ICSE Class 10

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

Objective

1. 1.
   Solve the inequation and determine its solution set: $4 - 5x > -11$, where $x \in I$. Which of the following represents the correct solution set?

   1. A. $\{x : x < 3, x \in I\}$
   2. B. $\{x : x > 3, x \in I\}$
   3. C. $\{x : x \leq 3, x \in I\}$
   4. D. $\{\dots, -1, 0, 1, 2, 3\}$
   [1]
2. 2.
   If $3x - 7 > 5x - 1 \forall x \in R$ then:

   1. A. $x > -3$
   2. B. $x < -3$
   3. C. $x \leq -3$
   4. D. $x \geq -3$
   [1]
3. 3.
   Solve the inequation and identify its solution set from the given options: $2x - 7 < 4$, $x \in \{1, 2, 3, 4, 5, 6, 7\}$.

   1. A. {1, 2, 3, 4, 5}
   2. B. {1, 2, 3, 4, 5, 6}
   3. C. {2, 3, 4, 5, 6}
   4. D. {1, 2, 3, 4}
   [1]
4. 4.
   The solution set for the inequation $2x + 7 \leq 17$, $x \in N$ is:

   1. A. $\{1,2,3,4,5\}$
   2. B. $\{0,1,2,3,4,5\}$
   3. C. $\{1,2,3,4\}$
   4. D. $\{0,1,2,3,4\}$
   [1]
5. 5.
   If $3(x - 1) \geq 2(x - 3)$ then:

   1. A. $x > -3$
   2. B. $x < -3$
   3. C. $x \leq -3$
   4. D. $x \geq -3$
   [1]
6. 6.
   The solution set representing the following number line is: $$- \infty \leftarrow \begin{array}{cccccccc}1 & 1 & 1 & 1 & 1 & 1 & 1 \\ -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3\end{array}$$

   1. A. $[x : x \in R, -3 \leq x < 2]$
   2. B. $[x : x \in R, -3 < x < 2]$
   3. C. $[x : x \in R, -3 \leq x \leq 2]$
   4. D. $[x : x \in R, -3 \leq x \leq 2]$
   [1]
7. 7.
   Find the value(s) of x ∈ W that satisfy the inequation: -2 5/6 < 1/2 - (2x)/3 ≤ 2. Which set represents the solution?

   1. A. {0, 1, 2, 3, 4, 5}
   2. B. {1, 2, 3, 4}
   3. C. {0, 1, 2, 3, 4}
   4. D. {-1, 0, 1, 2, 3, 4}
   [1]
8. 8.
   Solve the following inequation and select the correct solution set: $-2 + 10x \leq 13x + 10 \text{ < } 24 + 10x, x \in Z$. Represent this on the real number line as shown:

   1. A. $\{-4, -3, -2, -1, 0, 1, 2, 3\}$
   2. B. $\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$
   3. C. $\{-3, -2, -1, 0, 1, 2, 3, 4\}$
   4. D. $\{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4\}$
   [1]
9. 9.
   Solve the following inequation and identify the correct solution set: $$-2 < \frac{1}{2} - \frac{2x}{3} \leq \frac{11}{6}, x \in R.$$ Represent the solution set on the number line.

   1. A. $$\{x : -2 < x \leq \frac{15}{4}, x \in R\}$$
   2. B. $$\{x : -2 \leq x < \frac{15}{4}, x \in R\}$$
   3. C. $$\{x : x < -2, x \in R\}$$
   4. D. $$\{x : x \geq \frac{15}{4}, x \in R\}$$
   [1]
10. 10.
    Multiplying $-3$ on both sides of a linear inequation $\frac{-2x + 7}{3} \leq \frac{-4x + 5}{3}$ becomes

    1. A. $-2x + 7 \geq -4x + 5$
    2. B. $2x - 7 \geq 4x - 5$
    3. C. $-2x + 7 \leq 4x - 5$
    4. D. $2x - 7 \leq 4x - 5$
    [1]
11. 11.
    Solve the inequation and choose the correct solution set on a number line: $5(x - 2) > 4(x + 3) - 20$, where $x \in I$.

    1. A. $\{\dots, -3, -2, -1, 0, 1, 2, 3\}$
    2. B. $\{3, 4, 5, \dots\}$
    3. C. $\{\dots, -1, 0, 1, 2\}$
    4. D. $\{2, 3, 4, \dots\}$
    [1]
12. 12.
    Reshma has ₹120 and wants to buy some registers and pens. The cost of one register is ₹40 and that of a pen is ₹20. The mathematical formulation of the given problem is:

    1. A. $40x + 20y = 120$
    2. B. $40x + 20y > 120$
    3. C. $40x + 20y \leq 120$
    4. D. $40x + 20y \geq 120$
    [1]
13. 13.
    Solve the inequation -8½ < -½ - 4x ≤ 7½ for x ∈ I and identify the correct solution set.

    1. A. {-2, -1, 0, 1}
    2. B. {-3, -2, -1, 0, 1}
    3. C. {-1, 0, 1, 2}
    4. D. {x : -2 ≤ x < 2, x ∈ I}
    [1]
14. 14.
    Given the sets: $A = \{x: 11x - 5 > 7x + 3, x \in \mathbb{R}\}$ and $B = \{x: 18x - 9 \geq 15 + 12x, x \in \mathbb{R}\}$, their intersection $A \cap B$ is represented on the number line below. Which labeled part on the number line correctly shows the range of $A \cap B$?

    1. A. An open circle at 2 with a ray extending to the right
    2. B. A closed circle at 2 with a ray extending to the right
    3. C. An open circle at 2 with a ray extending to the left
    4. D. A closed circle at 2 with a segment extending to 3
    [1]
15. 15.
    The solution set of the inequations $-2 < 2x - 6$ or $-2x + 5 \geq 13$, where $x \in \mathbb{R}$, is represented on the number line below: Which labeled part correctly indicates the darkened circle at $-4$ with an arrow extending to the left?

    1. A. Label P: hollow circle at 2, arrow to the right
    2. B. Label Q: darkened circle at -4, arrow to the left
    3. C. Label R: hollow circle at -4, arrow to the right
    4. D. Label S: darkened circle at 2, arrow to the left
    [1]
16. 16.
    Solve the inequation and determine its solution set for the given condition: $-x > -3x - 5$, $x \in I$. Which of the following represents the correct solution set?

    1. A. $\{-3, -2, -1, 0, 1, 2, \dots\}$
    2. B. $\{-2, -1, 0, 1, 2, \dots\}$
    3. C. $\{\dots, -3, -2, -1, 0\}$
    4. D. $\{-1, 0, 1, 2, 3, \dots\}$
    [1]
17. 17.
    The number of pairs of consecutive odd natural numbers both of which are larger than 10, such that their sum is less than 40, is

    1. A. 8
    2. B. 6
    3. C. 4
    4. D. 3
    [1]
18. 18.
    Solve the following system of inequations and identify the correct solution set: $$3 - x \leq 2(x - 3) \text{ and } 5 - x > x - 7, x \in R.$$ Graph the solution set.

    1. A. $$\{x : 1 \leq x < 3, x \in R\}$$
    2. B. $$\{x : \frac{8}{5} \leq x < 3, x \in R\}$$
    3. C. $$\{x : x \geq 3, x \in R\}$$
    4. D. $$\{x : x < \frac{8}{5}, x \in R\}$$
    [1]
19. 19.
    The longest side of a triangle is 3 times the shortest side and the third side is $2\,\mathrm{cm}$ shorter than the longest side. If the perimeter of the triangle is at least $61\,\mathrm{cm}$, then minimum length of the shortest side is

    1. A. $7\,\mathrm{cm}$
    2. B. $8\,\mathrm{cm}$
    3. C. $9\,\mathrm{cm}$
    4. D. $10\,\mathrm{cm}$
    [1]
20. 20.
    Solve the following inequation and select the correct solution set from the options below: (-x)/3 ≤ x/2 - 1 1/3 < 1/6, x ∈ R.

    1. A. {x: 1.6 ≤ x < 3, x ∈ R}
    2. B. {x: x ≤ 1.6 or x > 3, x ∈ R}
    3. C. {x: 1.6 < x ≤ 3, x ∈ R}
    4. D. {x: x < 1.6 or x ≥ 3, x ∈ R}
    [1]