms of GP 3, 12, 48, 192, ...

    1. A. $4^n - 3$
    2. B. $2^n - 1$
    3. C. $2^n - 3$
    4. D. $4^n - 1$
    [1]
16. 16.
    If the first and the third terms of a G.P. are 2 and 8 respectively, find its second term.

    1. A. 2
    2. B. 4
    3. C. 6
    4. D. -4
    [1]
17. 17.
    The product of the 3ʳᵈ and 8ᵗʰ terms of a geometric progression is 243. If its 4ᵗʰ term is 3, what is the value of its 7ᵗʰ term?

    1. A. 9
    2. B. 27
    3. C. 81
    4. D. 3
    [1]
18. 18.
    The first term of a geometric progression (G.P.) is $-3$ and the square of the second term is equal to its $4^{\text{th}}$ term. Which of the following is the $7^{\text{th}}$ term of this G.P.?

    1. A. $-2187$
    2. B. $-243$
    3. C. $243$
    4. D. $729$
    [1]
19. 19.
    If $a$, $b$ and $c$ are in A.P. whereas $x$, $y$ and $z$ are in G.P., which of the following must be true?

    1. A. $x^{b - c} \cdot y^{c - a} \cdot z^{a - b} = 1$
    2. B. $x^{a - b} \cdot y^{b - c} \cdot z^{c - a} = 1$
    3. C. $x^{b - c} + y^{c - a} + z^{a - b} = 1$
    4. D. $x^{b - c} \cdot y^{c - a} \cdot z^{a - b} = 0$
    [1]
20. 20.
    If $y = x + x^2 + x^3 + \dots \dots \dots \infty$, where $|x| < 1$, which of the following expressions correctly represents $x$ in terms of $y$?

    1. A. $x = \frac{y}{1 - y}$
    2. B. $x = \frac{y}{1 + y}$
    3. C. $x = y(1 + y)$
    4. D. $x = \frac{1 + y}{y}$
    [1]