ICSE Class 10

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

Objective

1. 1.
   The sub-triplicate ratio of $64:125$ is:

   1. A. $16:17$
   2. B. $61:340$
   3. C. $4:5$
   4. D. $\frac{64}{3}:\frac{125}{3}$
   [1]
2. 2.
   If $x, y$ and $z$ are in continued proportion, which of the following expressions simplifies to $\frac{x}{z}$?

   1. A. $\frac{(x + y)^2}{(y + z)^2}$
   2. B. $\frac{(x - y)^2}{(y - z)^2}$
   3. C. $\frac{(x + y)^2}{(y - z)^2}$
   4. D. $\frac{(x - y)^2}{(y + z)^2}$
   [1]
3. 3.
   What must be subtracted from each of the numbers 23, 30, 57, and 78 so that the resulting remainders are in proportion?

   1. A. 3
   2. B. 6
   3. C. 9
   4. D. 12
   [1]
4. 4.
   Divide Rs. 1162 among A, B, and C in the ratio 35:28:20. What is A's share?

   1. A. Rs. 490
   2. B. Rs. 350
   3. C. Rs. 560
   4. D. Rs. 420
   [1]
5. 5.
   The ratio between $3.6\,\mathrm{m}$ and $75\,\mathrm{cm}$ is:

   1. a. $18:275$
   2. b. $24:5$
   3. c. $6:125$
   4. d. $4:5$
   [1]
6. 6.
   Reena reduces her weight in the ratio 3:2. What is her weight now if originally it was 81 kg?

   1. A. 27 kg
   2. B. 54 kg
   3. C. 60 kg
   4. D. 121.5 kg
   [1]
7. 7.
   In a regiment, the ratio of the number of officers to the number of soldiers was $3 : 31$ before a battle. In the battle, 6 officers and 22 soldiers were killed. The ratio between the number of officers and the number of soldiers after the battle is $1 : 13$. What was the total number of officers and soldiers in the regiment before the battle?

   1. A. 238
   2. B. 245
   3. C. 210
   4. D. 196
   [1]
8. 8.
   If $x: y = 3:2$, find the value of $(5x - 3y): (7x + 2y)$.

   1. A. 3 : 5
   2. B. 9 : 25
   3. C. 15 : 23
   4. D. 5 : 7
   [1]
9. 9.
   6 is the mean proportion between two numbers $x$ and $y$, and 48 is the third proportional of $x$ and $y$. Which pair of numbers satisfies these conditions?

   1. A. $x = 2$, $y = 18$
   2. B. $x = 3$, $y = 12$
   3. C. $x = 4$, $y = 9$
   4. D. $x = 6$, $y = 6$
   [1]
10. 10.
    If $a, b, c$ are in continued proportion, which of the following correctly shows that $a : c$ is the duplicate ratio of $a$ to $b$?

    1. A. $\frac{a}{c} = \frac{a^2}{b^2}$ because $a = ck^2$ and $b = ck$ for some constant $k$.
    2. B. $\frac{a}{c} = \frac{b}{a}$ since $a, b, c$ are in continued proportion.
    3. C. $a : c = b : a$ as the duplicate ratio implies equality of ratios.
    4. D. $\frac{a}{c} = \frac{b^2}{a^2}$ because $b^2 = ac$ in continued proportion.
    [1]
11. 11.
    If $A : B = 2 : 3$, $B : C = 4 : 5$ and $C : D = 6 : 7$, then what is the combined ratio $A : B : C : D$?

    1. A. 8:12:15:14
    2. B. 16:24:30:35
    3. C. 4:6:7:8
    4. D. 16:20:25:35
    [1]
12. 12.
    Find the reciprocal ratio of: $\frac{x}{3} : \frac{y}{5}$

    1. A. $5x : 3y$
    2. B. $3y : 5x$
    3. C. $15xy : xy$
    4. D. $\frac{5}{y} : \frac{3}{x}$
    [1]
13. 13.
    If $ x = \frac{2ab}{a + b} $, find the value of: $ \frac{x + a}{x - a} + \frac{x + b}{x - b} $.

    1. A. 0
    2. B. 1
    3. C. 2
    4. D. $\frac{a + b}{a - b}$
    [1]
14. 14.
    If $2x + 3y: 3x + 5y = 18:29$, then the ratio $x:y$ is:

    1. a. $2:3$
    2. b. $3:5$
    3. c. $3:4$
    4. d. $5:29$
    [1]
15. 15.
    If $\frac{a}{b} = \frac{c}{d}$, which of the following correctly shows that $\frac{ma + nb}{mc + nd} = \frac{ma - nb}{mc - nd}$?

    1. A. By applying alternendo and componendo-dividendo to the given proportion, the equality is proved.
    2. B. Cross-multiplying the given proportion yields $ad = bc$, which directly proves the equality.
    3. C. Adding and subtracting $ma$ and $nb$ on both sides of the proportion proves the result.
    4. D. The equality holds because $\frac{ma + nb}{mc + nd} = \frac{a}{c}$ and $\frac{ma - nb}{mc - nd} = \frac{b}{d}$.
    [1]
16. 16.
    If $\frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b}$ and $a + b + c = 0$, what is the value of each given ratio?

    1. A. $1$
    2. B. $-1$
    3. C. $0$
    4. D. $\frac{1}{2}$
    [1]
17. 17.
    Solve the equation: $\frac{\sqrt{3x} + \sqrt{2x - 1}}{\sqrt{3x} - \sqrt{2x - 1}} = 5$. What is the value of $x$?

    1. A. $\frac{3}{2}$
    2. B. $\frac{2}{3}$
    3. C. $\frac{1}{2}$
    4. D. $3$
    [1]
18. 18.
    If $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$, which of the following correctly proves the equality $\frac{2x^3 - 3y^3 + 4z^3}{2a^3 - 3b^3 + 4c^3} = \left(\frac{2x - 3y + 4z}{2a - 3b + 4c}\right)^3$?

    1. A. Let $\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = k$. Substitute $x = ak$, $y = bk$, $z = ck$ into both sides to show they equal $k^3$.
    2. B. Assume $x = a$, $y = b$, and $z = c$ directly, then simplify both sides to show they equal 1.
    3. C. Cross-multiply the given ratios to get $x = y = z$ and substitute into the equation to verify.
    4. D. Express $a$, $b$, and $c$ in terms of $x$, $y$, and $z$, then substitute into the left side only to simplify.
    [1]
19. 19.
    The monthly pocket money of Ravi and Sanjeev are in the ratio 5 : 7. Their expenditures are in the ratio 3 : 5. If each saves Rs. 80 every month, what is Ravi's monthly pocket money?

    1. A. Rs. 300
    2. B. Rs. 400
    3. C. Rs. 500
    4. D. Rs. 560
    [1]
20. 20.
    Given $\frac{x^3 + 12x}{6x^2 + 8} = \frac{y^3 + 27y}{9y^2 + 27}$, use componendo and dividendo to find the ratio $x:y$.

    1. A. $1:2$
    2. B. $2:3$
    3. C. $3:4$
    4. D. $4:5$
    [1]