Maths Formulas for Class 12

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Maths Formulas for Class 12
- Maths Formulas for Class 12 by Chapters
  - Summary of Maths Formulas class 12

Maths Formulas is very-very important for class 12th students because the future depends on the class 12 results. Your aim to get admission to the engineering course then math is more important with better scores in the exams. According to the future requirement of the class 12th student, we are going to share math formulas on a single page. Millions of Students are searching for class 12 maths formulas on a single page. You can find the class 12 maths formulas by chapters.

Class 12 Inverse Trigonometric Functions Maths Formulas

\(y=sin^{-1}x\Rightarrow x=sin\:y\)
\(x=sin\:y\Rightarrow y=sin^{-1}x\)
\(sin^{-1}\frac{1}{x}=cosec^{-1}x\)
\(cos^{-1}\frac{1}{x}=sec^{-1}x\)
\(tan^{-1}\frac{1}{x}=cot^{-1}x\)
\(cos^{-1}(-x)=\pi-cos^{-1}x\)
\(cot^{-1}(-x)=\pi-cot^{-1}x\)
\(sec^{-1}(-x)=\pi-sec^{-1}x\)
\(sin^{-1}(-x)=-sin^{-1}x\)
\(tan^{-1}(-x)=-tan^{-1}x\)
\(cosec^{-1}(-x)=-cosec^{-1}x\)
\(tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\)
\(sin^{-1}x+cos^{-1}x=\frac{\pi}{2}\)
\(cosec^{-1}x+sec^{-1}x=\frac{\pi}{2}\)
\(tan^{-1}x+tan^{-1}y=tan^{-1}\frac{x+y}{1-xy}\)
\(2\:tan^{-1}x=sin^{-1}\frac{2x}{1+x^2}=cos^{-1}\frac{1-x^2}{1+x^2}\)
\(2\:tan^{-1}x=tan^{-1}\frac{2x}{1-x^2}\)
\(tan^{-1}x+tan^{-1}y=\pi+tan^{-1}\left (\frac{x+y}{1-xy} \right )\); xy > 1; x, y > 0

Class 12 Matrices Maths Formulas

Addition of Matrix

If A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\\
a_{31} & a_{32}
\end{bmatrix} and B = \begin{bmatrix}
b_{11} & b_{12}\\
b_{21} & b_{22}\\
b_{31} & b_{32}
\end{bmatrix} let us calculate A + B.

Here, both matrices A and B are of same size (3 x 2).
This simplies

C = A + B = \begin{bmatrix}
a_{11} + b_{11} & a_{12} + b_{12}\\
a_{21} + b_{21} & a_{22} + b_{22}\\
a_{31} + b_{31} & a_{32} + b_{32}
\end{bmatrix}

Substraction of Matrix

Here, both matrices A and B are of same size (3 x 2).
This simplies

C = A – B = \begin{bmatrix}
a_{11} – b_{11} & a_{12} – b_{12}\\
a_{21} – b_{21} & a_{22} – b_{22}\\
a_{31} – b_{31} & a_{32} – b_{32}
\end{bmatrix}

Multiplication of Matrix

If A = \begin{bmatrix}
a_{1} & a_{2}\\
a_{3} & a_{4}\end{bmatrix} and B = \begin{bmatrix}
b_{1} & b_{2}\\
b_{3} & b_{4}\end{bmatrix} let us calculate A X B.

Here, both matrices A and B are of same size (3 x 2).
This simplies

C = A X B = \begin{bmatrix}
a_{1} b_{1} + a_{2} b_{3} & a_{1} b_{2} + a_{2} b_{4}\\
a_{3} b_{1} + a_{4} b_{3} & a_{3} b_{2} + a_{4} b_{4}\end{bmatrix}

The adjoint of a 2×2 matrix is given as,

If A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\end{bmatrix}

adj(A) = \begin{bmatrix}
a_{22} & -a_{12}\\
-a_{21} & a_{11}\end{bmatrix}

Covariance Matrix Formula

\( Cov(X,Y)=\sum \frac{(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{N}=\sum \frac{X_{i}Y_{i}}{N}\)

Where,

N = Number of scores in each set of data

X = Mean of the N scores in the first data set

X_i = i^{th^{raw score in the first set of scores}}

\( x_{i}\) = \(i^{th}\) deviation score in the first set of scores

Y = Mean of the N scores in the second data set

\( Y_{i}\) = \(i^{th}\) raw score in the second set of scores

\( y_{i}\) = \(i^{th}\) deviation score in the second set of scores

Cov(X, Y) = Covariance of corresponding scores in the two sets of data

The inverse of a 2×2 matrix is given as

\( A^{-1}=\frac{1}{|A|}\times adj(A)\)

Class 12 Determinants Maths Formulas

Determinant of a matrix \(A=\begin{bmatrix} a_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3 \end{bmatrix}\) can be expanded as:
|A| = \(\begin{vmatrix} a_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3 \end{vmatrix}=a_1 \begin{vmatrix} b_2& c_2\\ b_3& c_3 \end{vmatrix}-b_1 \begin{vmatrix} a_2& c_2\\ a_3& c_3 \end{vmatrix}+c_1 \begin{vmatrix} a_2& b_2\\ a_3& b_3 \end{vmatrix}\)
Area of triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is:
∆ = \(\frac{1}{2}\)\(\begin{vmatrix} x_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1 \end{vmatrix}\)
Cofactor of aij of given by A_ij = (– 1)^{i+ j} M_ij
If A = \(\begin{bmatrix} a_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33} \end{bmatrix}\), then adj A = \(\begin{bmatrix} A_{11}& A_{21}& A_{31}\\ A_{12}& A_{22}& A_{32}\\ A_{13}& A_{23}& A_{33} \end{bmatrix}\) ; where A_ij is the cofactor of a_ij.
\(A^{-1}=\frac{1}{|A|}(adj\:A)\)
If a₁x + b₁y + c₁z = d₁ a₂x + b₂y + c₂z = d₂ a₃x + b₃ y + c₃z = d₃ , then these equations can be written as A X = B, where:
A=\(\begin{bmatrix} a_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3 \end{bmatrix}\), X = \(\begin{bmatrix} x\\ y\\ z \end{bmatrix}\) and B = \(\begin{bmatrix} d_1\\ d_2\\ d_3 \end{bmatrix}\)
For a square matrix A in matrix equation AX = B
- (i) | A| ≠ 0, there exists unique solution
- (ii) | A| = 0 and (adj A) B ≠ 0, then there exists no solution
- (iii) | A| = 0 and (adj A) B = 0, then the system may or may not be consistent.

Class 12 Continuity and Differentiability Maths Formulas

Derivative	Formulas
\(\frac{\mathrm{d} }{\mathrm{d} x}(sin^{-1}x)\)	\(\frac{1}{\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(cos^{-1}x)\)	\(-\frac{1}{\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(tan^{-1}x)\)	\(\frac{1}{1+x^2}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(cot^{-1}x)\)	\(\frac{-1}{1+x^2}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(sec^{-1}x)\)	\(\frac{1}{x\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(cosec^{-1}x)\)	\(\frac{-1}{x\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(e^x)\)	\(e^x\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(log\:x)\)	\(\frac{1}{x}\)

Class 12 Integrals Maths Formulas

Formulas – Standard Integrals

\(\int x^ndx=\frac{x^{n+1}}{n+1}+C,n\neq -1\). Particularly, \(\int dx=x+C)\)
\(\int cos\:x\:dx=sin\:x+C\)
\(\int sin\:x\:dx=-cos\:x+C\)
\(\int sec^2x\:dx=tan\:x+C\)
\(\int cosec^2x\:dx=-cot\:x+C\)
\(\int sec\:x\:tan\:x\:dx=sec\:x+C\)
\(\int cosec\:x\:cot\:x\:dx=-cosec\:x+C\)
\(\int \frac{dx}{\sqrt{1-x^2}}=sin^{-1}x+C\)
\(\int \frac{dx}{\sqrt{1-x^2}}=-cos^{-1}x+C\)
\(\int \frac{dx}{1+x^2}=tan^{-1}x+C\)
\(\int \frac{dx}{1+x^2}=-cot^{-1}x+C\)
\(\int e^xdx=e^x+C\)
\(\int a^xdx=\frac{a^x}{log\:a}+C\)
\(\int \frac{dx}{x\sqrt{x^2-1}}=sec^{-1}x+C\)
\(\int \frac{dx}{x\sqrt{x^2-1}}=-cosec^{-1}x+C\)
\(\int \frac{1}{x}\:dx=log\:|x|+C\)

Formulas – Partial Fractions

Partial Fraction	Formulas
\(\frac{px+q}{(x-a)(x-b)}\)	\(\frac{A}{x-a}+\frac{B}{x-b},a\neq b\)
\(\frac{px+q}{(x-a)^2}\)	\(\frac{A}{x-a}+\frac{B}{(x-b)^2}\)
\(\frac{px^2+qx+r}{(x-a)(x-b)(x-c)}\)	\(\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\)
\(\frac{px^2+qx+r}{(x-a)^2(x-b)}\)	\(\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}\)
\(\frac{px^2+qx+r}{(x-a)(x^2+bx+c)}\)	\(\frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}\)

Formulas – Integration by Substitution

\(\int tan\:x\:dx=log\:|sec\:x|+C\)
\(\int cot\:x\:dx=log\:|sin\:x|+C\)
\(\int sec\:x\:dx=log\:|sec\:x+tan\:x|+C\)
\(\int cosec\:x\:dx=log\:|cosec\:x-cot\:x|+C\)

Formulas – Integrals (Special Functions)

\(\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\:log\:\left |\frac{x-a}{x+a} \right |+C\)
\(\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\:log\:\left |\frac{a+x}{a-x} \right |+C\)
\(\int \frac{dx}{x^2+a^2}=\frac{1}{a}\:tan^{-1}\frac{x}{a}+C\)
\(\int \frac{dx}{\sqrt{x^2-a^2}}=log\:\left |x+\sqrt{x^2-a^2} \right |+C\)
\(\int \frac{dx}{\sqrt{x^2+a^2}}=log\:\left |x+\sqrt{x^2+a^2} \right |+C\)
\(\int \frac{dx}{\sqrt{x^2-a^2}}=sin^{-1}\frac{x}{a}+C\)

Formulas – Integration by Parts

The integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}
\(\int f_1(x).f_2(x)=f_1(x)\int f_2(x)\:dx-\int \left [ \frac{\mathrm{d} }{\mathrm{d} x}f_1(x).\int f_2(x)\:dx \right ]dx\)
\(\int e^x\left [ f(x)+f'(x) \right ]\:dx=\int e^x\:f(x)\:dx+C\)

Formulas – Special Integrals

\(\int \sqrt{x^2-a^2}\:dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\:log\left | x+\sqrt{x^2-a^2} \right |+C\)
\(\int \sqrt{x^2+a^2}\:dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\:log\left | x+\sqrt{x^2+a^2} \right |+C\)
\(\int \sqrt{a^2-x^2}\:dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a}{2}\:sin^{-1}\frac{x}{a}+C\)
\(ax^2+bx+c=a\left [ x^2+\frac{b}{a}x+\frac{c}{a} \right ]=a\left [ \left ( x+\frac{b}{2a} \right )^2+\left ( \frac{c}{a}-\frac{b^2}{4a^2} \right ) \right ]\)

Class 12 Applications of Integrals Maths Formulas

The area enclosed by the curve y = f (x) ; x-axis and the lines x = a and x = b (b > a) is given by the formula:
\(Area=\int_{a}^{b}y\:dx=\int_{a}^{b}f(x)\:dx\)
Area of the region bounded by the curve x = φ (y) as its y-axis and the lines y = c, y = d is given by the formula:
\(Area=\int_{c}^{d}x\:dy=\int_{c}^{d}\phi (y)\:dy\)
The area enclosed in between the two given curves y = f (x), y = g (x) and the lines x = a, x = b is given by the following formula:
\(Area=\int_{a}^{b}[f(x)-g(x)]\:dx,\: where, f(x)\geq g(x)\:in\:[a,b]\)
If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then:
\(Area=\int_{a}^{c}[f(x)-g(x)]\:dx,+\int_{c}^{b}[g(x)-f(x)]\:dx\)

Class 12 Vector Algebra Maths Formulas

Position Vector: \( \overrightarrow{OP}=\vec{r}=\sqrt{x^{2}+y^{2}+z^{2}}\)
Direction Ratios= \( l=\frac{a}{r},m=\frac{m}{r},n=\frac{c}{r}\)
Vector Addition = \(\vec{PQ}+\vec{QR}=\vec{PR}\)
Properties of Vector Addition
Commutative Property = \(\vec{a}+\vec{b}=\vec{b}+\vec{a}\)

Associative Property = \( \left (\vec{a}+\vec{b} \right )\vec{c}+=\vec{a}+\left (\vec{b}+\vec{c} \right )\)
Vector Joining Two Points = \(\overrightarrow{P_{1}P_{2}}=\overrightarrow{OP_{1}}-\overrightarrow{OP_{1}}\)

If two vectors \(\vec{a}\) and \(\vec{b}\) are given in its component forms as \(\hat{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) and \(\hat{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) and λ as the scalar part; then:

(i) \(\vec{a}+\vec{b}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}\) ;
(ii) \(\lambda \vec{a}=(\lambda a_1)\hat{i}+(\lambda a_2)\hat{j}+(\lambda a_3)\hat{k}\) ;
(iii) \(\vec{a}\:.\:\vec{b}=(a_1b_1)+(a_2b_2)+(a_3b_3)\)
(iv) and \(\vec{a}\times \vec{b}= \begin{bmatrix} \hat{i}& \hat{j}& \hat{k}\\ a_{1}& b_{1}& c_{1}\\ a_{2}& b_{2}& c_{2} \end{bmatrix}\).

Class 12 Three dimensional Geometry Maths Formulas

The Direction cosines of a line joining two points P (x₁ , y₁ , z₁) and Q (x₂ , y₂ , z₂) are \(\frac{x_2-x_1}{PQ}\:,\:\frac{y_2-y_1}{PQ}\:,\frac{z_2-z_1}{PQ}\) where
PQ=\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
Equation of a line through a point (x₁ , y₁ , z₁ ) and having direction cosines l, m, n is: \(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\)
The vector equation of a line which passes through two points whose position vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{r}=\vec{a}+\lambda (\vec{b}-\vec{a})\)
The shortest distance between \(\vec{r}=\vec{a_1}+\lambda\: \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b_2}\) is:
\(\left | \frac{(\vec{b_1}\times \vec{b_2}).(\vec{a_2}-\vec{a_1})}{|\vec{b_1}\times \vec{b_2}|} \right |\)
The distance between parallel lines \(\vec{r}=\vec{a_1}+\lambda\: \vec{b}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b}\) is
\(\left | \frac{\vec{b}\times (\vec{a_2}-\vec{a_1})}{|\vec{b}|} \right |\)
The equation of a plane through a point whose position vector is \(\vec{a}\) and perpendicular to the vector \(\vec{N}\) is \((\vec{r}-\vec{a})\:.\:\vec{N}=0\)
Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x₁ , y₁ , z₁) is A (x – x₁) + B (y – y₁) + C (z – z₁) = 0
The equation of a plane passing through three non-collinear points (x₁ , y₁ , z₁); (x₂ , y₂ , z₂) and (x₃ , y₃ , z₃) is:
\(\begin{vmatrix} x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1 \end{vmatrix}=0\)
The two lines \(\vec{r}=\vec{a_1}+\lambda\: \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b_2}\) are coplanar if:
\((\vec{a_2}-\vec{a_1})\:.\:(\vec{b_1}\times \vec{b_2})=0\)
The angle φ between the line \(\vec{r}=\vec{a}+\lambda\: \vec{b}\) and the plane \(\vec{r}\:.\:\hat{n}=d\) is given by:
\(sin\:\phi =\left |\frac{\vec{b}\:.\:\hat{n}}{|\vec{b}||\hat{n}|} \right |\)
The angle θ between the planes A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0 is given by:
\(cos\:\theta =\left | \frac{A_1\:A_2+B_1\:B_2+C_1\:C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\:\sqrt{A_2^2+B_2^2+C_2^2}} \right |\)
The distance of a point whose position vector is \(\vec{a}\) from the plane \(\vec{r}\:.\:\hat{n}=d\) is given by: \(\left | d-\vec{a}\:.\:\hat{n} \right |\)
The distance from a point (x₁ , y₁ , z₁) to the plane Ax + By + Cz + D = 0:
\(\left | \frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}} \right |\)

Class 12 Probability Maths Formulas

The conditional probability of an event E holds the value of the occurrence of the event F as:
\(P(E\:|\:F)=\frac{E\cap F}{P(F)}\:,\:P(F)\neq 0\)
Total Probability: Let E₁ , E₂ , …. , E_n be the partition of a sample space and A be any event;
P(A) = P(E₁) P (A|E₁) + P (E₂) P (A|E₂) + … + P (E_n) . P(A|E_n)
Bayes Theorem: If E₁ , E₂ , …. , E_n are events contituting in a sample space S;
\(P(E_i\:|\:A)=\frac{P(E_i)\:P(A|E_i)}{\sum_{j=1}^{n}P(E_j)\:P(A|E_j)}\)
Var (X) = E (X²) – [E(X)]²

Maths Formulas for Class 12 by Chapters

We have also shared the Maths formulas for class 12 by chapter wise. You can easily get the maths formula by the chapters which are more effective for you.

Summary of Maths Formulas class 12

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