Learning all the fundamental principles of CBSE Class 8 Maths is very imperative. If students are serious about scoring high marks in the higher grades, then CBSE Class 8 Maths Practice Worksheets and Question Papers are beneficial for you. There are numerous advantages that Class 8 Maths Students can experience by studying maths with the help of these Worksheets, Practice papers and Multiple Choice Questions . To start with, one can practice each topic and questions by following a extensive detailed and stepwise approach. Moreover, Students gets to practice each chapter and gauge their understanding of a particular topic by attempting various types of questions present in worksheets and practice papers.
- A number that can be expressed in the form p/q, where p and q are integers and q≠0.
- All integers and all fractions are rational numbers.
- A rational number is said to be in its standard form if its denominator is positive and its denominator and numerator are prime to each other.
- 0 is also a rational number, but it is neither positive nor a negative rational number.
- Rational numbers are closed under addition, subtraction, multiplication and division (provided divisor is not 0).
- The commutativity of addition as well as that of multiplication is true for rational numbers i.e. if a/b and c/d are any two rational numbers, then a/b + c/d = c/d + a/b and a/b × c/d = c/d × a/b
- The associativity of addition is true for rational numbers i.e. is a/b, c/d and e/f are any three rational numbers, then (a/b +c/d) + e/f = a/b + (c/d + e/f)
- Subtraction is neither commutative nor associative for rational numbers.
- Commutativity and associativity of division is not true for rational numbers.
- The sum of 0 and any rational number is the rational number itself i.e. if a/b is any rational number, then a/b + 0 = a/b.
- The result of subtracting zero from a rational number is the rational number itself i.e. is a/b is a rational number, then a/b – 0 = a/b.
- The negative of a rational number p/q is –p/q.
- P/q and –p/q are called negative (or additive inverse) of each other.
- Zero is the identity element for addition of rational numbers.
- The product of 1 and any rational number is the rational number itself i.e. if p/q is a rational number, then p/q × 1 = p/q
- The product of 0 and any rational number is 0 i.e. p/q × 0 = 0.
- If a/b and c/d are two rational numbers such that a/b × c/d = 1, then each is called the multiplicative inverse or reciprocal of each other.
- Zero has no reciprocal.
- We can get countless rational numbers between any two given rational numbers.
- If x and y are two rational numbers, then (x + y)/2 is a rational number between x and y such that x < (x + y)/2 < y.
LINEAR EQUATIONS IN ONE VARIABLE
- An equation is a statement of equality which contains an unknown quantity.
- In an equation, the expression on the left of the equality sign is the Left-Hand Side (L.H.S). The expression on the right of the equality sign is the Right-Hand Side (R.H.S).
- Any value of the unknown, which makes L.H.S and R.H.S equal, is called its solution or not.
- The process of carrying terms from one side to another is called transporting.
- We can transpose a term from one side of an equation to the other side b merely changing its sign.
- A simple closed curve made up of only line segments is called a polygon.
- Triangle is a polygon having 3 sides (or vertices).
- Quadrilateral is a polygon having 4 sides (or vertices).
- Pentagon is a polygon having 5 sides (or vertices).
- Hexagon is a polygon having 6 sides (or vertices).
- Heptagon is a polygon having 7 sides (or vertices).
- Octagon is a polygon having 8 sides (or vertices).
- Nonagon is a polygon having 9 sides (or vertices).
- Decagon is a polygon having 10 sides (or vertices).
- N-gon is a polygon having n sides (or vertices).
- Diagonal is a line segment connecting two consecutive vertices of a polygon.
- A regular polygon has sides of equal length and angles of equal measure.
- The sum of the measures of the three angles of a triangle is 180°.
- The sum of the measures of the external angles of the polygon is 360°.
- The trapezium is a quadrilateral with a pair of parallel sides.
- If the non-parallel sides of a trapezium are of equal length, it is called an isosceles trapezium.
- A quadrilateral having exact two distinct consecutive pairs of sides of equal length is called a kite.
- A parallelogram is a quadrilateral whose opposite sides are parallel.
- The opposite sides of a parallelogram are of equal length and the opposite angles are of equal measure.
- The diagonals of a parallelogram bisect each other.
- The adjacent angles of a parallelogram are supplementary.
- A rhombus is a quadrilateral with sides of equal length.
- The diagonals of a rhombus are perpendicular bisectors of one another.
- A rectangle is an equiangular quadrilateral.
- Each angle of a rectangle is a right angle and its diagonals are of equal length.
- A square is and equilateral rectangle.
- The diagonals of square are perpendicular bisectors of one another.
- The sum of all the interior angles of a polygon of n-sides is given by (n – 2) × 180°.
- The numerical information collected in various cases is called data.
- A pictograph is a pictorial representation of data using symbols to represent a group of items.
- A bar-graph is a representation of information using bars of uniform width, the length (or height) of which is proportional to the given value.
- The frequency gives the number of times that a particular entry occurs.
- A bar graph having bars of equal width with no gaps in between is called histogram.
- The difference between its highest and the lowest value of the data is called its range.
- In circle graph (or a pie chart) the size of each sector is proportional to the information it represents.
- To draw a pie graph, we divide the whole angle (360°) at the centre of a circle in proportion of the fractions.
- The number of data items in a certain interval is called its frequency.
- The observations of an experiment are called its outcomes.
- If each outcome of an experiment is independent of the other and also the chances of getting any one of them are the same, then they are called equally likely outcomes.
- Probability is a measure of the likelihood of getting a certain outcome.
- The probability of an event is obtained by dividing the number of times a favourable outcome by the trial number of outcomes.
- The probability values of an event lie between 1 and 0.
- An event which can never happen is called an impossible event.
- The probability of an impossible event is 0.
- The event which will certainly happen is called sure event.
- The probability of a sure event is 1.
SQUARES AND SQUARE ROOTS
- If any number m can be expressed as n2, then m is a square number.
- The numbers 1, 4, 9, 16, … … … are square numbers. These numbers are called perfect square numbers.
- The sum of first n odd natural numbers is n2.
- Squares of odd numbers are always odd.
- Squares of even numbers are always even.
- The numbers x, y, z are called Pythagorean Triplets, if x2 + y2 = z2.
- A number ending in 2, 3, 7 or 8 is never a perfect square.
- A number ending in an odd number of zeros is never a perfect square.
- If n is a perfect square, then 2n can never be a perfect square.
- The inverse operation of squaring is called square roots.
- Every perfect square number has two square roots.
- If a perfect square of n-digits and n is even, then its square root will have (n + 1)/2 digits.
CUBES AND CUBE ROOTS
- The numbers 1, 8, 27, 64, 125 … are called Perfect cubes or cube numbers.
- First ten cube numbers are: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- The cubes of even numbers are even.
- The cubes of odd numbers are odd.
- In the prime factorisation of any number, if each factor appears three times, then the number if a perfect cube.
- Finding the cube root of a number is the inverse operation of cubing.
- The symbol 3√ denotes\ ‘cube root’.
- The ratio of two quantities, say a and b, in the same units a/b and written as a : b.
- A ratio is said to be in the simplest form, if the first and the second temr have no common factor, other than 1.
- Ratio has no units.
- An equality of two ratios is called proportion.
- A fraction with its denominator as 100 is equal to that much per cent as much is the numerator.
- x/100 = x%
- To change a fraction into a per cent, we multiply the fraction by 100.
- The money paid by the shopkeeper to buy an article is called its cost price.
- The money received by a shopkeeper on selling an article is called its selling price.
- Actual cost price = cost price + overhead charges.
- The money which is borrowed, is called principal.
- The period for the money is borrowed is called interest.
- The total money which is paid after the expiry of the time is called the amount.
- Amount = Principal + Interest
- If we denote principal by P, rate by R and time by T, then Simple Interest = (P×R×T)/100
- When the interest is added at the end of the year to calculate interest for the next year, the interest so obtained is called Compound Interest (C.I.)
- The period after which the interest is added to the principal is called the conversion period.
- The compound interest differs every year but simple interest remains the same.
- If ‘A’ stands for the amount, ‘P’ stands for period, ‘R’ stands for rate and ‘n’ stand for number of conversion periods, then A = P (1+R/100)
ALGEBRAIC EXPRESSION AND IDENTITIES
- Expressions are formed from variables and constants.
- A variable takes on different values.
- The numerical factor of a term is called its coefficient.
- An expression with one or more terms is called a polynomial.
- Expressions that contain only one term are called monomials.
- Expressions that contain two terms are called binomials.
- A three term expression is called trinomial.
- Algebraic terms with the same variable and same exponent are called like terms similar terms.
- Subtraction of a number is the same as addition of its additive inverse.
- To multiply a polynomial by a monomial multiply each term of the polynomial by the monomial and write the resulting terms with their proper signs.
- To multiply a polynomial by a polynomial, multiply each term of the multiplicand each term of the multiplier and take the algebraic sum of these partial products.
- An equation is a statement of equality of two algebraic expressions involve unknown quantity called a variable.
- An equation is not true for all values of the variable. It is true for only certain values of the variables in it.
- If equality is true for every value of the variables in it, then it is called an identity.
- (x+y)2 = x2 2xy + y2
- (x – y)2 = x2 – 2xy + y2
- (x – y) (x + y) = x2 – y2
- (x + a) (x + b) = x2 + (a + b)x+ ab
- A number which when substituted for the variable in an equation, make
- L.H.S. =R.H.S., is said to satisfy the equation and is called a root of the equation.
- Area of a plane figure is magnitude of the region enclosed by it.
- A closed plane figure formed by line segments is called a rectilinear figure.
- A rectilinear figure is said to be simple if no two sides of it intersect at a point other than the end point.
- The sum of the lengths of all sides of a rectilinear figure is called its perimeter.
- Area of a:
- rectangle = length x breadth
- square = side x side
- triangle = ½ x base x height
- rhombus = ½ x product of the diagonals
- quadrilateral = ½ x diagonal x (sum of offsets)
- trapezium = ½ (sum of parallel sides) x (Distance between parallel sides)
- circle = πr2
- In a cuboidal box there are three pairs of identical rectangular faces.
- In a cubical box, all six faces are squares and identical.
- Total surface area of a cuboid = 2[hxl + hxb + bxI], where h, I and b are the height, length and breadth respectively of the cuboid.
- Area of four walls of a rooms (= Lateral surface area of a cuboid) = 2h (l + b).
- Lateral surface area of a cylinder = 2πrh
- Total surface area of a cylinder = 2πr (r + h)
- Volume of a cuboid = area of base x height = I x b x h
- Volume of a cube =13
- Volume of a cylinder = πr2 x h
- Measures of area:
- 1 sq. cm = 100 mm2
- 1 sq. m = 10000 cm2
- 1 dm2 = 100 cm2
- 1 dam2 = 100 m2
- 1 km2 = 100 hm2
- 1 km2 = 1000000 m2
- Measures of volume:
- 1 cm3 = 1000 mm3
- 1 dm3 = 1000 cm3
- 1 m3 =1000 dm3
- 1 m3 = 1000 litres
- 1 m3 = 1000000 cm3
- 1 litre = 1000 millilitres
EXPONENTS AND POWERS
- If m is a positive integer and a is a rational number, then
- am = a x a x a x a …….. (m factors)
- am is called the mth power of ‘a’. Here ‘a’ is called base and ‘m’ is called exponent.
- A power with negative exponent (a-m) can also be expressed in terms of a power with positive exponent having the same base ‘a’ as 1/am.
- For any non-zero integer a, am x an = am+n Where m and n are natural numbers.
- For any non-zero integer a, am/an = am-n where m and n are natural numbers and ‘m > n’.
- If ‘a’ and ‘b’ are non-zero integers and ‘m’, ‘n’ are any integers, then
- (am)n = amn
- am x bm = (ab)m
- am/bm = (a/b)m
- (1)m = (1)n, for any integer m and n.
- For any rational (non-zero) number ‘a’, we have a0 = 1
- For comparing very large numbers or very small numbers, we have to change them in standard forms.
- The process of expressing a polynomial as a product of two or more polynomial of smaller degree is called factoring polynomials.
- Various types of polynomials to be factorized:
- Polynomial each of whose terms contains a common monomial factor am* bm * cm = m(a + b + c)
- Trinomial as a perfect square.
- x2 + 2xy+ y2 = (x + y)2 = (x + y) (x – y)
- x2 – 2xy + y2 = (x – y)2 =(x – y) (x – y)
- Polynomial as the difference of two squares.
- x2 – y2 = (x + y) (x – y)
- Second degree trinomial ax2 + bx + c = (x + p) (x + q) where p + q = b and pq = c
- A monomial multiplied by a monomial always gives a monomial. But a monomial divided by a monomial may not give a monomial.
- Expressions having variables in the denominator, are not polynomials.
- Expressions in which variables occur under root signs are not polynomials.
- If the remainder is zero, both, the divisor and the quotient, are factors of the dividend.
PLAYING WITH NUMBERS
- Think of any two-digit number and reverse it. Add the two numbers. Divide the sum by 11. The quotient will be equal to the sum of the digits of the number that was thought.
- Think of any two-digit number and reverse it. Subtract the smaller number from the larger. Divide the answer by 9. The quotient will be equal to the difference in digits of the number which was thought.
- Think of a three digit number. Write it by reversing the digits. Find their difference. Now this Difference + 99 = Difference between the hundreds digit and ones digit.
- A number is divisible by 10 just when its ones digit is 0.
- If we subtract the ones digit of a number from that number, the portion left over is a multiple of 10.
- If the ones digit of a number is 0 or 5, then it is divisible by 5.
- If the ones digit of a number is even, then the number itself is even.
- If the ones digit of a number is odd, then the number itself is odd.
- If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.
- If the sum of the digits of a number is divisible by 3, then number is divisible by 3.
- If a number is divisible by both 2 and 3, it must be divisible by 6.
- A number is divisible by 4, if the number formed by tens and units digits is divisible by 4.
- A number is divisible by 11, if the difference of the sum of digits at even places and the sum of digits at odd places (beginning from units place) is either 0 or a multiple of 11.
- A three-digit number is a multiple of 11, if the sum of its outer two digits minus its middle digit is a multiple of 11.
VISUALISING SOLID SHAPES
- 1. A polyhedron is a solid that is bounded by polygons which are called its faces. The faces meet at edges which are line segments. The edges meet at vertices which are points.
- 2. A polyhedron is convex if any two points on its surface can be joined by a line segment that entirely lies inside or on the polyhedrons.
- 3. A polyhedron is said to be regular if its faces are made up of regular polygons and the same number of faces meet at each vertex.
- 4. A prism is a polyhedron, two of whose faces are congruent polygons in parallel planes and whose other faces are parallelograms.
- 5. A pyramid is polyhedron whose base is a polygon (of any number of sides) and whose other faces are triangles with a common vertex.
- 6. According to Euler’s formula, the relation:
- [Number of faces (f)] + [Number of vertices (v)] = [Number of edges (e)] + 2 is true for any simple convex polyhedron.
- A map depicts the location of a particular object/place in relation to other objects/places
- 8. Maps use a scale which is fixed for a particular map. It reduces the actual distances proportionately to distances on the paper.
- 9. Visualising solid shapes (i.e., 3-D shapes) is a very useful skill which enables us to see its ‘hidden’ parts.
- 10. Various ways to visualise a solid shape are:
- by cutting or slicing the shape which would result in the cross-section of the solid.
- by observing a 2-D shadow of a 3-D shape.
- by looking at the shape from different angles, i.e., the front view, the side view and the top view.
- We require three measurements (of sides, angles) to draw a unique triangle.
- A unique quadrilateral can be constructed when:
- Four sides and one diagonal Or
- Two diagonals and three sides Or
- Two adjacent sides and three angles Or
- Three sides and two included angle are given.
- Before constructing the figure, its rough sketch is made using the given measurements.
DIRECT AND INVERSE PROPORTIONS
- Two quantities x and y are in ‘direct proportion’, if they change together in such a manner that the ratio x/y remains constant.
- Two quantities x and y being in direct proportion are written as x y.
- Two quantities x and y are said to be in inverse proportion, if they change together in such a manner that the product xy remains constant.
- If two quantities are in inverse-proportion then an increase in one causes a proportional decrease in the other (and vice-versa).
- Two quantities x and y being in inverse proportion are written as x 1/y.
INTRODUCTION TO GRAPHS
- Graphs represent a wide variety of data in an easy-to-read format for quick and ready comprehension.
- Three useful kinds of graphs are
- Line graphs
- Bar graphs
- Circle graphs
- Line graphs are usually used to show changes in amounts or results over a period of time.
- Bar graphs are generally used to compare amounts or results. They are used to show categories which are different from each other.
- Circle graphs are designed to show relative proportions. The angle of a sector of a circle is made proportional to the percent of an item of the data list to be displayed. They are used to represent data where all categories or groups are specified.
- The position of a point on a plane is fixed by specifying its distances from two fixed lines perpendicular to each other.
- In an ordered pair the x-coordinate is always stated first. The coordinates are always written in this order and thus they are called ordered pair.