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Class X Mathematics

Chapter 1: Real Numbers

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

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Chapter Overview

Welcome to Class X Mathematics: Real Numbers. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Number SystemsPlaying with Numbers

About This Chapter

This comprehensive study guide for Real Numbers is designed for Class X students following the CBSE and NCERT Mathematics curriculum. It covers 4 key subtopics including Fundamental Theorem of Arithmetic, Euclid division lemma overview, Rational and Irrational proofs (proving √2, √3, √5 are irrational), and 1 more essential concepts. Whether you are preparing for school examinations, CBSE board exams, or competitive tests, this resource provides everything you need to build a strong conceptual foundation and achieve mastery.

The chapter includes 1 key formulas and equations, 1 fully worked step-by-step example problems, interactive practice exercises across 9 difficulty categories, timed mock quizzes, and downloadable worksheets. Each topic is explained with detailed concept definitions, mathematical representations, and expert study guidelines to help you understand not just the "how" but the "why" behind every formula and method.

Mathematics is a subject that rewards consistent practice and conceptual clarity over rote memorization. As you work through this chapter on Real Numbers, focus on understanding the underlying principles first, then gradually increase problem difficulty. Use the practice sections to identify and strengthen weak areas, and refer to the common mistakes section to avoid the pitfalls that most students encounter.

What You'll Learn in This Chapter

By the end of studying Real Numbers for Class X, you will have developed proficiency in the following learning outcomes as outlined by the NCERT syllabus:

Verify prime factorization structures using product properties.

Prove irrationality of square root integers using contradiction proofs.

Determine if a rational fraction has terminating decimal expansion.

Prerequisites for This Chapter

Before studying Real Numbers, make sure you are comfortable with the following prerequisite concepts. A strong foundation in these areas will help you understand new topics faster and solve problems more confidently:

Number SystemsPlaying with Numbers

If any of these prerequisites feel unfamiliar, consider reviewing them first using the Related Chapters section at the bottom of this page. Building a solid base ensures you can tackle Real Numbers with full confidence.

Real-World Applications of Real Numbers

Students often wonder “Where will I use Real Numbersin real life?” The answer is: everywhere. The mathematical concepts you learn in this chapter have practical applications across science, engineering, technology, medicine, finance, and everyday problem-solving. Here are some notable examples:

Digital Signal Processing

Engineers use the unique prime factorization theorem (Fundamental Theorem of Arithmetic) in encryption algorithms that secure digital communications.

Computer Science & Cryptography

RSA encryption relies on properties of prime numbers and the difficulty of factoring large numbers — a direct application of number theory.

Musical Harmony & Frequencies

Musical intervals are based on rational number ratios between frequencies — the octave is 2:1, the perfect fifth is 3:2.

Measurement Precision

Understanding rational vs irrational numbers helps scientists express measurement precision and handle significant figures correctly.

Understanding the real-world relevance of Real Numbers not only makes learning more engaging but also helps you appreciate how mathematical thinking is a superpower that opens doors in virtually every career path — from engineering and medicine to finance and technology.

Study Tips for Real Numbers

Follow these expert study strategies to maximize your understanding and exam performance in this chapter. These tips are specifically tailored for the type of content covered in Real Numbers:

✏️

Practice Step-by-Step

Write out every intermediate step when solving problems. Skipping steps is the most common source of errors in calculation-heavy chapters. Build speed only after achieving consistent accuracy.

🔄

Verify by Back-Substitution

After finding your answer, substitute it back into the original equation to verify correctness. This simple habit catches most arithmetic and sign errors before they cost you marks.

📝

Maintain an Error Log

Keep a dedicated notebook of mistakes you make during practice. Review it weekly to identify patterns — you will notice the same types of errors recurring and can actively work to eliminate them.

Pro Tip: Consistency beats intensity. Studying Real Numbers for 30 minutes daily is far more effective than cramming for 5 hours before the exam. Use the practice sections below to build muscle memory through regular problem-solving.

Detailed Topic Breakdown

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Fundamental Theorem of Arithmetic

Concept Explanation

The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorized) as a product of prime numbers, and this factorization is unique, apart from the order in which the prime factors occur.

Mathematical Representation

n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k} \quad \text{where } p_i \text{ are distinct primes}

Study Guideline: Use prime factorization to find the HCF and LCM of numbers: HCF is the product of the lowest powers of common prime factors; LCM is the product of the highest powers of all prime factors.

2Euclid division lemma overview

Concept Explanation

Euclid's Division Lemma states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) satisfying a = bq + r, where the remainder r is non-negative and strictly less than the divisor b.

Mathematical Representation

a = bq + r \quad \text{where } 0 \le r < b

Study Guideline: Use this lemma repeatedly in Euclid's division algorithm to compute the HCF of two numbers by setting the divisor as the new dividend and the remainder as the new divisor until r = 0.

3Rational and Irrational proofs (proving √2, √3, √5 are irrational)

Concept Explanation

Proofs of irrationality show that numbers like √2, √3, or √5 cannot be written as a ratio of co-prime integers. These proofs use contradiction: assuming the number is rational (p/q), showing that both p and q must share a common factor (violating co-primality).

Mathematical Representation

\sqrt{2} = \frac{p}{q} \implies 2q^2 = p^2 \implies 2 \mid p \implies 4 \mid p^2 \implies 2 \mid q

Study Guideline: Remember that co-prime numbers share no common factors other than 1. The proof shows both numerator and denominator are even, which contradicts their co-primality.

4Decimal expansions of rational numbers

Concept Explanation

A rational number p/q has a terminating decimal expansion if the prime factorization of its denominator q is of the form 2^n * 5^m, where n and m are non-negative integers. Otherwise, it has a non-terminating repeating decimal expansion.

Mathematical Representation

q = 2^n \cdot 5^m \iff \text{decimal terminates}

Study Guideline: Simplify the fraction completely by canceling common factors before checking the prime factorization of the denominator.

Scientific Calculator Active: A custom solver for the specific subtopics of this chapter is currently in development. In the meantime, please use our comprehensive Scientific Functions Calculator below to verify your study equations.

Scientific Functions Calculator

Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.

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Value (X)45

Function

Calculated Results

Function Output

0.71

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NEXPROTOOLS COMPUTATIONAL REPORT

Deterministic Mathematical Simulation Engine • Verified Calculations

Date: 9/1/2026

Time: 8:48:00 AM

Scientific Functions Calculator

Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.

1. Input Parameters

Parameter	Value	Unit
Value (X)	45	—
Function	sin	—

2. Calculated Results

Metric	Calculated Output
Function Output	0.707

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Student Solver & Visualizer Guide

Real-time calculations

Step-by-step solving

1. Collect input parameters:

- operand1 = 45

- operation = sin

2. Feed parameters through standard algorithmic calculation methods.

3. Compile calculated results:

- calcResult = 0.7071

Student-friendly explanations

"This math utility processes input parameters using standard algebraic algorithms. By substituting variables into target formulas, the engine solves equations and computes results instantly with high arithmetic precision!"

Visual explanations

MATH SOLVER RUNNING:
 [Inputs] ──► [Mathematical Formula] ──► [Outputs]
  Processed elements successfully.

Class X CBSE Board Practice Questions

Review and solve 213 real Board paper questions from the 20 mock sample sets matching Real Numbers.

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Sample Paper 01 | Question 3Section A (1 Mark)

If two positive integers p and q can be expressed as p = 18 a2b4 and q = 20 a3b2, where a and b are prime numbers, then LCM (p, q) is :

Sample Paper 01 | Question 8Section A (1 Mark)

If ∆PQR ~ ∆ABC; PQ = 6 cm, AB = 8 cm and the perimeter of ∆ABC is 36 cm, then the perimeter of ∆PQR is

Sample Paper 01 | Question 10Section A (1 Mark)

ΔABC~ΔPQR. If AM and PN are altitudes of ΔABC and ΔPQR respectively and AB2 : PQ2 = 4 : 9, then AM : PN =

Sample Paper 01 | Question 16Section A (1 Mark)

ABCD is a trapezium with AD ∥ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Sample Paper 01 | Question 18Section A (1 Mark)

The middle most observation of every data arranged in order is called

Sample Paper 01 | Question 19Section A (1 Mark)

Assertion (A): If product of two numbers is 5780 and their HCF is 17, then their LCM is 340 Reason (R): HCF is always a factor of LCM

Sample Paper 01 | Question 22Section B (2 Marks)

The length of the minute hand of a clock is 6cm. Find the area swept by it when it moves from 7:05 p.m. to 7:40 p.m.

Sample Paper 01 | Question 26Section C (3 Marks)

Prove that √5 is an irrational number.

Sample Paper 01 | Question 31Section C (3 Marks)

Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old are Rashmi and Nazma now?

Sample Paper 01 | Question 33Section D (5 Marks)

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

Sample Paper 02 | Question 3Section A (1 Mark)

When 2120 is expressed as the product of its prime factors we get

Sample Paper 02 | Question 5Section A (1 Mark)

The ratio in which x-axis divides the join of (2, -3) and (5, 6) is:

Sample Paper 02 | Question 6Section A (1 Mark)

The solution of the following pair of equation is: x - 3y = 2, 3x - y = 14

Sample Paper 02 | Question 12Section A (1 Mark)

ABCD is a trapezium with AD ∥ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Sample Paper 02 | Question 18Section A (1 Mark)

The LCM of smallest two-digit composite number and smallest composite number is:

Sample Paper 02 | Question 19Section A (1 Mark)

Assertion (A): If HCF ( 90, 144) = 18, then LCM (90, 144) = 720 Reason (R): HCF (a, b) x LCM (a, b) = a x b

Sample Paper 02 | Question 20Section A (1 Mark)

Assertion (A): The point (0, 4) lies on y-axis. Reason (R): The y co-ordinate of the point on x-axis is zero.

Sample Paper 02 | Question 21Section B (2 Marks)

In the given figure below, AD/AE=AC/BD and ∠1=∠2. Show that Δ BAE~ ΔCAD .

Sample Paper 02 | Question 26Section C (3 Marks)

Prove that √2 is an irrational number.

Sample Paper 02 | Question 33Section D (5 Marks)

A motor boat whose speed is 15 km/hr in still water goes 30 km downstream and comes back in 4 hours 30 minutes. Find the speed of the stream.

Sample Paper 02 | Question 34Section D (5 Marks)

State and prove Basic Proportional Theorem.

Sample Paper 03 | Question 1Section A (1 Mark)

The solution of the following pair of equation is: x - 3y = 2, 3x - y = 14

Sample Paper 03 | Question 6Section A (1 Mark)

108 can be expressed as a product of its primes as ……………..

Sample Paper 03 | Question 16Section A (1 Mark)

In the ∆ABC, D and E are points on side AB and AC respectively such that DE || B

Sample Paper 03 | Question 17Section A (1 Mark)

What is the positive real root of 64x2 - 1 = 0?

Sample Paper 03 | Question 18Section A (1 Mark)

If two positive integers a and b are written as a = x3y2 and b = xy3, where x and y are prime numbers, then the HCF (a, b) is:

Sample Paper 03 | Question 19Section A (1 Mark)

Assertion (A): The HCF of two numbers is 9 and their LCM is 2016. If the one number is 54, then the other number is 336. Reason (R): Relation between numbers and their HCF and LCM is product of two numbers a, b = HCF (a, b) × LCM (a, b).

Sample Paper 03 | Question 24Section B (2 Marks)

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

Sample Paper 03 | Question 26Section C (3 Marks)

A part of monthly hostel charges in a college is fixed and the remaining depends on the number of days one has taken food in the mess. When a student ‘A’ takes food for 22 days, he has to pay Rs. 1380 as hostel charges; whereas a student ‘B’, who takes food for 28 days, pays Rs. 1680 as hostel charges. Find the fixed charges and the cost of food per day.

Sample Paper 03 | Question 31Section C (3 Marks)

Prove that √5 is an irrational number.

Sample Paper 03 | Question 32Section D (5 Marks)

Some students planned a picnic. The total budget for food was Rs. 2,000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by Rs. 20. How many students attended the picnic and how much did each student pay for the food?

Sample Paper 03 | Question 38Section E (4 Marks)

Case Study - 3 In the month of April to June 2022, the exports of passenger cars from India increased by 26% in the corresponding quarter of 2021 -22, as per a report. A car manufacturing company planned to produce 1800 cars in 4th year and 2600 cars in 8th year. Assuming that the production increases uniformly by a fixed number every year. Based on the above information

Sample Paper 04 | Question 8Section A (1 Mark)

If ∆ABC ~ ∆EDF and ∆ABC is not similar to ∆DEF, then which of the following is not true?

Sample Paper 04 | Question 9Section A (1 Mark)

ΔABC~ΔPQR. If AM and PN are altitudes of ΔABC and ΔPQR respectively and AB2: PQ2 = 4: 9, then AM: PN =

Sample Paper 04 | Question 10Section A (1 Mark)

If 2sin2 β - cos2 β = 2, then β is:

Sample Paper 04 | Question 12Section A (1 Mark)

If two positive integers p and q can be expressed as p = ab2 and q = a3b; a, b being prime numbers, then LCM (p, q) is

Sample Paper 04 | Question 13Section A (1 Mark)

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is

Sample Paper 04 | Question 20Section A (1 Mark)

Assertion: The HCF of two numbers is 9 and their LCM is 2016. If the one number is 54, then the other number is 336. Reason: Relation between numbers and their HCF and LCM is product of two numbers a, b = HCF (a, b) × LCM (a, b).

Sample Paper 04 | Question 21Section B (2 Marks)

If sin(A + B) = 1 and cos(A - B) = √3/2, 0°< A + B ≤ 90° and A > B, then find the measures of angles A and B.

Sample Paper 04 | Question 22Section B (2 Marks)

The length of the minute hand of a clock is 6cm. Find the area swept by it when it moves from 5:25 pm to 6:00 pm.

Sample Paper 04 | Question 24Section B (2 Marks)

In the given figure, AP = 3 cm, AR = 4.5 cm, AQ = 6 cm, AB = 5 cm, AC = 10 cm. Find the length of AD

Sample Paper 04 | Question 30Section C (3 Marks)

Given that √5 is irrational, prove that 2 + 3√5 is irrational.

Sample Paper 04 | Question 33Section D (5 Marks)

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Sample Paper 04 | Question 38Section E (4 Marks)

The top of a table is shown in the figure given below: On the basis of above information

Sample Paper 05 | Question 1Section A (1 Mark)

The ratio in which x-axis divides the join of (2, -3) and (5, 6) is:

Sample Paper 05 | Question 2Section A (1 Mark)

In ∆ABC and ∆DEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE. Then, the two triangles are

Sample Paper 05 | Question 3Section A (1 Mark)

ABCD is a trapezium with AD ∥ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Sample Paper 05 | Question 6Section A (1 Mark)

In the figure, if OA OD B OC O= , then which pair of angles are equal?

Sample Paper 05 | Question 10Section A (1 Mark)

The value of ‘a’, if HCF (a, 18) = 2 and LCM (a, 18) = 36, is: (1)

Sample Paper 05 | Question 11Section A (1 Mark)

10. The value of ‘a’, if HCF (a, 18) = 2 and LCM (a, 18) = 36, is: (1)

Sample Paper 05 | Question 14Section A (1 Mark)

The LCM of smallest two-digit composite number and smallest composite number is:

Sample Paper 05 | Question 15Section A (1 Mark)

The solution of the following pair of equation is: x - 3y = 2, 3x - y = 14

Sample Paper 05 | Question 17Section A (1 Mark)

A horse is tied to a pole with 28 m long rope. The perimeter of the field where the horse can graze is (Take π = 22/7)

Sample Paper 05 | Question 19Section A (1 Mark)

Assertion (A): The number 6n, n being a natural number, ends with the digit 5. Reason (R): The number 9n cannot end with digit 0 for any natural number n.

Sample Paper 05 | Question 22Section B (2 Marks)

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

Sample Paper 05 | Question 23Section B (2 Marks)

Evaluate: 3 cos2 60° sec2 30° - 2 sin2 30° tan2 60°.

Sample Paper 05 | Question 24Section B (2 Marks)

In the given figure below, AD/AE=AC/BD and ∠1=∠2. Show that Δ BAE~ ΔCAD .

Sample Paper 05 | Question 26Section C (3 Marks)

Prove that √5 is an irrational number.

Sample Paper 05 | Question 33Section D (5 Marks)

Sample Paper 05 | Question 35Section D (5 Marks)

State and prove Basic Proportional Theorem.

Sample Paper 05 | Question 38Section E (4 Marks)

Case Study - 3 Ananya saves Rs. 24 during the first month Rs. 30 in the second month and Rs. 36 in the third month. She continues to save in this manner. On the basis of above information

Sample Paper 06 | Question 3Section A (1 Mark)

In ΔABC right angled at B, if tanA = √3 , then then cosA cosC - sinAsinC =

Sample Paper 06 | Question 4Section A (1 Mark)

If LCM(x, 18) = 36 and HCF(x, 18) = 2, then x is:

Sample Paper 06 | Question 13Section A (1 Mark)

What is the positive real root of 64x2 - 1 = 0?

Sample Paper 06 | Question 14Section A (1 Mark)

In the figure, if DE || BC, AD = 3 cm, BD = 4 cm and BC = 14 cm, then DE equals :

Sample Paper 06 | Question 15Section A (1 Mark)

The ratio of LCM and HCF of the least composite and the least prime numbers is:

Sample Paper 06 | Question 19Section A (1 Mark)

Assertion (A): 6n never ends with the digit zero, where n is natural number. Reason (R): Any number ends with digit zero, if its prime factor is of the form 2m × 5n, where m, n are natural numbers.

Sample Paper 06 | Question 25Section B (2 Marks)

In the below left figure, two chords AB and CD intersect each other at the point P. Prove that (i) ΔAPC ~ ΔDPB (ii) AP. PB = CP. DP

Sample Paper 06 | Question 28Section C (3 Marks)

A part of monthly hostel charges in a college is fixed and the remaining depends on the numb er of days one has taken food in the mess. When a student ‘A’ takes food for 22 days, he has to pay Rs. 1380 as hostel charges; whereas a student ‘B’, who takes food for 28 days, pays Rs. 1680 as hostel charges. Find the fixed charges and the cost of food per day.

Sample Paper 06 | Question 29Section C (3 Marks)

Four bells toll at an interval of 8, 12, 15 and 18 seconds respectively. All the four begin to toll together. Find the number of times they toll together in one hour excluding the one at the start.

Sample Paper 06 | Question 32Section D (5 Marks)

Two pipes running together can fill a cistern in 1313 hours. If one pipe takes 3 hours more than the other to fill it, find the time in which each pipe would fill the cistern.

Sample Paper 07 | Question 1Section A (1 Mark)

The pair of equations x + 2y + 5 = 0 and -3x - 6y + 1 = 0 have

Sample Paper 07 | Question 5Section A (1 Mark)

Which of the following relationship is correct ?

Sample Paper 07 | Question 7Section A (1 Mark)

7. The distance of the point P (2, 3) from the x-axis is

Sample Paper 07 | Question 8Section A (1 Mark)

The sum of exponents of prime factors in the prime-factorisation of 196 is:

Sample Paper 07 | Question 13Section A (1 Mark)

A circus artist is climbing a 30 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the distance of the pole to the peg in the ground, if the angle made by the rope with the ground level is 30⁰.

Sample Paper 07 | Question 14Section A (1 Mark)

The solution of the equations x - y = 2 and x + y = 4 is:

Sample Paper 07 | Question 16Section A (1 Mark)

If 1080 = 2x × 3y × 5, then (x - y) is equal to :

Sample Paper 07 | Question 19Section A (1 Mark)

Assertion (A): The largest number that divide 70 and125 which leaves remainder 5 and 8 is 13 Reason (R): HCF (65,117) =13

Sample Paper 07 | Question 20Section A (1 Mark)

Assertion (A): If two triangles are similar and have an equal area, then they are congruent. Reason (R): Corresponding sides of two triangles are equal, then triangles are congruent.

Sample Paper 07 | Question 21Section B (2 Marks)

Find the points on the x-axis which are at a distance of 2√5 from the point (7, -4). How many such points are there?

Sample Paper 07 | Question 24Section B (2 Marks)

Solve for x and y: 71x + 37y = 253, 37x + 71y = 287

Sample Paper 07 | Question 27Section C (3 Marks)

Prove that 2 + 5√3 is an irrational number, if it is given that √3 is an irrational number.

Sample Paper 07 | Question 29Section C (3 Marks)

In the below figure, if ∠1 =∠2 and ΔNSQ = ΔMTR, then prove that ΔPTS ~ ΔPRQ.

Sample Paper 07 | Question 32Section D (5 Marks)

A motorboat whose speed in still water is 9 km/h, goes 15km downstream and comes back to the same spot, in a total time of 3 hours 45 minutes. Find the speed of the stream.

Sample Paper 08 | Question 1Section A (1 Mark)

HCF of (23 x 32 x 5), (22 x 33 x 52) and (24 x 3 x 53 x 7) is

Sample Paper 08 | Question 3Section A (1 Mark)

The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is:

Sample Paper 08 | Question 15Section A (1 Mark)

In what ratio does the x-axis divide the join of A(2, -3) and B(5, 6)?

Sample Paper 08 | Question 18Section A (1 Mark)

The distance between the points (2, - 1) and (- 1, - 5) is :

Sample Paper 08 | Question 20Section A (1 Mark)

Assertion : The HCF of two numbers is 18 and their product is 3072. Then their LCM = 169. Reason : If a, b are two positive integers, then HCF x LCM = a x b.

Sample Paper 08 | Question 23Section B (2 Marks)

If the point P(k - 1, 2) is equidistant from the points A(3, k) and B(k, 5), find the values of k.

Sample Paper 08 | Question 28Section C (3 Marks)

If 2x + y = 23 and 4x - y = 19, find the values of 5y - 2x and y/x - 2.

Sample Paper 08 | Question 31Section C (3 Marks)

Given that √3 is irrational, prove that (2 + √3) is an irrational number.

Sample Paper 08 | Question 32Section D (5 Marks)

Out of a group of swans, 7/2 times the square root of the total number of swans are playing on the shore of a tank. Remaining two are playing, with amorous fight, in the water. What is the total number of swans?

Sample Paper 09 | Question 2Section A (1 Mark)

The pair of equations x = 2a and y = 3b (a, b ≠ 0) graphically represents straight lines which are :

Sample Paper 09 | Question 4Section A (1 Mark)

In the given figure, if M and N are points on the sides OP and OS respectively of ∆OPS, such that MN || PS, then the length of OP is :

Sample Paper 09 | Question 5Section A (1 Mark)

The point on x-axis which is equidistant from the points (5, - 3) and (4, 2) is :

Sample Paper 09 | Question 6Section A (1 Mark)

The 7th term from the end of the A.P. : - 8, - 5, - 2, ..., 49 is :

Sample Paper 09 | Question 7Section A (1 Mark)

After an examination, a teacher wants to know the marks obtained by maximum number of the students in her class. She requires to calculate ................. of marks.

Sample Paper 09 | Question 8Section A (1 Mark)

Two positive integers m and n are expressed as m = p5q2 and n = p3q4, where p and q are prime numbers. The LCM of m and n is :

Sample Paper 09 | Question 16Section A (1 Mark)

XOYZ is a rectangle with vertices X(-3, 0), O(0, 0), Y(0, 4) and Z(x, y). The length of its each diagonal is

Sample Paper 09 | Question 17Section A (1 Mark)

Which term of the A.P. -29, -26, -23, ..., 61 is 16?

Sample Paper 09 | Question 21Section B (2 Marks)

If sin(A + B) = 1 and cos(A - B) = √3/2, 0°< A + B ≤ 90° and A > B, then find the measures of angles A and B.

Sample Paper 09 | Question 24Section B (2 Marks)

Given that √3 is irrational, prove that 5 + 2√3 is irrational.

Sample Paper 09 | Question 25Section B (2 Marks)

Find the type of triangle ABC formed whose vertices are A(1, 0), B(-5, 0) and C(-2, 5).

Sample Paper 09 | Question 30Section C (3 Marks)

If Nidhi were 7 years younger than what she actually is, then the square of her age (in years) would be 1 more than 5 times her actual age. What is her present age?

Sample Paper 09 | Question 35Section D (5 Marks)

(a) Using graphical method, solve the following system of equations: 3x + y + 4 = 0 and 3x - y + 2 = 0

Sample Paper 09 | Question 38Section E (4 Marks)

In a survey on holidays, 120 people were asked to state which type of transport they used on their last holiday. The following pie chart shows the results of the survey. Observe the pie chart and

Sample Paper 10 | Question 1Section A (1 Mark)

Let a and b be two positive integers such that a = p3q4 and b = p2q3 , where p and q are prime numbers. If HCF(a, b) = pmqn and LCM(a, b) = prqs, then (m + n)(r + s) =

Sample Paper 10 | Question 14Section A (1 Mark)

If the distance between the points (3, -5) and (x, -5) is 15 units, then the values of x are :

Sample Paper 10 | Question 23Section B (2 Marks)

In the given figure below, AD/AE=AC/BD and ∠1=∠2. Show that Δ BAE~ ΔCAD .

Sample Paper 10 | Question 24Section B (2 Marks)

Can the number (15)n, n being a natural number, end with the digit 0? Give reasons.

Sample Paper 10 | Question 26Section C (3 Marks)

Prove that √5 is an irrational number.

Sample Paper 10 | Question 38Section E (4 Marks)

A backyard is in the shape of a triangle ABC with right angle at B. AB = 7 m and BC 15 m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that AP = x m. Based on the above information,

Sample Paper 11 | Question 1Section A (1 Mark)

If 2 32 50p = , then p is a/an

Sample Paper 11 | Question 5Section A (1 Mark)

What is the ratio of their radii ?

Sample Paper 11 | Question 6Section A (1 Mark)

The distance of the point (-1, 7) from x-axis is:

Sample Paper 11 | Question 8Section A (1 Mark)

The distance of the point (-6, 8) from origin is:

Sample Paper 11 | Question 18Section A (1 Mark)

In ∆ABC, PQ || B

Sample Paper 11 | Question 24Section B (2 Marks)

Two number are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers?

Sample Paper 11 | Question 28Section C (3 Marks)

How many terms are there in an A.P. whose first and fifth terms are -14 and 2, respectively and the last term is 62. OR Which term of the A.P.: 65, 61, 57, 53, .................. is the first negative term?

Sample Paper 11 | Question 31Section C (3 Marks)

Find the HCF and LCM of 26, 65 and 117, using prime factorisation.

Sample Paper 11 | Question 35Section D (5 Marks)

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC, prove that CA2 = CB. CD

Sample Paper 11 | Question 38Section E (4 Marks)

Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey Rs. x per student and Cricket Rs. y per student. School 'P' decided to award a total of Rs. 9,500 for the two games to 5 and 4 students respectively; while school 'Q' decided to award Rs. 7,370 for the two games to 4 and 3 students respectively. Based on the above information,

Sample Paper 12 | Question 2Section A (1 Mark)

In a formula racing competition, the time taken by two racing cars A and B to complete 1 round of the track is 30 minutes and p minutes respectively. If the cars meet again at the starting point for the first time after 90 minutes and the HCF (30, p) = 15, then the value of p is

Sample Paper 12 | Question 3Section A (1 Mark)

For the following distribution: Marks Below 10 Below 20 Below 30 Below 40 Below 50 Below 60 No. of Students 3 12 27 57 75 80 the modal class is

Sample Paper 12 | Question 5Section A (1 Mark)

In ΔABC right angled at B, if cotC = √3 , then then cosAsinC + sinAcosC =

Sample Paper 12 | Question 9Section A (1 Mark)

In ABC, DE || AB, If CD = 3 cm, EC = 4 cm, BE = 6 cm, then DA is equal to

Sample Paper 12 | Question 15Section A (1 Mark)

If in two triangles, DEF and PQR, ∠D=∠Q and ∠R=∠Ethen which of the following is not true?

Sample Paper 12 | Question 16Section A (1 Mark)

The pair of equations x + 2y + 5 = 0 and -3x - 6y + 1 = 0 have

Sample Paper 12 | Question 17Section A (1 Mark)

If p and q are positive integers such that p = a3b2 and q = a2b, where ‘a’ and ‘b’ are prime numbers, then the LCM ( p, q) is …..

Sample Paper 12 | Question 18Section A (1 Mark)

The ratio in which x-axis divides the join of (2, -3) and (5, 6) is:

Sample Paper 12 | Question 20Section A (1 Mark)

Assertion (A): The length of the minute hand of a clock is 7 cm. Then the area swept by the minute hand in 5 minute is 77/6 cm2. Reason (R): The length of an arc of a sector of angle q and radius r is given by 0 2360l r θ π= ×

Sample Paper 12 | Question 23Section B (2 Marks)

Find the point on y-axis which is equidistant from the points (5, - 2) and (-3, 2).

Sample Paper 12 | Question 30Section C (3 Marks)

If its 25th term is three times its 8th term, find the AP.

Sample Paper 12 | Question 31Section C (3 Marks)

Prove that (sinA + cosecA)2 + (cosA + secA)2 = 7 + tan2A + cot2A

Sample Paper 12 | Question 32Section D (5 Marks)

A person on tour has Rs.360 for his expenses. If he extends his tour for 4 days, he has to cut down his daily expenses by Rs.3. Find the original duration of the tour.

Sample Paper 12 | Question 35Section D (5 Marks)

State and prove Basic Proportional Theorem.

Sample Paper 13 | Question 7Section A (1 Mark)

If two positive integers a and b are written as a = x3y2 and b = xy3; x, y are prime numbers, then HCF (a, b) is

Sample Paper 13 | Question 8Section A (1 Mark)

In ∆ABC and ∆DEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE. Then, the two triangles are

Sample Paper 13 | Question 12Section A (1 Mark)

In the figure, if OA OD B OC O= , then which pair of angles are equal?

Sample Paper 13 | Question 13Section A (1 Mark)

If the distance between the points (2, -2) and (-1, x) is 5, one of the values of x is

Sample Paper 13 | Question 15Section A (1 Mark)

The value of ‘a’, if HCF (a, 18) = 2 and LCM (a, 18) = 36, is: (1)

Sample Paper 13 | Question 20Section A (1 Mark)

Assertion (A): If product of two numbers is 5780 and their HCF is 17, then their LCM is 340 Reason (R) : HCF is always a factor of LCM

Sample Paper 13 | Question 22Section B (2 Marks)

In the below left figure, two chords AB and CD intersect each other at the point P. Prove that (i) ΔAPC ~ ΔDPB (ii) AP. PB = CP. DP

Sample Paper 13 | Question 23Section B (2 Marks)

Evaluate: 3 cos2 60° sec2 30° - 2 sin2 30° tan2 60°.

Sample Paper 13 | Question 25Section B (2 Marks)

If the perimeter of a protractor is 72 cm, calculate its area. (Use π = 22 7 )

Sample Paper 13 | Question 30Section C (3 Marks)

Two numbers are in the ratio of 1 : 3. If 5 is added to both the numbers, the ratio becomes 1 : 2. Find the numbers.

Sample Paper 13 | Question 31Section C (3 Marks)

Given that √3 is irrational, prove that 2 + 5√3 is irrational. So √3 = 2 5 p q q - …(i) RHS of equation (i) is rational.

Sample Paper 13 | Question 34Section D (5 Marks)

Two pipes running together can fill a cistern in 3 1 13 hours. If one pipe takes 3 hours more than the other to fill it, find the time in which each pipe would fill the cistern.

Sample Paper 13 | Question 38Section E (4 Marks)

Case Study-3 Mohan is an auto driver. His autorickshaw was too old and he had to spend a lot of money on repair and maintenance every now and then. One day he got to know about the EV scheme of the Government of India where he can not only get a good exchange bonus but also avail heavy discounts on the purchase of an electric vehicle. So, he took a loan of 1,18,000 from a reputed bank and purchased a new autorickshaw. Mohan repays his total loan of 118000 rupees by paying every month starting with the first instalment of 1000 rupees. (i) If he increases the instalment by 100 rupees every month, then what amount will be paid by him in the 30th instalment? [1] (ii) If he increases the instalment by 100 rupees every month, then what amount of loan does he still have to pay after 30th instalment? [2] OR (ii)If he increases the instalment by 200 rupees every month, then what amount would he pay in 40th instalment? [2] (iii) If he increases the instalment by 100 rupees every month, then what amount will be paid by him in the 100th instalment

Sample Paper 14 | Question 1Section A (1 Mark)

If two positive integers p and q can be expressed as p = ab3 and q = a3 b; a, b being prime numbers, then HCF (p, q) is

Sample Paper 14 | Question 4Section A (1 Mark)

If triangles ABC and DEF are similar and AB=4 cm, DE=6 cm, EF=9 cm and FD=12 cm, the perimeter of triangle ABC is:

Sample Paper 14 | Question 9Section A (1 Mark)

4 tan2 A - 4 sec2 A is equal to:

Sample Paper 14 | Question 10Section A (1 Mark)

Which of the following equations has 2 as a root?

Sample Paper 14 | Question 11Section A (1 Mark)

In the below figure, AD = 3 cm, AE = 5 cm, BD = 4 cm, CE = 4 cm, CF = 2 cm, BF = 2.5 cm, then

Sample Paper 14 | Question 15Section A (1 Mark)

The LCM of smallest two digit composite number and smallest composite number is:

Sample Paper 14 | Question 19Section A (1 Mark)

Assertion (A): The number 6n, n being a natural number, ends with the digit 5. Reason (R): The number 9n cannot end with digit 0 for any natural number n.

Sample Paper 14 | Question 20Section A (1 Mark)

Assertion (A): The point (3, 0) lies on x -axis. Reason (R): The x co-ordinate on the point on y -axis is zero.

Sample Paper 14 | Question 28Section C (3 Marks)

A man wished to give Rs. 12 to each person and found that he fell short of Rs. 6 when he wanted to give to all the persons present. He, therefore, distributed Rs. 9 to each person and found that Rs. 9 were left over. How much money did he have and how many persons were there?

Sample Paper 14 | Question 31Section C (3 Marks)

Given that √3 is irrational, prove that 5 + 2√3 is irrational. This contradiction has arisen due to our wrong assumption that 5 + 2√3 is rational So, 5 + 2√3 is irrational. SECTION - D Questions 32 to 35 carry 5 marks each.

Sample Paper 14 | Question 33Section D (5 Marks)

A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete tota l journey, what is the original average speed?

Sample Paper 14 | Question 38Section E (4 Marks)

Anita’s mother start a new shoe shop. To display the shoes, she put 3 pairs of shoes in 1st row, 5 pairs in 2nd row, 7 pairs in 3rd row and so on. On the basis of above information,

Sample Paper 15 | Question 1Section A (1 Mark)

If the HCF of two positive integers is 3 and their product is 9, what is their LCM?

Sample Paper 15 | Question 10Section A (1 Mark)

If two positive integers a and b are written as a = p^3 q^4 and b = p^2 q^3, where p and q are prime numbers, then LCM(a, b) is:

Sample Paper 15 | Question 19Section A (1 Mark)

Assertion (A): The HCF of two numbers is 5 and their LCM is 200. Product of numbers is 1000. Reason (R): For any two positive integers, HCF * LCM = Product of numbers.

Sample Paper 15 | Question 27Section C (3 Marks)

Prove that √2 - √5 is an irrational number.

Sample Paper 16 | Question 1Section A (1 Mark)

If two positive integers a and b are written as a = x^3 y^2 and b = x y^3, where x, y are prime numbers, then HCF(a, b) is:

Sample Paper 16 | Question 19Section A (1 Mark)

Assertion (A): The polynomial p(x) = x^2 + 3x + 2 has real zeroes. Reason (R): A quadratic polynomial ax^2 + bx + c has real zeroes if b^2 - 4ac ≥ 0.

Sample Paper 16 | Question 23Section B (2 Marks)

Explain why 2 * 5 * 7 * 11 + 11 * 13 is a composite number.

Sample Paper 16 | Question 26Section C (3 Marks)

Prove that √5 is an irrational number.

Sample Paper 17 | Question 8Section A (1 Mark)

If the diagonals of a quadrilateral divide each other proportionally, then it is a:

Sample Paper 17 | Question 9Section A (1 Mark)

In ∆ABC, DE || BC(as shown in the figure). If AD = 2cm, BD = 3cm, BC = 7.5cm, then the length of DE(in cm) is:

Sample Paper 17 | Question 10Section A (1 Mark)

The HCF and the LCM of 12, 21, 15 respectively are

Sample Paper 17 | Question 11Section A (1 Mark)

A pair of irrational numbers whose product is a rational number is:

Sample Paper 17 | Question 16Section A (1 Mark)

If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF, then the product of two numbers is

Sample Paper 17 | Question 21Section B (2 Marks)

If sin(A - B) = 1 2 , cos(A + B) = 1 2 , 00< A + B ≤900 , A > B. Find A and B.

Sample Paper 17 | Question 24Section B (2 Marks)

(a) Find a relation between x and y such that the point P(x, y) is equidistant from the points A(7,1) and B(3,5).

Sample Paper 17 | Question 25Section B (2 Marks)

In the figure, QR QT QS PR= and ∠1 = ∠2, Show that ∆PQS ∼ ∆TQR.

Sample Paper 17 | Question 33Section D (5 Marks)

A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.

Sample Paper 17 | Question 36Section E (4 Marks)

Mutual Fund : A mutual fund is a type of investment vehicle that pools money from multiple investors to invest in securities like stocks, bonds or other securities. Mutual funds are operated by professional money managers, who allocate the fund’s assets and attempt to produce capital gains or income for the fund’s investors. Net Asset Value (NAV) represents a fund’s per share market value. It is the price at which the investors buy fund shares from a fund company and sell them to a fund company. The following table shows the Net Asset Value (NAV) per unit of mutual fund of ICICI mutual funds: NAV (in Rs.) 0 - 5 5 - 10 10 - 15 15 - 20 20 - 25 Number of mutual funds 13 16 22 18 11 Based on the above information,

Sample Paper 17 | Question 37Section E (4 Marks)

A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze. Based on the above,

Sample Paper 17 | Question 38Section E (4 Marks)

Teaching Mathematics through activities is a powerful approach that enhances students' understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multipl y it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250. Now, Mukta asked some questions as given below to the students : (i) What is the least prime number used by students? (1) (ii) (a) How many students are in the class? (2) OR (b) What is the highest prime number used by students? (2) (iii) Which prime number has been used maximum times ? (1)

Sample Paper 18 | Question 5Section A (1 Mark)

For the following distribution: Marks Below 10 Below 20 Below 30 Below 40 Below 50 Below 60 No. of students 3 12 27 57 75 80 the modal class is

Sample Paper 18 | Question 9Section A (1 Mark)

If x = 2sin2θ and y = 2cos2θ + 1 then x + y is:

Sample Paper 18 | Question 11Section A (1 Mark)

If the LCM of a and 18 is 36 and the HCF of a and 18 is 2, then a =

Sample Paper 18 | Question 13Section A (1 Mark)

The point on the x-axis which is equidistant from (- 4, 0) and (10, 0) is:

Sample Paper 18 | Question 22Section B (2 Marks)

(a) In the given figure, PQ || RS. Prove that OP × OR = OQ × OS.

Sample Paper 18 | Question 23Section B (2 Marks)

Given that HCF (306, 1314) = 18, find LCM of (306, 1314).

Sample Paper 18 | Question 26Section C (3 Marks)

Show that 5 + 2√7 is an irrational number, where √7 is given to be an irrational number.

Sample Paper 18 | Question 29Section C (3 Marks)

(a) If Ritu were younger by 5 years than what she really is, then the square of her age would have been 11 more than five times her present age. What is her present age?

Sample Paper 18 | Question 32Section D (5 Marks)

A contractor plans to install two slides for the children to play in a park. For the children below the age of 6 years, he prefers to have a slide whose top is at a height of 2·0 m and is inclined at an angle of 30° to the ground, whereas for older children, he wants to have a steep slide at a height o f 4·0 m and inclined at an angle of 60° to the ground. What would be the length of the slide in each case ?

Sample Paper 18 | Question 35Section D (5 Marks)

A manufacturer of TV sets produced 720 TV sets in the fourth year and 880 TV sets in the eighth year. Assuming that the production increases uniformly by a fixed number every year, find the production in the tenth year and the total production in the first seven years.

Sample Paper 19 | Question 4Section A (1 Mark)

If ∆ABC ~ ∆EDF and ∆ABC is not similar to ∆DEF, then which of the following is not true?

Sample Paper 19 | Question 11Section A (1 Mark)

Given HCF(2520, 6600) = 40, LCM(2520, 6600) = 252 × k, then the value of k is:

Sample Paper 19 | Question 14Section A (1 Mark)

If cosθ = √3/2 and sinϕ = 1/2, then tan(θ + ϕ) is:

Sample Paper 19 | Question 17Section A (1 Mark)

Consider the following distribution: Marks obtained Number of students More than or equal to 0 63 More than or equal to 10 58 More than or equal to 20 55 More than or equal to 30 51 More than or equal to 40 48 More than or equal to 50 42 the frequency of the class 30-40 is

Sample Paper 19 | Question 20Section A (1 Mark)

Assertion (A): For any two positive integers a and b, HCF(a, b) x LCM(a, b) = a x b Reason (R): The HCF of two numbers is 5 and their product is 150. Then their LCM is 40.

Sample Paper 19 | Question 23Section B (2 Marks)

In the given figure, EA/EC = EB/ED, prove that ∆EAB ~ ∆ECD

Sample Paper 19 | Question 27Section C (3 Marks)

4 Bells toll together at 9.00 am. They toll after 7, 8, 11 and 12 seconds respectively. How many times will they toll together again in the next 3 hours?

Sample Paper 19 | Question 38Section E (4 Marks)

Deepankar bought 3 notebooks and 2 pens for Rs. 80 and his friend Suryansh bought 4 notebooks and 3 pens for Rs. 110 from the school bookshop. Based on the above information,

Sample Paper 20 | Question 1Section A (1 Mark)

In the ∆ABC, D and E are points on side AB and AC respectively such that DE || B

Sample Paper 20 | Question 3Section A (1 Mark)

The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, then the other number is

Sample Paper 20 | Question 8Section A (1 Mark)

ABCD is a trapezium with AD ∥ BC and AD = 4cm. If the diagonals AC and BD intersect each other at O such that AO/OC = DO/OB =1/2, then BC =

Sample Paper 20 | Question 9Section A (1 Mark)

When 2120 is expressed as the product of its prime factors we get

Sample Paper 20 | Question 14Section A (1 Mark)

The number of revolutions made by a circular wheel of radius 0.7 m in rolling a distance of 176 m is

Sample Paper 20 | Question 17Section A (1 Mark)

If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?

Sample Paper 20 | Question 20Section A (1 Mark)

Assertion (A): The number 6n never end with digit 0 for any natural number n.. Reason (R): The number 9n never end with digit 0 for any natural number n.

Sample Paper 20 | Question 23Section B (2 Marks)

In figure, ABCD is a rectangle. Find the values of x and y.

Sample Paper 20 | Question 30Section C (3 Marks)

A part of monthly hostel charges in a college is fixed and the remaining depends on the number of days one has taken food in the mess. When a student ‘A’ takes food for 22 days, he has to pay Rs. 1380 as hostel charges; whereas a student ‘B’, who takes foo d for 28 days, pays Rs. 1680 as hostel charges. Find the fixed charges and the cost of food per day.

Sample Paper 20 | Question 31Section C (3 Marks)

Prove that √5 is an irrational number.

Sample Paper 20 | Question 34Section D (5 Marks)

Two water taps together can fill a tank in 39 8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Sample Paper 20 | Question 38Section E (4 Marks)

Raj is an electrician in a village. One day power was not there in entire village and villagers called Raj to repair the fault. After thorough inspection he found an electric fault in one of the electric pole of height 5 m and he has to repair it. He needs to reach a point 1.3m below the top of the pole to undertake the repair work. Based on the above information

Chapter 1: Real Numbers

Chapter Overview

Prerequisite Concepts

About This Chapter

What You'll Learn in This Chapter

Prerequisites for This Chapter

Real-World Applications of Real Numbers

Digital Signal Processing

Computer Science & Cryptography

Musical Harmony & Frequencies

Measurement Precision

Study Tips for Real Numbers

Practice Step-by-Step

Verify by Back-Substitution

Maintain an Error Log

Detailed Subtopics Study Guide

1Fundamental Theorem of Arithmetic

Concept Explanation

Mathematical Representation

2Euclid division lemma overview

Concept Explanation

Mathematical Representation

3Rational and Irrational proofs (proving √2, √3, √5 are irrational)

Concept Explanation

Mathematical Representation

4Decimal expansions of rational numbers

Concept Explanation

Mathematical Representation

Learning Outcomes

Sub-Topics Directory

Key Formulas & Equations

Fundamental Theorem of Arithmetic

Related Formulas from Adjacent Chapters

Quadratic Roots Sum

Quadratic Roots Product

Interactive Problem Solver

Scientific Functions Calculator

Adjust Inputs

Calculated ResultsCopy Output

NEXPROTOOLS COMPUTATIONAL REPORT

Scientific Functions Calculator

1. Input Parameters

2. Calculated Results

Student Solver & Visualizer Guide

Step-by-step solving

Student-friendly explanations

Visual explanations

Core Metrics

Focus Weak Areas

Select Practice Mode

Easy Practice

Medium Practice

Hard Practice

Timed Chapter Quiz

Interactive Worksheet

Chapter Syllabus Test

Prerequisite & Mixed Revision

Previous Year style (Board Exam)

Olympiad & JEE Challenge

Class X Quiz Challenge

Recommended Quizzes from Other Chapters

Mock Quiz: Real Numbers

Adjacent Mock Quiz: Polynomials

Class X CBSE Board Practice Questions

Critical Mistakes to Avoid

!Sign Arithmetic Errors

!Division by Zero

!Incorrect LCM in Fractions

Revision Summary Cards

1Number Types

2Equivalent Fractions

3Reciprocal Rules

4Prerequisites Checked

5Step-by-Step Checking

Frequently Asked Questions

Recommended Study Syllabus

Polynomials

Pair of Linear Equations in Two Variables

Quadratic Equations

Recommended Mathematical Solvers

Calculated Results