# Class X Maths Formulas

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Oct 19 • Board Sample Papers • 181177 Views • 635 Comments on Class X Maths Formulas

Introduction
Here is a quick review of class X maths formulas chapter wise for all those students of class X. It will help the students to memories the formulas very quickly. The ppt given below will help the students to remember the formulas quickly.

• Real number
• Polynomials
• Remainder theorem
• Linear equations in two variables
• Ratio and proportion
• Similarity
• Distance and section formulae
• Equation of a line
• Circle and tangents
• Circumference and area of a circle
• Solids
• Trigonometric identities
• Graphical representation
• Measures of central tendency
• probability

Class X Maths Formulas

Real number

Euclid’s division algorithm (lemma): Given positive integers ‘a’ and ‘b’, there exists unique integers q and r such that a= b.q+r, where 0 <=r< b ( where a= dividend, b= divisor, q= quotient, and r= remainder.

Polynomials

In step 1 : Factorize the given polynomials,

a)  Either by splitting the terms, ( OR )

b)  Using these identities :

1. (a+b)^2 = a*a + 2ab+b*b
2. (a-b)^2 = a*a – 2ab +b*b
3. a*a – b*b = (a+b)( a-b)
4. a^4 – b^4 = (a^2)^2 –( b^2)^2= (a*a+ b*b)(a*a – b*b) = (a*a + b*b)(a-b)(a+b)
5. (a+b)^3 = a^3 + b^3 + 3ab(a+b)
6. a^3 + b^3 =( a+b)(a*a +ab + b*b)
7. (a-b)^3 = a^3- b^3 – 3ab (a-b)
8. a^3- b^3= (a-b) (a*a+ ab +b*b)
9. (a+b+c)^2 = a*a + b*b +c*c + 2ab+2bc+2ca
10. a^3+b^3+c^3- 3abc = (a+b+c)(a*a + b*b + c*c –ab – bc- ac)

Trial and error method

In step 2 : take a product of ‘ Common terms’ as their HCF.

In step3 : Take the product of All the terms, Omit the HCF value which gives you the value of LCM.

Product of  LCM* HCF=Product of the two polynomials

Note: If cubical expression is given, it may be factorized by using ‘trial and error’ method.

Remainder theorem

If (x-2) is a factor of the given expression, then take x-2 = 0 , therefore x = 2 , then substitute this value in p(x) = 5 x*x + 3 x-6

P(2) : 5(2*2) + 3(2)-6 =0 (here taking = 0 is very important. If not taken, answer can’t be found)

If (x-2) leaves a remainder of 4

P(2) : 5(2*2) + 3 (2)-6 =4 ( Here taking = 4 is very important. If not taken, answer can’t be found)

Linear equations in two variables

If pair of linear equation is : a1 + b1y +c1 =0  and a2x + b2 y + c2=0

Then nature of roots/zeroes/solutions :

i.  If a1/a2 is not equal to b1/b2 then, system has unique solution, is consistent OR graph is two intersecting lines

ii.  If a1/a2 = b1/b2 is not equal to c1/c2 , then system has no solution, is inconsistent OR graph is parallel lines.

iii.  If a1/a2=b1/b2=c1/c2, then system has infinite solution, is consistent OR graph are coincident lines.

Note: To find the value of ‘x’ you may adopt either ‘splitting the middle term’ or ‘formula method’.

X=( -b +-(D)^.5)/2a (where D= b*b – 4ac) Hence x= (-b+- (b*b- 4ac)^.5)/2a

• Sum of the roots = -b/a & Product of roots= c/a
• If roots of an equation are given, then :

Quadratic equation : x*x – (Sum of roots).x + (product of the roots) =0

If Discriminant > 0, then the roots are Real & unequal or unique, lines are intersecting.

Discriminant = 0, then the roots are real & equal , lines are coincident.

Discriminant< 0 , then the roots are imaginary (not real), parallel lines.

Ratio & proportion

•  Duplicate ratio of a : b is a*a : b*b (Incase of Sub-duplicate ratio you have to take ‘Square root’)
• Triplicate ratio of a: b is a^3 : b^3 (Incase of Sub-triplicate ratio you have to take ‘cube root’)
• Proportion a:b =c:d, continued proportion a :b = b : c, (Middle value is repeated)
• Product of ‘Means’( Middle values)= Product of ‘Extremes’ (Either end values)
• If a/b= c/d is given then Componendo & dividend is (a+b)/ (a-b) = (c+d) / (c-d)

Note : “Where to take “K” method ? “ You may adopt it in the following situations.

If a/b = c/d = e/f are given , then you may assume as a/b= c/d= e/f =k

Therefore a = b.k, c= d.k, e= f.k, then substitute the values of ‘a’ ‘b’ and c’c’ in the given problem.

Incase of continued proportion : a/b = b/c = k , hence , a=bk, b= ck therefore putting the value of b we can get a= c k*k & b= ck.( putting these values equation can be solved)

Similarity

•  If two triangles are similar then, ratio of their sides are equal.

i.e if triangle ABC~ triangle PQR then AB/PQ = BC/QR = AC/PR

• If triangle ABC ~ triangle PQR then (Area of triangle ABC)/ Area of triangle PQR) = (side*side) /(side*side) = ( AB*AB) /( PQ*PQ) =( BC*BC)/( QR*QR) = (AC*AC)/ (PR*PR)

Distance and section formulae

• Distance =(( x2- x1 )^2 + y2-y1)^2))^.5  ( The same formula is to be used to find the length of line segment, sides of a triangle , square , rectangle, parallelogram etc.)
• To prove co-linearity of the given three points A,B and C, you have to find length of  AB, BC, AC then use the condition AB + BC = AC .OR use this condition to solve the question easily :

Area of triangle formed by these points : 0.5 [x1 ( y2-y3)+ x2(y3 – y1) + x3(y1- y2)]=0

• Section formula : point (x,y)=[ ( m1x2 + m2x1)/( m1 + m2) , (m1y2 + m2y1)/ (m1 + m2)]
• Mid-point =[( x1 + x2 )/2 ,( y1 +y2 )/2 ]
• Centroid of a triangle=[( x1+x2+x3)/3 ,( y1+y2+y3)/3]
• If line is trisected then take m:n ratio as 1:2 and find co- ordinates of point p(x,y).

Equation of a line

• If two points are given, then Slope (m) = (y2-y1)/(x2-x1)
• If a point , and slope are given , then Slope (m)= (y-y1)/(x-x1)
• If two lines are ‘Parallel’ to each other then their slopes are equal i.e m1=m2.
• If two lines are ‘Perpendicular’ to each other then product of their slopes is -1 i.e m1 *m2 = -1
• Depending upon the question You may have to use equation of straight line as

a)      Y=mx + c, where ‘c’ is the y- intercept. OR    b) (y-y1)= m.( x-x1)

Circles and Tangents

• Equal chords of a circle are equidistant from the centre .( Chord Property)
• The perpendicular drawn from the centre of a circle, bisects the chord of the circle. (Chord Property)
• The angle subtended at the centre by an arc= Double the angle at any part of the circumference of the circle .(Angle Property)
• Angles subtended by the same arc in the same segment are equal.(Angle property)
• To a circle, if a tangent is drawn and a chord is drawn from the point of contact , then angle made between the chord and the tangent = angle made in the alternate segment(Tangent property)
• The sum of opposite angles of a cyclic quadrilateral is always 180 degree.

Circumference and area of a circle

• Area of a circle = pi(r*r)
• Perimeter of a circle = 2*pi*r
• Area of sector = theta/360 ( pi*r*r)
• Length of an arc = theta / 360 (2*pi*r)
• Area of ring = pi (R*R- r*r)
• Distance moved by a wheel in one revolution = Circumference of the wheel.
• Number of revolutions=Total distance moved/Circumference of the wheel.

Note: While solving ‘Mensuration’ problems, take care of the following.

1. If diameter of a circle is given , then find the radius first

2. Check the units of the entire data. If the units are different , then convert them to the same units.

Solids

Cylinder  : Volume of a cylinder= pi* r*r *h

Curved surface area = 2*pi*r*h

Total surface area = 2*pi*r*h + 2*pi*r*r = 2*pi*r ( h+r)

Volume of hollow cylinder = pi * R*R*h- pi*r*r*h= pi(R*R-r*r) h

TSA of hollow cylinder = Outer CSA + Inner CSA+ 2. Area of ring .

Cone: Volume  of a cone =1/3  pi*r*r*h

CSA of a cone = pi*r*l( here ‘l’ refers to ‘slant height’) [where l= [(h*h + r*r)]^.5

TSA of a cone = pi*r*l + pi*r*r = pi*r (l+r)

Sphere  :  Surface area of a sphere = 4*pi*r*r( Incase of sphere , CSA=TSA i.e they are same)

Volume of hemisphere = 2/3 pi*r*r*r [take half the volume of a sphere]

CSA of hemisphere = 2*pi*r*r [Take half the SA of a sphere]

TSA of hemisphere= 2*pi*r*r+pi*r*r = 3*pi*r*r

Volume of a sphere = 4/3 pi*r*r*r

Volume of spherical shell= Outer volume-Inner volume = 4/3*pi*(R^3-r^3)

Trigonometric identities

Wherever ‘square’ appears think of using the identities

•  sin^2(x) + cos^2(x) = 1
• tan^2(x) + 1 = sec^2(x)
• cot^2(x) + 1 = csc^2(x)
• tan(x y) = (tan x tan y) / (1 tan x tan y)
• sin(2x) = 2 sin x cos x
• cos(2x) = cos^2(x) – sin^2(x) = 2 cos^2(x) – 1 = 1 – 2 sin^2(x)
• tan(2x) = 2 tan(x) / (1 – tan^2(x))
• sin^2(x) = 1/2 – 1/2 cos(2x)
• cos^2(x) = 1/2 + 1/2 cos(2x)
• sin x – sin y = 2 sin( (x – y)/2 ) cos( (x + y)/2 )
• cos x – cos y = -2 sin( (x – y)/2 ) sin( (x + y)/2 )

Measures of Central tendency

For un-grouped data

• Arithmatic Mean = Sum of observations/ no. of observations
• Mode = the most frequently occurred value of the raw data
• To find the Median first of all arrange the data in ‘Ascending’ or ‘Descending’ order, then

Median= N+1)/2 term value of the given data, in case of the data is having odd no of observations.

Median= (N/2) +(N+1)/2]/2 term value of the given data, in case of the datais having even number of observation.

Probability

Probability of an event : P(event)= Number of favourable outcomes/ Total number of outcomes

If probability of happening an event is x then probability of not happening that event is (1-x).

(Spades in black colour ) having A,2,3,4,5,6,7,8,9,10,J,K and Q total 13 cards

(Clubs in black colour) having A,2,3,4,5,6,7,8,9,10,J,K and Q total 13 cards.

(Hearts in red colour) having A,2,3,4,5,6,7,8,9,10,J,K and Q total 13 cards.

(Diamond in red colour) having A,2,3,4,5,6,7,8,9,10,J,K and Q total 13 cards.

• Jack , king and queen are known as face cards.
• If one coin is tossed the total number of outcomes are 2 either a head or a tail.
• If two coins are tossed the total number of outcomes are 2*2 = 4
• If three coins are tossed the total number of outcomes are 2*2*2 = 8
• Similarly for dice, in a single roll total number of outcomes are 6
• If two Dices are rolled , total number of outcomes are 6*6 = 36.

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