CLASS 10 - MATHS
FORMULA BOOK
FOR CBSE BOARD
REAL NUMBERS If , and are the zeroes of a cubic polynomial
Euclid’s Division Lemma ax3 + bx 2 + cx + d, then
a = b × q + r, 0 r < b. b c
For any two positive integers a and b + = , + + = and
a a
HCF(a, b) × LCM(a, b) = a × b
d
For three numbers a, b & c =
a
(i) HCF (a, b, c) LCM(a, b, c) a b c where a,
SOME USEFUL IDENTITIES
b, c are positive integers.
(i) (x + y)2 = x2 + y2 + 2xy
a b c HCF( a , b , c )
(ii) LCM (a, b, c) = (ii) (x – y)2 = x2 + y2 – 2xy
HCF( a , b) HCF( b , c ) HCF( a , c )
(iii) (x + y) (x – y) = x2 – y2
a b c LCM( a , b , c ) (iv) (x + a) (x + b) = x2 + (a + b)x + ab
(iii) HCF (a, b, c) =
LCM( a , b) LCM( b , c ) LCM( a , c ) (v) (x + y + z)2 = x2 + y 2 + z2 + 2xy + 2yz + 2zx
(vi) (x + y)3 = x3 + y3 + 3xy (x + y)
POLYNOMIALS = x 3 + 3x2y + 3xy 2 + y 3
Remainder Theorem : Let p(x) be any (vii) (x – y) = x – y 3 – 3xy(x – y)
3 3
polynomial of degree greater than or equal = x 3 – 3x2y + 3xy 2 – y 3
to 1 and a be any real number, if p(x) be divided (viii) x + y + z – 3xyz = (x + y + z) (x2 + y2 + z2
3 3 3
by linear polynomial (x – a), then the remainder – xy – yz – zx)
is equal to p(a). If x + y + z = 0, then x3 + y3 + z3 = 3xyz
Factor Theorem : If p(x) is a polynomial of (ix) x3 + y3 = (x + y)(x 2 – xy + y 2)
degree greater than or equal to 1 and a be (x) x3 – y3 = (x – y)(x 2 + xy + y 2)
any real number such that
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
(i) if p(a) = 0 then (x – a) is a factor of p(x) and
(ii) if (x – a) is a factor of p(x), then p(a) = 0 If a pair of linear equations a1x + b1y + c1 = 0
Division Algorithm for Polynomial : p (x) = and a2x + b2y + c2 = 0 represents :
q (x) × g (x) + r (x), where r (x) = 0 or a1 b1
degree of r (x) < degree of q (x). (i) Intersecting lines then
a2 b2
If and are the zeroes of a quadratic (one solution)
polynomial ax2 + bx + c, then
a1 b1 c1
b c (ii) Parallel lines, then
+ = and = a2 b2 c2
a a (no solution)
, AB AC A
a b c (ii)
(iii) Coincident lines, then 1 1 1 AD AE
a2 b2 c2
(infinitely many solutions)
D E
AB AC
QUADRATIC EQUATIONS (iii) .
DB EC
Roots of the quadratic equation B C
ax2 + bx + c = 0, a, b, c R and a 0 is given
by
AAA Similarity Criterion : If two triangles
are equiangular, then they are similar.
b b2 4 ac AA Similarity Criterion : If two angles of one
x= ; D b2 4 ac triangle are respectively equal to two angles
2a
of another triangle, then the two triangles are
Nature of Roots similar.
(i) If D > 0, distinct and unequal real roots. SSS Similarity Criterion : If the corresponding
(ii) If D is a perfect square, the equation has sides of two triangles are proportional, then
unequal-rational roots. they are similar.
(iii) If D = 0, real and equal roots and each SAS Similarity Criterion : If in two triangles,
b one pair of corresponding sides are
root is .
2a proportional and the included angles are equal,
(iv) If D < 0, no real roots. then the two triangles are similar.
Formation of a quadratic equation Area of Similar Triangles : The ratio of the
x2 – (sum of roots) x + product of roots = 0 areas of two similar triangles is equal to the
ratio of the squares of their corresponding
ARITHMETIC PROGRESSIONS sides, altitudes, medians, angle bisector
The nth term an of an A.P. is segments.
an = a + (n – 1) d; The Pythagoras Theorem :
A
a = first term In a right triangle, the
n = number of terms square of the hypotenuse
d = common difference is equal to the sum of the
The sum to n terms of an A.P. square of other two sides.
In the given figure,
n B C
Sn = {2a + (n – 1)d} AC2 = AB2 + BC 2.
2
n CO-ORDINATE GEOMETRY
Also, Sn = {a + l}
2 If x y, then (x, y) (y, x).
TRIANGLES If (x, y) = (y, x), then x = y.
Distance between the points A(x 1 , y 1 ),
Basic Proportionality Theorem (B.P.T.) (Thales
Theorem) : In a triangle, a line drawn paral- B(x2, y2) is AB = ( x2 x1 )2 ( y2 y1 )2 .
lel to one side, to intersect the other sides in If A, B and C are collinear, then AB + BC = AC
distinct points, divides the two sides in the same or AC + CB = AB or BA + AC = BC.
ratio. In ABC, if DE||BC. The points which divides the line segment
AD AE joining the points A(x1, y1), B(x2, y2) in the ratio
Then (i) l:m
DB EC
FORMULA BOOK
FOR CBSE BOARD
REAL NUMBERS If , and are the zeroes of a cubic polynomial
Euclid’s Division Lemma ax3 + bx 2 + cx + d, then
a = b × q + r, 0 r < b. b c
For any two positive integers a and b + = , + + = and
a a
HCF(a, b) × LCM(a, b) = a × b
d
For three numbers a, b & c =
a
(i) HCF (a, b, c) LCM(a, b, c) a b c where a,
SOME USEFUL IDENTITIES
b, c are positive integers.
(i) (x + y)2 = x2 + y2 + 2xy
a b c HCF( a , b , c )
(ii) LCM (a, b, c) = (ii) (x – y)2 = x2 + y2 – 2xy
HCF( a , b) HCF( b , c ) HCF( a , c )
(iii) (x + y) (x – y) = x2 – y2
a b c LCM( a , b , c ) (iv) (x + a) (x + b) = x2 + (a + b)x + ab
(iii) HCF (a, b, c) =
LCM( a , b) LCM( b , c ) LCM( a , c ) (v) (x + y + z)2 = x2 + y 2 + z2 + 2xy + 2yz + 2zx
(vi) (x + y)3 = x3 + y3 + 3xy (x + y)
POLYNOMIALS = x 3 + 3x2y + 3xy 2 + y 3
Remainder Theorem : Let p(x) be any (vii) (x – y) = x – y 3 – 3xy(x – y)
3 3
polynomial of degree greater than or equal = x 3 – 3x2y + 3xy 2 – y 3
to 1 and a be any real number, if p(x) be divided (viii) x + y + z – 3xyz = (x + y + z) (x2 + y2 + z2
3 3 3
by linear polynomial (x – a), then the remainder – xy – yz – zx)
is equal to p(a). If x + y + z = 0, then x3 + y3 + z3 = 3xyz
Factor Theorem : If p(x) is a polynomial of (ix) x3 + y3 = (x + y)(x 2 – xy + y 2)
degree greater than or equal to 1 and a be (x) x3 – y3 = (x – y)(x 2 + xy + y 2)
any real number such that
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
(i) if p(a) = 0 then (x – a) is a factor of p(x) and
(ii) if (x – a) is a factor of p(x), then p(a) = 0 If a pair of linear equations a1x + b1y + c1 = 0
Division Algorithm for Polynomial : p (x) = and a2x + b2y + c2 = 0 represents :
q (x) × g (x) + r (x), where r (x) = 0 or a1 b1
degree of r (x) < degree of q (x). (i) Intersecting lines then
a2 b2
If and are the zeroes of a quadratic (one solution)
polynomial ax2 + bx + c, then
a1 b1 c1
b c (ii) Parallel lines, then
+ = and = a2 b2 c2
a a (no solution)
, AB AC A
a b c (ii)
(iii) Coincident lines, then 1 1 1 AD AE
a2 b2 c2
(infinitely many solutions)
D E
AB AC
QUADRATIC EQUATIONS (iii) .
DB EC
Roots of the quadratic equation B C
ax2 + bx + c = 0, a, b, c R and a 0 is given
by
AAA Similarity Criterion : If two triangles
are equiangular, then they are similar.
b b2 4 ac AA Similarity Criterion : If two angles of one
x= ; D b2 4 ac triangle are respectively equal to two angles
2a
of another triangle, then the two triangles are
Nature of Roots similar.
(i) If D > 0, distinct and unequal real roots. SSS Similarity Criterion : If the corresponding
(ii) If D is a perfect square, the equation has sides of two triangles are proportional, then
unequal-rational roots. they are similar.
(iii) If D = 0, real and equal roots and each SAS Similarity Criterion : If in two triangles,
b one pair of corresponding sides are
root is .
2a proportional and the included angles are equal,
(iv) If D < 0, no real roots. then the two triangles are similar.
Formation of a quadratic equation Area of Similar Triangles : The ratio of the
x2 – (sum of roots) x + product of roots = 0 areas of two similar triangles is equal to the
ratio of the squares of their corresponding
ARITHMETIC PROGRESSIONS sides, altitudes, medians, angle bisector
The nth term an of an A.P. is segments.
an = a + (n – 1) d; The Pythagoras Theorem :
A
a = first term In a right triangle, the
n = number of terms square of the hypotenuse
d = common difference is equal to the sum of the
The sum to n terms of an A.P. square of other two sides.
In the given figure,
n B C
Sn = {2a + (n – 1)d} AC2 = AB2 + BC 2.
2
n CO-ORDINATE GEOMETRY
Also, Sn = {a + l}
2 If x y, then (x, y) (y, x).
TRIANGLES If (x, y) = (y, x), then x = y.
Distance between the points A(x 1 , y 1 ),
Basic Proportionality Theorem (B.P.T.) (Thales
Theorem) : In a triangle, a line drawn paral- B(x2, y2) is AB = ( x2 x1 )2 ( y2 y1 )2 .
lel to one side, to intersect the other sides in If A, B and C are collinear, then AB + BC = AC
distinct points, divides the two sides in the same or AC + CB = AB or BA + AC = BC.
ratio. In ABC, if DE||BC. The points which divides the line segment
AD AE joining the points A(x1, y1), B(x2, y2) in the ratio
Then (i) l:m
DB EC
