Real Numbers : Assessment for classes IX and X

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25 Question MCQ Chapter on Real Numbers for CBSE, ICSE, Madhyamik and other State Boards Class IX & X

👨‍🏫 Author: Singh

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Real Numbers Quiz

Test your understanding of rational, irrational numbers and their properties. The study of Real Numbers in Classes IX and X establishes the foundation for all higher-level mathematics.Real Numbe: The set including all Rational and Irrational numbers. Rational Numbers: Numbers expressible as p/q, q is non-zero . Their decimal expansions are either terminating or non-terminating recurring.Irrational Numbers: Numbers that cannot be written as fractions. Their decimal expansions are non-terminating non-recurring.Number Line: Every point on a number line represents a unique real number.

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• This quiz contains 25 multiple choice questions on Real Numbers.
• Select only one correct answer per question.
• Use the navigator to jump between questions.
• Submit when you are finished to see results.

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Question 1

Which of the following is an irrational number?

A\(\sqrt{25}\) B\(\sqrt{2} + \sqrt{3}\) C\(\frac{3}{7}\) D\(0.\overline{3}\)

Question 2

The decimal expansion of \(\frac{13}{625}\) will terminate after how many decimal places?

A2 B3 C4 D5

Question 3

Which of the following has a terminating decimal expansion?

A\(\frac{77}{210}\) B\(\frac{23}{30}\) C\(\frac{125}{441}\) D\(\frac{129}{2^2 \times 5^7}\)

Question 4

The simplified value of \(\sqrt{12} + \sqrt{27} - \sqrt{75}\) is:

A\(\sqrt{3}\) B\(2\sqrt{3}\) C\(3\sqrt{3}\) D\(0\)

Question 5

If \(p\) and \(q\) are co-prime numbers, then \(p^2\) and \(q^2\) are:

ACo-prime BNot co-prime CEven numbers DOdd numbers

Question 6

A rational number between \(\sqrt{2}\) and \(\sqrt{3}\) is:

A\(\frac{\sqrt{2} + \sqrt{3}}{2}\) B\(\frac{\sqrt{2} \times \sqrt{3}}{2}\) C\(1.5\) D\(1.8\)

Question 7

The value of \(1.\overline{36} + 0.\overline{9}\) is:

A\(2.\overline{27}\) B\(2.\overline{26}\) C\(2.\overline{3}\) D\(2.3\)

Question 8

Which of the following is not an irrational number?

A\(\sqrt{8} \times \sqrt{2}\) B\(\sqrt{18} \div \sqrt{2}\) C\(\sqrt{5} + \sqrt{10}\) D\(\sqrt{7} \times \sqrt{28}\)

Question 9

If \(n\) is a natural number, then \(\sqrt{n}\) is:

AAlways rational BAlways irrational CRational if \(n\) is a perfect square DIrrational if \(n\) is a perfect square

Question 10

The product of a non-zero rational and an irrational number is:

AAlways rational BAlways irrational CCan be rational or irrational DAlways an integer

Question 11

Which of the following rational numbers have terminating decimal representation?

A\(\frac{16}{125}\) B\(\frac{27}{250}\) C\(\frac{343}{2^3 \times 5^3}\) DAll of these

Question 12

The ascending order of \(\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5}\) is:

A\(\sqrt[3]{2} < \sqrt[6]{5} < \sqrt{3}\) B\(\sqrt[6]{5} < \sqrt[3]{2} < \sqrt{3}\) C\(\sqrt[6]{5} < \sqrt{3} < \sqrt[3]{2}\) D\(\sqrt{3} < \sqrt[3]{2} < \sqrt[6]{5}\)

Question 13

The decimal expansion of the rational number \(\frac{14587}{1250}\) will terminate after:

AOne decimal place BTwo decimal places CThree decimal places DFour decimal places

Question 14

After rationalizing the denominator of \(\frac{7}{3\sqrt{3} - 2\sqrt{2}}\), we get:

A\(3\sqrt{3} + 2\sqrt{2}\) B\(3\sqrt{3} - 2\sqrt{2}\) C\(7(3\sqrt{3} + 2\sqrt{2})\) D\(7(3\sqrt{3} - 2\sqrt{2})\)

Question 15

An irrational number between \(\frac{1}{7}\) and \(\frac{2}{7}\) is:

A\(\frac{1}{2}\left(\frac{1}{7} + \frac{2}{7}\right)\) B\(\sqrt{\frac{1}{7} \times \frac{2}{7}}\) C\(\frac{1}{\sqrt{14}}\) D\(\sqrt{0.14}\)

Question 16

If \(x = 2 + \sqrt{3}\), then \(x + \frac{1}{x}\) equals:

A\(4\) B\(2\sqrt{3}\) C\(-4\) D\(-2\sqrt{3}\)

Question 17

Which of the following is a rational number?

A\(\pi\) B\(\sqrt{2}\) C\(0.\overline{101}\) D\(\sqrt{5} + 1\)

Question 18

The simplest rationalizing factor of \(\sqrt[3]{54}\) is:

A\(\sqrt[3]{2}\) B\(\sqrt[3]{3}\) C\(\sqrt[3]{4}\) D\(\sqrt[3]{9}\)

Question 19

If \(\frac{p}{q}\) is a rational number with terminating decimal expansion, where \(p\) and \(q\) are co-prime, then \(q\) must be of the form:

A\(2^m \times 3^n\) B\(2^m \times 5^n\) C\(3^m \times 5^n\) D\(2^m \times 3^n \times 5^k\)

Question 20

The sum of two irrational numbers is:

AAlways irrational BAlways rational CCan be rational or irrational DAlways an integer

Question 21

The value of \(\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}}\) is:

A\(1\) B\(9\) C\(10\) D\(99\)

Question 22

If \(a\) and \(b\) are rational numbers and \(\frac{3 + 2\sqrt{3}}{3 - 2\sqrt{3}} = a + b\sqrt{3}\), then \(a + b =\)

A\(11\) B\(-11\) C\(13\) D\(-13\)

Question 23

Which of the following is not true?

AEvery integer is a rational number BEvery whole number is a natural number CEvery irrational number is a real number DEvery rational number is a real number

Question 24

The product \(\sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32}\) equals:

A\(\sqrt{2}\) B\(2\) C\(\sqrt[12]{2}\) D\(\sqrt[12]{32}\)

Question 25

The value of \(0.\overline{6} + 0.\overline{7} + 0.\overline{4}\) is:

A\(2\) B\(\frac{17}{9}\) C\(\frac{53}{22}\) D\(\frac{59}{22}\)

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Madhyamik Practice Test Paper

Test Examination - Practice Test Paper - Prime Maths

Class X - Mathematics

Examination Overview

This comprehensive mathematics examination covers key concepts from the Class X curriculum including algebra, geometry, trigonometry, mensuration, statistics, and commercial mathematics. The paper is designed to test conceptual understanding, problem-solving skills, and application of mathematical principles in real-world scenarios.

Duration: 3 Hours

Total Marks: 90

Class: X (Secondary)

Year: 2023

No Calculator

Use of calculators is not permitted in this examination.

Geometrical Tools

Geometrical instruments are required for construction problems.

Comprehensive Coverage

Covers all major topics from the Class X mathematics syllabus.

General Instructions

• All questions are compulsory unless specified otherwise.
• Marks for each question are indicated in brackets.
• Use of calculators is not permitted.
• Draw neat diagrams wherever required.
• For fill in the blanks and true/false questions, answer any five.
• Show all working steps for full marks.
• Write answers clearly and legibly.

Section 1: Multiple Choice Questions

[1 × 6 = 6 Marks]

Choose the correct answer in each of the following:

i In a partnership business, the ratio of profits received by two friends is \(\frac{1}{2} : \frac{1}{3}\). The ratio of their capitals is:

(a) 2 : 3

(b) 3 : 2

(c) 1 : 1

(d) 5 : 3

ii If \(p+q=\sqrt{13}\) and \(p-q=\sqrt{5}\), then the value of \(pq\) is:

(a) 2

(b) 18

(c) 9

(d) 8

iii O is the center of a circle and AB is a chord. ABCD is a cyclic quadrilateral. If \(\angle ABC = 65^\circ\) and \(\angle DAC = 40^\circ\), then the value of \(\angle BCD\) is:

(a) 75°

(b) 105°

(c) 115°

(d) 80°

iv If \(\tan \alpha + \cot \alpha = 2\), then the value of \(\tan^{13} \alpha + \cot^{13} \alpha\) is:

(a) 13

(b) 2

(c) 1

(d) 0

v Two cubes each with side \(2\sqrt{6}\) cm are placed side by side to form a cuboid. The length of the diagonal of the cuboid is:

(a) 10 cm

(b) 6 cm

(c) 2 cm

(d) 12 cm

vi If the mean of \(x_1, x_2, x_3, \dots, x_{10}\) is 20, then the mean of \(x_1 + 4\), \(x_2 + 4\), \(x_3 + 4\), \(\dots\), \(x_{10} + 4\) is:

(a) 20

(b) 24

(c) 40

(d) 10

Section 2: Fill in the Blanks

[1 × 5 = 5 Marks]

Answer any five of the following:

i If the annual compound interest rate is \(r\%\) and the principal in the first year is \(P\) rupees, then the principal in the second year is ______.

ii \(7\sqrt{11}\) is a ______ number.

iii If the radius of a sphere is \(r\) and its volume is \(v\), then \(v \propto\) ______.

iv Two triangles are similar if their corresponding sides are ______.

v If the opposite angles of a quadrilateral are supplementary, then the vertices of the quadrilateral are ______.

vi If the length, breadth, and height of a cuboid are equal, then the name of that solid is ______.

Section 3: True or False

[1 × 5 = 5 Marks]

Answer any five of the following:

i In compound interest, if the interest rates for the first, second, and third years are \(r_1\%\), \(r_2\%\), and \(r_3\%\) respectively, then the amount for \(P\) rupees after 3 years is \[P\left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)\left(1 + \frac{r_3}{100}\right) \text{ rupees.}\]

ii The values of cos 36° and sin 54° are equal.

iii Only one tangent can be drawn to a circle from an external point.

iv The compound ratio of \(2ab : c^2\), \(bc : a^2\), and \(ca : 2b^2\) is 1 : 1.

v If the numerical values of the surface area and volume of a sphere are equal, then the radius is 3 units.

vi The mode of the numbers 5, 2, 4, 3, 5, 2, 5, 2, 5, 2 is 2.

Section 4: Short Answer Questions

[2 × 10 = 20 Marks]

Answer any ten of the following:

vii The radii of two circles are 8 cm and 3 cm. The distance between their centers is 13 cm. What is the length of the direct common tangent of the circles?

viii What is the radian measure of the angle swept by the hour hand of a clock in 1 hour?

ix If \(\tan 4\theta \tan 6\theta = 1\) and \(6\theta\) is positive acute, determine the value of \(\theta\).

x The height of a right circular cylinder is 12 cm and its volume is 100π cubic cm. Determine the radius of the base of the cylinder.

xi If the ratio of the surface areas of two spheres is 1 : 4, determine the ratio of their volumes.

xii If \(u_1 = \frac{x_1 - 35}{10}\), \(\sum f_1 u_1 = 30\), and \(\sum f_1 = 60\), then determine the value of \(\bar{x}\).

Section 5: Compound Interest or Partnership

[5 Marks]

Answer any one of the following:

i Determine the compound interest on ₹40,000 for 3 years at 5% per annum compound interest.

ii Rajib started a business with a capital of ₹3,750. After 6 months, Sayan joined the business with a capital of ₹15,000. At the end of the year, the profit was ₹6,900. Determine each one's share of the profit.

Section 6: Algebra

[3 Marks]

Answer any one of the following:

i Solve: \(\frac{x}{x+1} + \frac{x+1}{x} = 2 \frac{1}{12}\), where \(x \neq 0, -1\)

ii A train travels 200 km at a uniform speed. If the speed of the train is increased by 5 km/h, it takes 2 hours less to cover the same distance. Determine the speed of the train.

Section 7: Algebra and Proportion

[3 Marks]

Answer any one of the following:

i If \(x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\) and \(y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\), then find the value of \(x^2 - xy + y^2\).

ii If \(x \propto y\) and \(y \propto z\), prove that \((x^2 + y^2 + z^2) \propto (xy + yz + zx)\).

Section 8: Proportions

[3 Marks]

Answer any one of the following:

i If \(a, b, c, d\) are in continued proportion, show that \[(b^2 + d^2) : (b^2 - d^2) = (ab + cd) : (ab - cd)\]

ii If \(\frac{a+b-c}{a+b} = \frac{b+c-a}{b+c} = \frac{c+a-b}{c+a}\) and \(a+b+c \neq 0\), then prove that \(a = b = c\).

Section 9: Geometry Proofs

[5 Marks]

Answer any one of the following:

i Prove that all angles in the same segment of a circle are equal.

ii Prove that if a perpendicular is drawn from the right angle vertex of a right triangle to the hypotenuse, then the two triangles formed on either side of the perpendicular are similar to the original triangle and to each other.

Section 10: Geometry

[3 Marks]

Answer any one of the following:

i O is the center of a circle and QR is a chord. Tangents drawn at Q and R intersect at P. If QM is a diameter of the circle, prove that \(\frac{QR}{QM} = 2\).

ii In triangle ABC, \(\angle C = 90^\circ\). If CD is the median, prove that \(BC^2 = CD^2 + 3AD^2\).

Section 11: Construction

[5 Marks]

Answer any one of the following:

i Construct a triangle ABC with BC = 7 cm, AB = 5 cm, and AC = 6 cm. Draw the circumcircle of the triangle. (Only construction traces are required)

ii Determine geometrically the mean proportional between 4 cm and 3 cm. (Only construction traces are required)

Section 12: Trigonometry

[3 × 2 = 6 Marks]

Answer any two of the following:

i In \(\triangle ABC\), \(\angle C = 90^\circ\). If \(BC = m\) and \(AC = n\), show that \[m \sin A + n \sin B = \sqrt{m^2 + n^2}\]

ii Find the value: \[ \frac{4}{3} \cot^2 30^\circ + 3 \sin^2 60^\circ - 2 \cos^2 60^\circ - \frac{3}{4} \tan^2 30^\circ \]

iii If \(\angle P + \angle Q = 90^\circ\), then show that \[ \sqrt{\frac{\sin P}{\cos Q}} - \sin P \cos Q = \cos^2 P \]

Section 13: Height and Distance

[5 Marks]

Answer any one of the following:

i The width of a river is 600 m. Two boats start from the same bank of the river to go to the opposite bank. If the first boat makes an angle of 30° with the bank and the second boat makes an angle of 90° with the direction of the first boat to reach the opposite bank, what will be the distance between the two boats after they reach the opposite bank?

ii On the roof of a three-story building, there is a flagpole of length 3.6 m. From a point on the road, the angles of elevation of the top and bottom of the flagpole are 50° and 45° respectively. What is the height of the building? (Take tan 50° = 1.2)

Section 14: Mensuration

[4 × 2 = 8 Marks]

Answer any two of the following:

i The ratio of length, breadth, and height of a solid cuboid is 4:3:2 and its total surface area is 468 sq. cm. Find the volume of the cuboid.

ii The radius of the base of a solid cylindrical rod is 3.2 dm. The rod is melted to form 21 solid spheres. If the radius of each sphere is 8 cm, find the length of the rod.

iii The slant height of a solid right circular cone is 7 cm and its total surface area is 147.84 sq. cm. Find the radius of the base of the cone.

Section 15: Statistics

[4 × 2 = 8 Marks]

Answer any two of the following:

i Calculate the arithmetic mean from the following frequency distribution:

Class Limit	10-20	20-30	30-40	40-50	50-60	60-70
Frequency	10	16	20	30	13	11

ii Find the median of the data below:

Class Limit	1-5	6-10	11-15	16-20	21-25	26-30	31-35
No. of Workers	2	4	6	8	10	5	4

iii Determine the mode of the following frequency distribution:

Class Limit	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
Frequency	2	6	10	16	22	11	8	5

Class X Mathematics Test Practice Paper | Total Marks: 90 | Duration: 3 Hours

This examination paper covers the complete Class X Mathematics syllabus as prescribed by the board.