Mathematics Formulae
Essential formulae for CBSE Class 10 โข First 2 chapters FREE!
Chapter 1: Real Numbers
Euclid's Division Lemma
a = bq + r, where 0 โค r < b
For any two positive integers a and b
๐ก Example:
To find HCF of 135 and 225: 225 = 135ร1 + 90
HCF ร LCM Formula
HCF(a,b) ร LCM(a,b) = a ร b
Product of two numbers equals product of their HCF and LCM
๐ก Example:
For 12 and 18: HCF=6, LCM=36, 6ร36 = 12ร18 = 216
Prime Factorization
n = pโ^aโ ร pโ^aโ ร ... ร pโ^aโ
Any composite number expressed as product of primes
๐ก Example:
360 = 2ยณ ร 3ยฒ ร 5
Euclid's Division Lemma
a = bq + r, where 0 โค r < b
For any two positive integers a and b, there exist unique integers q and r
๐ก Example:
To find HCF of 135 and 225: 225 = 135ร1 + 90
HCF ร LCM Formula
HCF(a,b) ร LCM(a,b) = a ร b
Product of two numbers equals product of their HCF and LCM
๐ก Example:
For 12 and 18: HCF=6, LCM=36, so 6ร36 = 12ร18 = 216
Prime Factorization
n = pโ^aโ ร pโ^aโ ร ... ร pโ^aโ
Any composite number can be expressed as product of prime powers
๐ก Example:
360 = 2ยณ ร 3ยฒ ร 5
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes uniquely
Unique factorization theorem - order doesn't matter
๐ก Example:
12 = 2ยฒ ร 3 (only one way)
HCF by Euclid's Algorithm
HCF(a,b) = HCF(b,r) where a = bq + r
Keep dividing until remainder is 0
๐ก Example:
HCF(252, 105): 252 = 105ร2 + 42, then 105 = 42ร2 + 21, then 42 = 21ร2 + 0, so HCF = 21
LCM Formula
LCM(a,b) = (a ร b) / HCF(a,b)
LCM can be found using HCF
๐ก Example:
LCM(12,18) = (12ร18)/6 = 36
Divisibility Test for 2
A number is divisible by 2 if last digit is even
Check unit's place
๐ก Example:
248 is divisible by 2 (ends in 8)
Divisibility Test for 3
A number is divisible by 3 if sum of digits is divisible by 3
Add all digits
๐ก Example:
123: 1+2+3 = 6, divisible by 3
Divisibility Test for 5
A number is divisible by 5 if last digit is 0 or 5
Check unit's place
๐ก Example:
125, 340 are divisible by 5
Chapter 2: Polynomials
Sum of Zeros (Quadratic)
ฮฑ + ฮฒ = -b/a
Product of Zeros (Quadratic)
ฮฑฮฒ = c/a
Division Algorithm
p(x) = g(x) ร q(x) + r(x)
Sum of Zeros (Quadratic)
ฮฑ + ฮฒ = -b/a
Mathematics Questions
Practice with CBSE pattern questions โข Free chapters unlocked!
Ch 1: Real Numbers
Q. Prove that โ2 is an irrational number.
Ch 1: Real Numbers
Q. Find the HCF of 96 and 404 by prime factorization method.
Ch 1: Real Numbers
Q. Express 3825 as a product of its prime factors.
Ch 1: Real Numbers
Q. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.
Ch 1: Real Numbers
Q. Find the LCM and HCF of 12, 15 and 21 by prime factorization method.
Ch 1: Real Numbers
Q. The HCF of 95 and 152 is:
Ch 1: Real Numbers
Q. The decimal expansion of 17/8 is:
Ch 1: Real Numbers
Q. The LCM of two numbers is 182 and their HCF is 13. If one number is 26, the other number is:
Ch 1: Real Numbers
Q. If two positive integers p and q are written as p = aยฒbยณ and q = aยณb where a and b are prime numbers, then HCF(p,q) is:
Ch 1: Real Numbers
Q. The product of a non-zero rational and an irrational number is:
Ch 2: Polynomials
Q. If ฮฑ and ฮฒ are the zeros of the polynomial xยฒ - 5x + 6, find the value of ฮฑยฒ + ฮฒยฒ.
Ch 2: Polynomials
Q. Divide the polynomial 3xโด + 5xยณ - 7xยฒ + 2x + 2 by xยฒ + 3x + 1.
Ch 2: Polynomials
Q. Find a quadratic polynomial whose zeros are 5 and 3.
Ch 2: Polynomials
Q. If one zero of the polynomial 3xยฒ + 8x + k is the reciprocal of the other, find the value of k.
Sample Papers - Class 10
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CBSE Sample Paper 2026 - Mathematics Set 1
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Attempt Now โCBSE Sample Paper 2026 - Mathematics Set 2
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Attempt Now โCBSE Sample Paper 2026 - Mathematics Set 3
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CBSE Sample Paper 2026 - Mathematics Set 4
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CBSE Sample Paper 2026 - Mathematics Set 5
PREMIUMComplete CBSE Board pattern paper with detailed solutions for comprehensive practice
CBSE Sample Paper 2026 - Mathematics Set 6
PREMIUMComplete CBSE Board pattern paper with detailed solutions for comprehensive practice