Exercise 1.2

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TRIGONOMETRIC FUNCTIONS-Exercise 3.2

Exercise • Jan 2026

Trigonometric Functions form a crucial foundation of higher mathematics and play a vital role in physics, engineering, astronomy, and real-life proble...

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Maths

TRIGONOMETRIC FUNCTIONS-Exercise 3.1

Exercise • Jan 2026

Trigonometric Functions form a crucial foundation of higher mathematics and play a vital role in physics, engineering, astronomy, and real-life proble...

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September 5, 2025  |  By Academia Aeternum

Exercise 1.2

Maths - Exercise
1. State whether the following statements are true or false. Justify your answers.
  1. Every irrational number is a real number. True, since collection of real numbers is made up of rational and irrational numbers.
  2. Every point on the number line is of the form \(\sqrt{m}\) , where m is a natural number. False, no negative number can be the square root of any natural number.
  3. Every real number is an irrational number. False, for example 2 is real but not irrational.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.False, Squre root of 4 is 2 which is a rational number.

Frequently Asked Questions

A number system is a way of expressing numbers using symbols and rules. It includes natural numbers, whole numbers, integers, rational, and irrational numbers.

Real numbers include both rational and irrational numbers that can be represented on the number line.

Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and \(q \neq 0.\)

Irrational numbers cannot be written as a simple fraction and have non-terminating, non-repeating decimals, like v2 or p.

Rational numbers can be expressed as p/q, while irrational numbers cannot. Rational decimals terminate or repeat; irrational decimals do not.

Natural numbers are counting numbers starting from 1, 2, 3, and so on.

Whole numbers include all natural numbers and 0, i.e., 0, 1, 2, 3, 4, ...

Integers include all whole numbers and their negatives, such as … -3, -2, -1, 0, 1, 2, 3 …

The decimal expansion of rational numbers is either terminating or non-terminating repeating.

The decimal expansion of irrational numbers is non-terminating and non-repeating.

Yes, every real number, whether rational or irrational, can be represented on the number line.

All rational numbers are real, but not all real numbers are rational. Real numbers include both rational and irrational types.

Construct a right-angled triangle with both legs of 1 unit each; the hypotenuse represents v2 when plotted on the number line.

A non-terminating decimal continues infinitely without ending, like 0.333... or 0.142857142857...

A repeating decimal has digits that repeat in a pattern, for example, 0.666… or 0.142857142857…

A terminating decimal has a finite number of digits after the decimal point, like 0.5 or 0.125.

The laws of exponents include: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), (a^m)^n = a^(mn), and a^0 = 1.

For any non-zero number, a° = 1.

The product of a number and its reciprocal is always 1.

A number line visually represents all real numbers in order, showing their relative positions.

Yes, 0 is a rational number because it can be expressed as 0/1.

A rational number is in standard form when its denominator is positive, and the numerator and denominator have no common factors except 1.

Surds are irrational numbers that can be expressed in root form, such as v2, v3, and v5.

Natural ? Whole ? Integers — meaning each set is contained in the next larger one.

Closure property states that the result of an operation on numbers in a set remains within that set.

Yes, the sum of two rational numbers is always a rational number.

No, the sum of two irrational numbers may or may not be irrational (e.g., v2 + (-v2) = 0).

Yes, all terminating decimals can be expressed as fractions, so they are rational.

No, p is an irrational number because its decimal expansion is non-terminating and non-repeating.

Real numbers are used in measurement, money, temperature, and scientific calculations involving both rational and irrational values.

A number represents quantity or value, while a numeral is the symbol used to express that number.

Prime numbers are natural numbers greater than 1 that have only two factors — 1 and the number itself.

Composite numbers have more than two factors, such as 4, 6, 8, and 9.

Euclid’s division lemma states that for any integers a and b, there exist unique integers q and r such that a = bq + r, where 0 = r < b.

By repeatedly applying a = bq + r, the last non-zero remainder gives the HCF of two numbers.

For two numbers a and b, HCF × LCM = a × b.

Terminating decimals are rational numbers whose denominators (in lowest form) have prime factors 2, 5, or both.

Rational numbers whose denominators (in lowest form) have prime factors other than 2 or 5 give non-terminating repeating decimals.

The base is the number of unique digits used to represent numbers in a system. For example, base 10 uses digits 0–9.

The binary system is a base-2 system using only 0 and 1, commonly used in computers.

The decimal system is a base-10 system using digits from 0 to 9, most commonly used in mathematics.

Examples include v2, v3, p, and e, which have non-repeating, non-terminating decimals.

If a number can be expressed as p/q, it’s rational; otherwise, it’s irrational.

No, dividing two integers may not always result in an integer, e.g., 3 ÷ 2 = 1.5.

It means plotting each real number at a specific position corresponding to its value on a line.

A negative exponent means reciprocal of the base raised to the positive exponent, e.g., a?n = 1/an.

Laws include: a^(1/n) = nva and a^(m/n) = (nva)^m.

Number systems help perform calculations, comparisons, and quantitative measurements efficiently.

It is used to compute the highest common factor (HCF) of two given numbers.

Construct a right triangle with sides 2 units and 1 unit; the hypotenuse equals v5 when placed on the number line.

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