Mathematics Formulae
Essential formulae for CBSE Class 12 โข First 2 chapters FREE!
Chapter 1: Relations and Functions
One-One Function
f(xโ) = f(xโ) โน xโ = xโ
Injective function - different inputs give different outputs
๐ก Example:
f(x) = 2x is one-one
Onto Function
For every y in codomain, โ x: f(x) = y
Surjective function - covers entire codomain
๐ก Example:
f: R โ R, f(x) = x is onto
Bijective Function
Both one-one and onto
One-to-one correspondence
๐ก Example:
f(x) = 2x + 3 from R to R
Composite Function
(fog)(x) = f(g(x))
Function of a function
๐ก Example:
If f(x)=xยฒ and g(x)=x+1, then (fog)(x)=(x+1)ยฒ
Identity Function
f(x) = x
Output equals input
๐ก Example:
I(5) = 5
Inverse Function
fโปยน(f(x)) = x and f(fโปยน(x)) = x
Reverses the function
๐ก Example:
If f(x) = 2x, then fโปยน(x) = x/2
Domain of fโปยน
Domain of fโปยน = Range of f
Inverse function domain-range relation
๐ก Example:
Swap domain and range
Range of fโปยน
Range of fโปยน = Domain of f
Inverse function range-domain relation
๐ก Example:
Swap domain and range
Reflexive Relation
aRa for all a
Every element related to itself
๐ก Example:
Equality relation is reflexive
Symmetric Relation
aRb โน bRa
Relation works both ways
๐ก Example:
"Is sibling of" is symmetric
Transitive Relation
aRb and bRc โน aRc
Relation carries through
๐ก Example:
"Greater than" is transitive
Equivalence Relation
Reflexive + Symmetric + Transitive
All three properties together
๐ก Example:
Equality is an equivalence relation
Chapter 2: Inverse Trigonometry
sinโปยนx Domain
-1 โค x โค 1
sinโปยนx Range
[-ฯ/2, ฯ/2]
cosโปยนx Domain
-1 โค x โค 1
cosโปยนx Range
[0, ฯ]
Mathematics Questions
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Ch 1: Relations and Functions
Q. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a - b| is divisible by 2} is an equivalence relation.
Ch 1: Relations and Functions
Q. Check whether the function f: R โ R defined by f(x) = xยฒ is one-one or not.
Ch 1: Relations and Functions
Q. Show that the function f: R โ R defined by f(x) = x/(xยฒ + 1) is neither one-one nor onto.
Ch 1: Relations and Functions
Q. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
Ch 1: Relations and Functions
Q. Find the number of all one-one functions from set A = {1, 2, 3} to itself.
Ch 1: Relations and Functions
Q. The number of bijective functions from set A to itself when A contains 106 elements is:
Ch 1: Relations and Functions
Q. The function f: R โ R defined by f(x) = 3x + 2 is:
Ch 1: Relations and Functions
Q. The domain of f(x) = โ(x-1) is:
Ch 1: Relations and Functions
Q. If f(x) = xยฒ and g(x) = x + 1, then fog(x) equals:
Ch 1: Relations and Functions
Q. Let f: R โ R be defined by f(x) = xยฒ + 1. Pre-images of 17 and -3 are:
Ch 2: Inverse Trigonometry
Q. Prove that: sinโปยน(3/5) + sinโปยน(8/17) = sinโปยน(77/85)
Ch 2: Inverse Trigonometry
Q. Write the principal value of tanโปยน(1) + cosโปยน(-1/2).
Ch 2: Inverse Trigonometry
Q. Find the value of: tanโปยน(โ3) - secโปยน(-2)
Ch 2: Inverse Trigonometry
Q. Prove that: 2tanโปยน(1/2) + tanโปยน(1/7) = tanโปยน(31/17)
Sample Papers - Class 12
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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