X Class Maths Formula`s
We are here to provide you the most important and basic formula`s for Mathematics.
- Real Numbers
Euclid Division Algorithm
For given positive integers a and b, there exist unique integers q and r satisfying a = bq + rRational Form
If a rational number is of the form p/q and q is not of the form 2m5n, then its decimal representation is non-terminating but repeating.Irrational Form
A number is called irrational if it cannot be expressed in the form p/q where p and q are integers - Polynomials
Linear polynomial
A polynomial of degree 1 is called a linear polynomial p(x) = ax + b, a ≠ 0 is a general linear polynomial.
Quadratic polynomial
A polynomial of degree 2 is called a quadratic polynomial p(x) = ax2 + bx + c, a ≠ 0 a,b and c are real numbers is a general quadratic polynomial. polynomial.Cubic roots
If α,β and γ are the zeros of the polynomial ax3 + bx2 + cx + d , then
α + β + γ = - b / a
αβ + βγ + γα = c / a
αβγ = - d / a - Linear equation in two variableLinear Equation
Equation of the form ax + by + c = 0, a ≠ 0, b ≠ 0 where a,b,c are real numbers is called a linear equation in two variables x and y.
Conclusion on solving equation
Compare the ratio Consistency Graphical representation a1 / a2≠b1 / b2Consistent Intersecting lines a1 / a2=b1 / b2=c1 / c2Consistent Coincident lines a1 / a2=b1 / b2≠c1 / c2Inconsistent Parallel lines - Quadratic Equations» General Formula
The general formula of a quadratic equation is ax2 + bx + c = 0, where a,b,c are real numbers.
» Quadratic Formula
The root of the equation ax2 + bx + c = 0, are- b + √b2 - 4ac / 2aand- b - √b2 - 4ac / 2a
» Nature of Roots
A quadratic equation is ax2 + bx + c = 0, has
(i) two distinct real roots if D = b2 - 4ac > 0
(ii) two equal real roots if D = b2 - 4ac = 0
(iii) no real roots if D = b2 - 4ac < 0
» Discriminant
The expression b2 - 4ac is called discriminant of the equation ax2 + bx + c = 0, and is usually denoted By D.
Thus discriminant D = b2 - 4ac. - Arithmetic ProgressionGeneral Formula
The general formula of an A.P is a, a + d, a + 2d, ...... where "a" is called first term and "d", the common difference.
nth term of an A.P
The nth term of an A.P a, a + d, a + 2d .... is given by tn = a + (n - 1)d
Sum of first n terms of an A.P
The sum of Sn of first n terms of an A.P is obtained by the following formulae :-
Sn =n[a + l] / 2where l = a + (n-1)d
and Sn =n[2a + (n-1)d] / 2where a is the firs term, d is the common difference, l is the nth term and n is the number of terms.
Sum of series
If Sn is the sum of first n terms of an A.P, then its nth term is given by tn = Sn - Sn - 1 - TrianglesAbout Similarity
Two figures having the same shape but not necessarily the same size are called similar figure.
About Congruency
All the congruent figures are similar but the converse is not true.
Similar Triangles
Two triangles are said to be similar , if their corresponding angles are equal and their corresponding sides are proportional.
Similarity Criteria
The triangle are said to be similar if their corresponding angles and corresponding sides in proportion.
(i) ∠A = ∠P ; ∠B = ∠Q ; ∠C = ∠R
(ii)AB / PQ=BC / QR=AC / PR
Then ▲ ABC ~ ▲ PQR
A B C P Q R
5.Congruent Criteria
The triangles are said to be congruent . If their corresponding angles and corresponding sides are equal.
(i) ∠A = ∠P ; ∠B = ∠Q ; ∠C = ∠R
(ii) AB = PQ ; BC = QR ; AC = PR
Then ▲ ABC ≅ ▲ PQR
A B C P Q R Cordinate geometryDistance formula
The distance between the points P(x1,y1) and Q(x2,y2) is
PQ = √( X2 - X1 )2 + ( Y2 - Y1 )2
About axis & coordinate
x-coordinate of a point is called abscissa and y-coordinate is called the ordinate.
Mid point
Mid point of a line segment joining the points (x1,y1) and (x2,y2) is given by
(x1 + x2) / 2and(y1 + y2) / 2
Collinear Points
Points A, B and C are collinear if they lie on the same straight line
B lie between A and C if AB + BC = AC
Parallelogram properties
(i) Opposite sides are equal
(ii) diagonals bisect each other and they are not equal.
Rectangle properties
(i) diagonals are equal and bisect each other.
Square properties
(i) diagonals are equal.
(ii) sides are equal.
Rhombus properties
(i) diagonals are not equal and bisect each other at Right angle.
(ii) opposite angles are equal.Introduction to TrigonometryRight triangle concept
(i) In Right triangle, the side in front of 900 is called Hypotenuse.
(ii) In Right triangle, the side in front of θ is called Perpendicular.
(ii) Now the remaining side is called Base.Some Application to TrigonometryAngle of elevation
If the object is above the horizontal level of eyes, then the angle formed by the object to the eye of observer is called the angle of elevation.
Angle of depression
If the object is below the horizontal level of eyes, then the observer has to viewed downward to view the object through an angle which is called angle of depression.CirclesSecant
(i)If a line intersects a circle at two points, called a secant of the circle.
Tangent
A line which intersect a circle in only one point is called a tangent to a circle.
Point of Contact
The common point of the tangent and the circle is called the point of contact.
Theorem 1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Theorem 2
The length of tangents drawn from an external point to a circle are equal.
Note
1. Only one tangent can be drawn from a point of contact of the circle.
2. No tangent can pass through the point lying inside a circle.
3. Exactly two tangents can be drawn on the circle from a point lying outside a circle.ConstructionIn this chapter we will study
(i) to divide a given line segment in the given ratio.
(ii) to draw a triangle similar to the given triangle.
(iii) to draw tangents from an external point to the circle.Area related to circleSemi-circle
(i) Area of semi-circle =πr2 / 4
(ii) Perimeter of semi-circle = πr + 2r
Quadrant of a circle
(i) Area of quadrant of circle =πr2 / 4
(ii) Perimeter of semi-circle = 2r +πr / 2
Sector
(i) Length of the arc, L =πrθ / 180
(ii) Area of sector, A =lr / 2
(iii) Perimeter of sector = l + 2rSurface Area volumeCuboid
(i) Volume = l x b x h
(ii) Total Surface area = 2(lb + bh + hl)
(iii) Lateral/Curved Surface area = 2h(l + b)
(iv) Diagonal of the cuboid = √l2 + b2 + h2
Cube
(i) Volume = a3
(ii) Total Surface area = 6a2
(iii) Lateral/Curved Surface area = 4a2
(iv) Diagonal of the cuboid = a√3
Right Circular cylinder
(i) Volume = πr2h
(ii) Total Surface area = 2πr(h + r)
(iii) Lateral/Curved Surface area = 2πrh
Right Circular cone
(i) Volume =1 / 3πr2h
(ii) Total Surface area = πr(l + r)
(iii) Lateral/Curved Surface area = πrl where l = √r2 + h2
Sphere
(i) Volume =4 / 3πr3
(ii) Total Surface area = 4πr2
(iii) Lateral/Curved Surface area = 4πr2
HemiSphere
(i) Volume =2 / 3πr3
(ii) Total Surface area = 3πr2
(iii) Lateral/Curved Surface area = 2πr2StaticsEmprical Formula for Mode
Mode = 3 Median - 2 Mean
Ogives graphs
Representation of cumulative frequency distribution graphically is called cumulative frequency curve or ogive which are of two types viz. less than and more than type.
Gruouping of gives graphs
The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of two ogives.ProbabilityEvent
Each possible ocutcome is called an event and is denoted by E. The number of event is denoted by n(E)
Sample space
A set of outcomes obtained by performing an experiment is called sample space. Then number of elements in sample space denoted by n (S)
Probability of occurence of an event E,
P(E) =n(E) / n(S)
0 ≤ P(E) ≤ 1
Probability of non-occurence of event E,
P(Ē) = 1 - P(E)
Formula
P(E) =Number of outcomes favourable to E / Number of all possible outcomes to the Experiment
Happenings Events
If p is the probability of an events E, then q, the probability of the event not E i.e Ē is given by 1 - p, i.e P(E) + P(Ē) = 1
Complementary Events
E and P(Ē) are called complementary events.
Distribution
A pack of cards consists of 52 cards which are divided into 4 suits of 13 cards each, spades(♠),hearts(♥),diamonds(♦) and clubs . Clubs and spades are of black colour while hearts and diamonds are of red colour.
Concept
Kings, queens and jacks are called face cards. Thus there are 12 face cards in a deck of cards.
