Maths Formulas For Class 11

Table of Contents

Maths Formulas For Class 11

Are you looking Maths formulas for class 11 on a single page then you are the right place to get all the Maths formulas for class 11 in a single page according to the chapter names? If you are studying in class 11 and want to prepare an engineering entrance exam then maths formulas are a vital role. I would like to suggest you remember all the class 11 maths formulas for future support. These maths formulas are most important according to the class 11 exam. NCERT Maths formulas for class 11 are also most important for a revision point of view. You can easily prepare the CBSE class 11 maths exam paper through maths formulas of class 11.

Class 11 Sets Formulas

Intersection of Associativity Sets: A∩(B∩C) = (A∩B)∩C
Distributive: A∪(B∩C) = (A∪B)∩(A∪C) and A∩(B∪C) = (A∩B)∪(A∩C)
Idempotency: A∪A = A, A∪A = A
Domination: A∩∅ = ∅, A∪I = I
Identity: A∪∅ = A, A∩I = A
Complement = A’= {X∈I|X∉A}
Complement of intersection and Union: A∪A’ = I, A∩A’ = ∅
De Morgan’s Law: (A∪B)’ = A’∩B’, (A∩B)’ = A’∪B’

Class 11 Relations and Functions Formulas

A cartesian product A × B of two sets A and B is given by:
A × B = { \((a,b):a\epsilon A, b\epsilon B\) }
If (a , b) = (x , y); then a = x and b = y
If n(A) = x and n(B) = y, then n(A × B) = xy
A × \(\phi\) = \(\phi\)
The cartesian product: A × B ≠ B × A
A function f from the set A to the set B considers a specific relation type where every element x in the set A has one and only one image in the set B.
A function can be denoted as f: A → B, where f(x) = y
Algebra of functions: If the function f: X → R and g: X → R; we have:
- (i) \((f + g) (x) = f (x) + g(x), x\epsilon X\)
- (ii) \((f – g) (x) = f (x) – g(x), x\epsilon X\)
- (iii) \((f.g)(x) = f (x) .g (x), x\epsilon X\)
- (iv) \((kf) (x) = k ( f (x) ), x\epsilon X\), where k is a real number
- (v)\( \frac{f}{g}(x) = \frac{f(x)}{g(x)}, x\epsilon X, g(x)\neq 0\)

Class 11 Trigonometric Functions Formulas

\(\sin \theta = \frac{Opposite}{Hypotenuse}\)
\(\sec \theta = \frac{Hypotenuse}{Adjacent}\)
\(\cos\theta = \frac{Adjacent}{Hypotenuse}\)
\(\tan \theta =\frac{Opposite}{Adjacent}\)
\(csc \theta = \frac{Hypotenuse}{Opposite}\)
\(cot \theta = \frac{Adjacent}{Opposite}\)

The Reciprocal Identities are given as:

\(cosec\theta =\frac{1}{\sin\theta }\)
\(sec\theta =\frac{1}{\cos\theta }\)
\(cot\theta =\frac{1}{\tan\theta }\)
\(sin\theta =\frac{1}{csc\theta }\)
\(cos\theta =\frac{1}{\sec\theta }\)
\(tan\theta =\frac{1}{cot\theta }\)

Trigonometric Values of Special Angles

Degree	sin	cos	tan	cot	sec	cosec
0∘	0	1	0	Not Defined	1	Not Defined
30∘	\[\frac{1}{2}\]	\[\frac{√3}{2}\]	\[\frac{1}{√3}\]	√3	\[\frac{2}{√3}\]	2
45∘	\[\frac{1}{√2}\]	\[\frac{1}{√2}\]	1	1	√2	√2
60∘	\[\frac{√3}{2}\]	\[\frac{1}{2}\]	√3	\[\frac{1}{√3}\]	2	\[\frac{2}{√3}\]
90∘	1	0	Not Defined	0	Not Defined	1

Product:

\(\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(x-y)}{2}\)
\(\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(x-y)}{2}\)
\(\sin\: x\cdot \sin\:y=\frac{\cos(x+y)-\cos(x-y)}{2}\)

Sum to Product:

\(\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\)
\(\sin\: x-\sin\: y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\)
\(\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\)
\(\cos\: x-\cos\: y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\)<

Pythagorean

Sin²x + Cos²x = 1
1 + tan²x = sec²x
1 + cot²x = cosec²x

Pythagorean in Radical Form

sinx = ∓√1–cos2x
tanx = ∓√sec2x-1
cosx = ∓√1–sin2x

Odd-Even

Sin(-x)=-sin x
cos(-x)=-cos x
tan(-x)=-tan x
cot(-x)=-cot x
sec(-x)=-sec x
cosec(-x)=-cosec x

Ratio or Quotient Identities are given as:

\( Sinθ = Cosθ \times Tanθ \)
\( Cosθ = Sinθ \times Cotθ \)
\( Tanθ = \frac{Sinθ}{Cosθ} \)
\( Cotθ = \frac{Cosθ}{Sinθ} \)

Periodicity

\(sin(x+2\pi )=sin\; x\)
\(cos(x+2\pi )=cos\; x\)
\(tan(x+\pi )=tan\; x\)
\(cot(x+\pi )=cot\; x\)

Co-function:

\(sin(90^{\circ}-x)=cos\; x\)
\(cos(90^{\circ}-x)=sin\; x\)
\(tan(90^{\circ}-x)=cot\; x\)
\(cot(90^{\circ}-x)=tan\; x\)

Sum/Difference:

\( \sin (x + y) = \sin(x) \cos(y) + \cos(x) \sin(y)\)
\(\cos(x + y) = \cos(x) \cos(y) – \sin(x) \sin(y)\)
\(\tan(x+y)=\frac{\tan\: x+\tan\: y}{1-\tan\: x\cdot \tan\: y}\)
\(\sin(x – y) = \sin(x) \cos(y) – \cos(x) \sin(y)\)
\(\cos(x – y) = \cos(x) \cos(y) + \sin(x) \sin(y)\)
\(\tan(x-y)=\frac{\tan\: x – \tan\: y}{1+\tan\: x\cdot tan\: y}\)

Double Angle:

\(\sin(2x) = 2\sin(x).\cos(x)\)
\(\cos(2x) = \cos^{2}(x) – \sin^{2}(x)\)
\(\cos(2x) = 2 \cos^{2}(x) -1\)
\(\cos(2x) = 1 – 2 \sin^{2}(x)\)
\(\tan(2x) = \frac{[2\: \tan(x)]}{[1 -\tan^{2}(x)]}\)

Half Angle:

\(\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}\)
\(\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}\)
\(\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\)

Class 11 Principle of Mathematical Induction Formulas

\( 1 + 2 + 3 + . . . + n = \frac{1}{2} \times n(n+1) \)

If
1. when a statement is true for a natural number n = k, then it will also be true for its successor, n = k + 1;

and
2. the statement is true for n = 1; then the statement will be true for every natural number n.

Class 11 Complex Numbers and Quadratic Equations Formulas

Complex number = \( z=a+bi \)
Conjugate = \( \bar z =a−bi \)
Symmetry = \( −z=−a−bi \)
Equality = a+bi=c+di⇔a=c and b=d
Addition = \( (a+bi)+(c+di)=(a+c)+(b+d)i \)
Subtraction = \( (a+bi)−(c+di)=(a−c)+(b−d)i \)
Multiplication = \( (a+bi)\times(c+di)=(ac−bd)+(ad+bc)i \)
Division = \( \frac{(a+bi)}{(c+di)} = \frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{ac+bd}{c^{2}+d^{2}} + \frac{bc-ad}{c^{2}+d^{2}}\times i \)
Multiplication Conjugates = \( (a+bi)(a+bi)=a^{2}+b^{2} \)

Powers of Complex Numbers

iⁿ= i, if n = 4a+1, i.e. one more than the multiple of 4.

Example – i¹=i; i⁵=i; i⁹=i; i^4a+1;

iⁿ= -1, if n = 4a+2, i.e. two more than the multiple of 4.

Example – i²= -1; i⁶= -1; i¹⁰= -1; i^4a+2;

iⁿ= -i, if n = 4a+3, i.e. Three more than the multiple of 4.

Example – i³= -i; i⁷= -i; i¹¹= -i; i^4a+3;

iⁿ= -1, if n = 4a, i.e. the multiple of 4.

Example – i⁴= 1; i⁸= 1; i¹²= 1; i^4a;

\(x^2+bx+c=0 \;where\; a ≠ 0\)

\(x=-b\pm\frac{\sqrt{b^2-4ac}}{2a}\)

Class 11 Permutations and Combinations Formulas

The number of permutations of n different things taken r at a time is given by \({}^{n}\textrm{P}{r}\) \(=\frac{n!}{(n-r)!}\) where 0 ≤ r ≤ n
\(n!=1\times 2\times 3\times …\times n\)
\(n!=n\times (n-1)!\)
The number of permutations of n different things taken r at a time with repetition being allowed is given as: n^r
The number of permutations of n objects taken all at a time, where p₁ objects are of one kind, p₂ objects of the second kind, …., p_k objects of kth kind are given as: \(\frac{n!}{p_{1}!\:p_{2}!\:…\:p_{k}!}\)
The number of permutations of n different things taken r at a time is given by \({}^{n}\textrm{C}{r}\) \(=\frac{n!}{r!(n-r)!}\) where 0 ≤ r ≤ n

Class 11 Binomial Theorem Formulas

A Binomial Theorem helps to expand a binomial given for any positive integral n.
\((a+b)^n={}^{n}\textrm{C}_{0}\:a^n+{}^{n}\textrm{C}_{1}\:a^{n-1}.b+{}^{n}\textrm{C}_{2}\:a^{n-2}.b^2+…+{}^{n}\textrm{C}_{n-1}\:a.b^{n-1}+{}^{n}\textrm{C}_{n}\:b^n\)

The general term of an expansion (a + b)ⁿ is \(T_{r+1}={}^{n}\textrm{C}_{r}\:a^{n-r}.b^r\)
In the expansion of (a + b)ⁿ; if n is even, then the middle term is \((\frac{n}{2}+1)^{th}\) term.
In the expansion of (a + b)ⁿ; if n is odd, then the middle terms are \((\frac{n+1}{2})^{th}\) and \((\frac{n+1}{2}+1)^{th}\) terms

Class 11 Sequences and Series Formulas

The nth term a_n of the Arithmetic Progression (A.P) \(a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots \) is given by \({a_n} = a + (n – 1)d\).
The arithmetic mean between a and b is given by \(A.M = \frac{{a + b}}{2}\).
If S_n denotes the sum up to n terms of A.P. \(a,{\text{ }}a + d,{\text{ }}a + 2d, \ldots \) then \({S_n} = \frac{n}{2}(a + l)\) where l stands for the last term, \({S_n} = \frac{n}{2}[2a + (n – 1)d]\)
The sum of n A.M’s between a and b is \( = \frac{{n(a + b)}}{2}\).
The nth term a_n of the geometric progression \(a,{\text{ }}ar,{\text{ }}a{r^2},{\text{ }}a{r^3}, \ldots \) is \({a_n} = a{r^{n – 1}}\).
The geometric mean between a and b is \( G.M = \pm \sqrt {ab} \).
If \({S_n}\) denotes the sum up to n terms of G.P is \({S_n} = \frac{{a(1 – {r^n})}}{{1 – r}};{\text{ }}r \ne 1\), \({S_n} = \frac{{a – rl}}{{1 – r}};{\text{ }}l = a{r^n}\) where \(\left| r \right| < 1\)
The sum S of infinite geometric series is \(S = \frac{a}{{1 – r}};{\text{ }}\left| r \right| < 1\)
The nth term a_n of the harmonic progression is \({a_n} = \frac{1}{{a + (n – 1)d}}\).
The harmonic mean between a and b is \(H.M = \frac{{2ab}}{{a + b}}\).

Class 11 Straight lines Formulas

Slope (m) of the intersecting lines through the points (x₁, y₁) and x₂, y₂) is given by \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\); where x₁ ≠ x₂
An acute angle θ between lines L1 and L2 with slopes m1 and m2 is given by \(tan\:\theta =\left | \frac{m_{2}-m_{1}}{1+m_{1}.m_{2}} \right |\); 1 + m₁.m₂ ≠ 0.
Equation of the line passing through the points (x₁, y₁) and (x₂, y₂) is given by: \(y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})\)
Equation of the line making a and b intercepts on the x- and y-axis respectively is: \(\frac{x}{a}+\frac{y}{b}=1\)
The perpendicular distance d of a line Ax + By + C = 0 from a point (x₁, y₁) is: \(d=\frac{\left | Ax_{1}+By_{1}+C \right |}{\sqrt{A^2+B^2}}\)
The distance between the two parallel lines Ax + By + C₁ and Ax + By + C₂ is given by: d=\(\frac{\left | C_{1}-C_{2} \right |}{\sqrt{A^2+B^2}}\)

Class 11 Conic Sections Formulas

The equation of the circle with the centre point (h, k) and radius r is given by (x – h)² + (y – k)² = r²
The equation of the parabola having focus at (a, 0) where a > 0 and directrix x = – a is given by: y² = 4ax
The equation of an ellipse with foci on the x-axis is \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)
Length of the latus rectum of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is given by: \(\frac{2b^2}{a}\)
The equation of a hyperbola with foci on the x-axis is \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)
Length of the latus rectum of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) is given by: \(\frac{2b^2}{a}\)

Class 11 Introduction to Three Dimensional Geometry Formulas

The distance of two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is:
\(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
The coordinates of a point R that divides the line segment joined by two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally as well as externally in the ratio m : n is given by:
\(\left ( \frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n} \right )\:and\:\left ( \frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n},\frac{mz_2-nz_1}{m-n} \right )\);
The coordinates of the mid-point of a given line segment joined by two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are \(\left ( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2} \right )\)
The coordinates of the centroid of a given triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) are \(\left ( \frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3} \right )\)

Class 11 Limits and Derivatives Formulas

For functions f and g, the following property holds true:
- (i) \(\lim\limits_{x \to a} \left [ f(x)\pm g(x) \right ]= \lim\limits_{x \to a}f(x) \pm \lim\limits_{x \to a}g(x)\)
- (ii) \(\lim\limits_{x \to a} \left [ f(x) .g(x) \right ]= \lim\limits_{x \to a}f(x) . \lim\limits_{x \to a}g(x)\)
- (iii) \(\large \lim\limits_{x \to a} \left [ \frac{f(x)}{g(x)} \right ] = \frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}\)
Standard Limits
- (i) \(\lim\limits_{x \to a}\frac{x^n-a^n}{x-a}= n\:a^{n-1}\)
- (ii) \(\lim\limits_{x \to a}\frac{sin\:x}{x}=1\)
- (iii) \(\lim\limits_{x \to a}\frac{1-cos\:x}{x}=0\)
The derivative of a function f at a holds as: \({f}'(a)=\lim\limits_{x \to a}\frac{f(a+h)-f(a)}{h}\)
The derivative of a function f at a given point x holds as: \({f}'(x)=\frac{\partial f(x)}{\partial x}=\lim\limits_{x \to a}\frac{f(x+h)-f(x)}{h}\)
For the functions u and v, the following holds true:
- (i) \((u\pm v)’=u’\pm v’\)
- (ii) \((uv)’=u’v+uv’\)
- (iii) \(\left ( \frac{u}{v} \right )’=\frac{u’v-uv’}{v^2}\)
Standard Derivatives
- (i) \(\frac{\partial}{\partial x}(x^n)=nx^{n-1}\)
- (ii) \(\frac{\partial}{\partial x}(sin\:x)=cos\:x\)
- (iii) \(\frac{\partial}{\partial x}(cos\:x)=-sin\:x\)

Class 11 Statistics Formulas

Mean Deviation for the ungrouped data:
- (i) \(M.D.(\bar x)=\frac{\sum \left | x_i-\bar x \right |}{n}\)
- (ii) \(M.D.(M)=\frac{\sum \left | x_i-M \right |}{n}\)
Mean Deviation for the grouped data:
- (i) \(M.D.(\bar x)=\frac{\sum f_i|x_i-\bar x|}{N}\)
- (ii) \(M.D.(M)=\frac{\sum f_i|x_i-M|}{N}\)
Variance and Standard Deviation for the ungrouped data:
- (i) \(\sigma ^2=\frac{1}{N}\sum (x_i-\bar x)^2\)
- (ii) \(\sigma=\sqrt{\frac{1}{N}\sum (x_i-\bar x)^2}\)
Variance and Standard Deviation of a frequency distribution (discrete):
- (i) \(\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2\)
- (ii) \(\sigma=\sqrt{\frac{1}{N}\sum f_i(x_i-\bar x)^2}\)
Variance and Standard Deviation of a frequency distribution (continuous):
- (i) \(\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2\)
- (ii) \(\sigma=\frac{1}{N}\sqrt{N\sum f_ix_i^2-(\sum f_ix_i)^2}\)
Coefficient of variation (C.V.) = \(\frac{\sigma}{\bar x}\times 100\) ; where \(\bar x\neq 0\)

Class 11 Probability Formulas

\(P(A) = \frac{Number\; of\; favorable\; outcome}{Total\; number \;of\; favorable \;outcomes}\)
Probability Range: 0 ≤ P(A) ≤ 1
Rule of Complementary Events: P(A^C) + P(A) = 1
Rule of Addition: P(A∪B) = P(A) + P(B) – P(A∩B)
Disjoint Events: Events A and B are disjoint if: P(A∩B) = 0
Conditional Probability: P(A | B) = P(A∩B) / P(B)
Bayes Formula: P(A | B) = P(B | A) ⋅ P(A) / P(B)
Independent Events: Events A and B are independent if: P(A∩B) = P(A) ⋅ P(B)
Cumulative Distribution Function: F_X(x) = P(X ≤ x)

NCERT Solutions Class 11 Maths by Chapters

Now, we are going to share NCERT solutions class 11 maths by chapter. If you are facing any problem to solve maths questions of class 11 then you can find the solutions of class 11 maths by chapter. this is the most important part of any class to get solutions to maths by chapters.

Chapter 1 Sets
Chapter 2 Relations and Functions
Chapter 3 Trigonometric Functions
Chapter 4 Principle of Mathematical Induction
Chapter 5 Complex Numbers and Quadratic Equations
Chapter 6 Linear Inequalities
Chapter 7 Permutations and Combinations
Chapter 8 Binomial Theorem
Chapter 9 Sequences and Series
Chapter 10 Straight lines
Chapter 11 Conic Sections
Chapter 12 Introduction to Three Dimensional Geometry
Chapter 13 Limits and Derivatives
Chapter 14 Mathematical Reasoning
Chapter 15 Statistics
Chapter 16 Probability

NCERT Class 11 Maths Formulas By chapters

We have already shared the Maths formulas for class 11 above. Here you can find the maths formulas by chapters. If you have any issue in the particular chapter then you can visit our maths formulas by chapters.

Summary Class 11 Maths Formulas

We have shared all the maths formulas and maths solutions for class 11 by chapters. Excepted for this information, if I losing any things which are needed to add please let me know through comment and mail or social media. If you have any questions related to maths formulas class 11 please let me know. I expert teams will be the reply to your questions.