# Maths Formulas For Class 11

## 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

1. A cartesian product A × B of two sets A and B is given by:
A × B = { $$(a,b):a\epsilon A, b\epsilon B$$ }
2. If (a , b) = (x , y); then a = x and b = y
3. If n(A) = x and n(B) = y, then n(A × B) = xy
4. A × $$\phi$$ = $$\phi$$
5. The cartesian product: A × B ≠ B × A
6. 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
7. 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

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

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; i4a+1;

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

Example – i2= -1; i6= -1; i10= -1; i4a+2;

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

Example – i3= -i; i7= -i; i11= -i; i4a+3;

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

Example – i4= 1; i8= 1; i12= 1; i4a;

 $$x^2+bx+c=0 \;where\; a ≠ 0$$ $$x=-b\pm\frac{\sqrt{b^2-4ac}}{2a}$$

#### Class 11 Permutations and Combinations Formulas

1. 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
2. $$n!=1\times 2\times 3\times …\times n$$
3. $$n!=n\times (n-1)!$$
4. The number of permutations of n different things taken r at a time with repetition being allowed is given as: nr
5. 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, …., pk objects of kth kind are given as: $$\frac{n!}{p_{1}!\:p_{2}!\:…\:p_{k}!}$$
6. 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.
$$(a+b)^n={}^{n}\textrm{C}_{0}\:a^n+{}^{n}\textrm{C}_{1}\:a^{n-1}.b+{}^{n}\textrm{C}_{2}\:a^{n-2}.b^2+…+{}^{n}\textrm{C}_{n-1}\:a.b^{n-1}+{}^{n}\textrm{C}_{n}\:b^n$$

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

1. 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
2. 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.
3. Equation of the line passing through the points (x1, y1) and (x2, y2) is given by: $$y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})$$
4. Equation of the line making a and b intercepts on the x- and y-axis respectively is: $$\frac{x}{a}+\frac{y}{b}=1$$
5. The perpendicular distance d of a line Ax + By + C = 0 from a point (x1, y1) is: $$d=\frac{\left | Ax_{1}+By_{1}+C \right |}{\sqrt{A^2+B^2}}$$
6. 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

1. The equation of the circle with the centre point (h, k) and radius r is given by (x – h)2 + (y – k)2 = r2
2. The equation of the parabola having focus at (a, 0) where a > 0 and directrix x = – a is given by: y2 = 4ax
3. The equation of an ellipse with foci on the x-axis is $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
4. Length of the latus rectum of the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ is given by: $$\frac{2b^2}{a}$$
5. The equation of a hyperbola with foci on the x-axis is $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$
6. 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

1. The distance of two points P(x1, y1, z1) and Q(x2, y2, z2) is:
$$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$
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:
$$\left ( \frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n} \right )\:and\:\left ( \frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n},\frac{mz_2-nz_1}{m-n} \right )$$;
3. 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 $$\left ( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2} \right )$$
4. 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

1. 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) $$\large \lim\limits_{x \to a} \left [ \frac{f(x)}{g(x)} \right ] = \frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$$
2. 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) $$\lim\limits_{x \to a}\frac{1-cos\:x}{x}=0$$
3. The derivative of a function f at a holds as: $${f}'(a)=\lim\limits_{x \to a}\frac{f(a+h)-f(a)}{h}$$
4. The derivative of a function f at a given point x holds as: $${f}'(x)=\frac{\partial f(x)}{\partial x}=\lim\limits_{x \to a}\frac{f(x+h)-f(x)}{h}$$
5. For the functions u and v, the following holds true:
• (i) $$(u\pm v)’=u’\pm v’$$
• (ii) $$(uv)’=u’v+uv’$$
• (iii) $$\left ( \frac{u}{v} \right )’=\frac{u’v-uv’}{v^2}$$
6. 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

1. Mean Deviation for the ungrouped data:
• (i) $$M.D.(\bar x)=\frac{\sum \left | x_i-\bar x \right |}{n}$$
• (ii) $$M.D.(M)=\frac{\sum \left | x_i-M \right |}{n}$$
2. Mean Deviation for the grouped data:
• (i) $$M.D.(\bar x)=\frac{\sum f_i|x_i-\bar x|}{N}$$
• (ii) $$M.D.(M)=\frac{\sum f_i|x_i-M|}{N}$$
3. Variance and Standard Deviation for the ungrouped data:
• (i) $$\sigma ^2=\frac{1}{N}\sum (x_i-\bar x)^2$$
• (ii) $$\sigma=\sqrt{\frac{1}{N}\sum (x_i-\bar x)^2}$$
4. Variance and Standard Deviation of a frequency distribution (discrete):
• (i) $$\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2$$
• (ii) $$\sigma=\sqrt{\frac{1}{N}\sum f_i(x_i-\bar x)^2}$$
5. Variance and Standard Deviation of a frequency distribution (continuous):
• (i) $$\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2$$
• (ii) $$\sigma=\frac{1}{N}\sqrt{N\sum f_ix_i^2-(\sum f_ix_i)^2}$$
6. 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(AC) + 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: FX(x) = P(Xx)

### 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

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#### Summary Class 11 Maths Formulas

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