ICSE Class 10

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

Objective

1. 1.
   Find the two natural numbers which differ by 5 and the sum of whose squares is 97. Which of the following pairs satisfies these conditions?

   1. A. 3 and 8
   2. B. 4 and 9
   3. C. 5 and 10
   4. D. 2 and 7
   [1]
2. 2.
   The roots of the quadratic equation $x^2 - 0.04 = 0$ are:

   1. A. $\pm 0.2$
   2. B. $\pm 0.02$
   3. C. 0.4
   4. D. 2
   [1]
3. 3.
   Solve the equation using the factorisation method: $2x^{2} - \frac{1}{2}x = 0$. Which of the following gives the correct solutions for $x$?

   1. A. $x = 0$ or $x = \frac{1}{4}$
   2. B. $x = 0$ or $x = -\frac{1}{4}$
   3. C. $x = 0$ or $x = \frac{1}{2}$
   4. D. $x = 0$ or $x = -\frac{1}{2}$
   [1]
4. 4.
   The roots of the quadratic equation $p x^{2} - q x + r = 0$ are real and equal if:

   1. A. $p^2 = 4qr$
   2. B. $q^2 = 4pr$
   3. C. $-q^{2} = 4pr$
   4. D. $p^2 > 4qr$
   [1]
5. 5.
   Five times a certain whole number is equal to three less than twice the square of the number. Which of the following is the number?

   1. A. 1
   2. B. 3
   3. C. -0.5
   4. D. 5
   [1]
6. 6.
   The sum of a number and its reciprocal is 4.25. Which of the following represents the possible value(s) of the number?

   1. A. 4 only
   2. B. 1/4 only
   3. C. 4 or 1/4
   4. D. 2.125 only
   [1]
7. 7.
   The hypotenuse of a right-angled triangle is $26\,\mathrm{cm}$ and the sum of the other two sides is $34\,\mathrm{cm}$. What are the lengths of its sides?

   1. A. 8 cm and 26 cm
   2. B. 10 cm and 24 cm
   3. C. 12 cm and 22 cm
   4. D. 14 cm and 20 cm
   [1]
8. 8.
   Solve the following equation by using the formula: $\sqrt{3} x^2 + 11x + 6\sqrt{3} = 0$. Which of the following is a correct solution for $x$?

   1. A. $x = -\frac{2\sqrt{3}}{3}$
   2. B. $x = -\frac{2}{3}$
   3. C. $x = -\frac{11\sqrt{3}}{6}$
   4. D. $x = -\frac{\sqrt{3}}{3}$
   [1]
9. 9.
   The area of a right-angled triangle is 96 m². If the base is three times its altitude, find the base.

   1. A. 8 m
   2. B. 16 m
   3. C. 24 m
   4. D. 32 m
   [1]
10. 10.
    The sides of a right-angled triangle containing the right angle are $4x\,\mathrm{cm}$ and $(2x - 1)\,\mathrm{cm}$. If the area of the triangle is $30\,\mathrm{cm}^2$, what are the lengths of its sides?

    1. A. 6 cm, 8 cm, and 10 cm (hypotenuse)
    2. B. 8 cm, 15 cm, and 17 cm (hypotenuse)
    3. C. 5 cm, 12 cm, and 13 cm (hypotenuse)
    4. D. 9 cm, 12 cm, and 15 cm (hypotenuse)
    [1]
11. 11.
    ₹ 250 is divided equally among a certain number of children. If there were 25 children more, each would have received 50 paise less. Find the number of children.

    1. A. 50
    2. B. 100
    3. C. 125
    4. D. 200
    [1]
12. 12.
    Solve the equation $4x^2 - 5x - 3 = 0$ and select the root that is correct to two decimal places.

    1. A. $x = 1.25$
    2. B. $x = 1.69$
    3. C. $x = 1.75$
    4. D. $x = 2.00$
    [1]
13. 13.
    Solve the equation $x^2 - x = 156$ by factorisation. What is the solution set?

    1. A. $\{-13, 12\}$
    2. B. $\{13, -12\}$
    3. C. $\{12, -13\}$
    4. D. $\{-12, -13\}$
    [1]
14. 14.
    The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their ages is 550. What are their ages?

    1. A. 20 and 27 years
    2. B. 22 and 25 years
    3. C. 23 and 24 years
    4. D. 19 and 28 years
    [1]
15. 15.
    Discuss the nature of the roots of the equation $x^{2} - 8x + 7 = 0$ without actually solving it. Which of the following correctly describes the roots?

    1. A. Roots are irrational and distinct.
    2. B. Roots are rational and distinct.
    3. C. Roots are equal and rational.
    4. D. Roots are complex conjugates.
    [1]
16. 16.
    Solve the equation: $\left(\frac{x}{x + 2}\right)^2 - 7\left(\frac{x}{x + 2}\right) + 12 = 0; x \neq -2$. What are the values of $x$?

    1. A. $x = -3 \text{ or } x = -\frac{8}{3}$
    2. B. $x = -4 \text{ or } x = -\frac{4}{3}$
    3. C. $x = -6 \text{ or } x = -\frac{6}{5}$
    4. D. $x = -2 \text{ or } x = -\frac{3}{4}$
    [1]
17. 17.
    A bus covers a distance of $240\,\mathrm{km}$ at a uniform speed. Due to heavy rain, its speed is reduced by $10\,\mathrm{km/h}$, causing it to take two hours longer to cover the distance. If the original speed is $x\,\mathrm{km/h}$, what is the value of $x$?

    1. A. 30 km/h
    2. B. 40 km/h
    3. C. 50 km/h
    4. D. 60 km/h
    [1]
18. 18.
    A piece of cloth costs Rs. 35. If the length of the piece would have been 4 m longer and each metre costs Re. 1 less, the cost would have remained unchanged. What is the length of the piece?

    1. A. 6 m
    2. B. 10 m
    3. C. 14 m
    4. D. 20 m
    [1]
19. 19.
    A student scored a total of 32 marks in mathematics and science. If he had scored 2 marks less in science and 4 more in mathematics, the product of his marks would have been 253. What are his marks in mathematics?

    1. A. 7
    2. B. 13
    3. C. 19
    4. D. 25
    [1]
20. 20.
    A train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. What is the usual speed of the train?

    1. A. 20 km/hr
    2. B. 25 km/hr
    3. C. 30 km/hr
    4. D. 35 km/hr
    [1]