## Maths Formulas for Class 12

## Relations And Functions

*Definition/Theorems*

- Empty relation holds a specific relation R in X as:
**R = φ ⊂ X × X**. - A Symmetric relation R in X satisfies a certain relation as:
**(a, b) ∈ R****implies (b, a) ∈ R**. - A Reflexive relation R in X can be given as:
**(a, a) ∈ R; for all ∀ a ∈ X**. - A Transitive relation R in X can be given as:
**(a, b) ∈ R and (b, c) ∈ R, thereby, implying (a, c) ∈ R**. - A Universal relation is the relation R in X can be given by R = X × X.
- Equivalence relation R in X is a relation that shows all the reflexive, symmetric and transitive relations.

*Properties*

- A function f: X → Y is one-one/injective; if f(x
_{1}) = f(x_{2}) ⇒ x_{1}= x_{2}∀ x_{1}, x_{2}∈ X. - A function f: X → Y is onto/surjective; if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
- A function f: X → Y is one-one and onto or bijective; if f follows both the one-one and onto properties.
- A function f: X → Y is invertible if ∃ g: Y → X such that gof = I
_{X}and fog = I_{Y}. This can happen only if f is one-one and onto. - A binary operation ∗∗ performed on a set A is a function ∗∗ from A × A to A.
- An element e ∈ X possess the identity element for binary operation ∗∗ : X × X → X, if a ∗∗ e = a = e ∗∗ a; ∀ a ∈ X.
- An element a ∈ X shows the invertible property for binary operation ∗∗ : X × X → X, if there exists b ∈ X such that a ∗∗ b = e = b ∗∗ a where e is said to be the identity for the binary operation ∗∗. The element b is called the inverse of a and is denoted by a
^{–1}. - An operation ∗∗ on X is said to be commutative if a ∗∗ b = b ∗∗ a; ∀ a, b in X.
- An operation ∗∗ on X is said to associative if (a ∗∗ b) ∗∗ c = a ∗∗ (b ∗∗ c); ∀ a, b, c in X.

## Inverse Trigonometric Functions

Inverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here.

**Properties/Theorems**

The domain and range of inverse trigonometric functions are given below:

*Formulas*

## Matrices

*Definition/Theorems*

- A matrix is said to have an ordered rectangular array of functions or numbers. A matrix of order m × n consists of m rows and n columns.
- An m × n matrix will be known as a square matrix when m = n.
- A = [a
_{ij}]_{m × m}will be known as diagonal matrix if a_{ij}= 0, when i ≠ j. - A = [a
_{ij}]_{n × n}is a scalar matrix if a_{ij}= 0, when i ≠ j, a_{ij}= k, (where k is some constant); and i = j. - A = [a
_{ij}]_{n × n}is an identity matrix, if a_{ij}= 1, when i = j and a_{ij}= 0, when i ≠ j. - A zero matrix will contain all its element as zero.
- A = [a
_{ij}] = [b_{ij}] = B if and only if:- (i) A and B are of the same order
- (ii) a
_{ij}= b_{ij}for all the certain values of i and j

*Elementary Operations*

- Some basic operations of matrices:
- (i) kA = k[a
_{ij}]_{m × n}= [k(a_{ij})]_{m × n} - (ii) – A = (– 1)A
- (iii) A – B = A + (– 1)B
- (iv) A + B = B + A
- (v) (A + B) + C = A + (B + C); where A, B and C all are of the same order
- (vi) k(A + B) = kA + kB; where A and B are of the same order; k is constant
- (vii) (k + l)A = kA + lA; where k and l are the constant

- (i) kA = k[a

### Determinants

*Definition/Theorems*

- The determinant of a matrix A = [a
_{11}]_{1 × 1}can be given as: |a_{11}| = a_{11}. - For any square matrix A, the |A| will satisfy the following properties:
- (i) |A′| = |A|, where A′ = transpose of A.
- (ii) If we interchange any two rows (or columns), then sign of determinant changes.
- (iii) If any two rows or any two columns are identical or proportional, then the value of the determinant is zero.
- (iv) If we multiply each element of a row or a column of a determinant by constant k, then the value of the determinant is multiplied by k.

*Formulas*

## Continuity And Differentiability

*Definition/Properties*

- A function is said to be continuous at a given point if the limit of that function at the point is equal to the value of the function at the same point.
- Properties related to the functions:
- (i) (f±g)(x)=f(x)±g(x)(f±g)(x)=f(x)±g(x) is continuous.
- (ii) (f.g)(x)=f(x).g(x)(f.g)(x)=f(x).g(x) is continuous.

*Formulas*

Given below are the standard derivatives:

## Integrals

*Definition/Properties*

*Formulas – Standard Integrals*

*Formulas – Partial Fractions*

*Formulas – Integration by Parts*

## Application Of Integrals

## Vector Algebra

*Definition/Properties*

- Vector is a certain quantity that has both the magnitude and the direction. The position vector of a point P (x, y, z) is given by:

*Formulas*

## Geometry

*Definition/Properties*

- Direction cosines of a line are the cosines of the angle made by a particular line with the positive directions on coordinate axes.
- Skew lines are lines in space which are neither parallel nor intersecting. These lines lie in separate planes.
- If l, m and n are the direction cosines of a line, then l
^{2}+ m^{2}+ n^{2}= 1.

*Formulas*

- The Direction cosines of a line joining two points P (x
_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2})

## Probability

*Definition/Properties*

- The conditional probability of an event E holds the value of the occurrence of the event F as:

*Please Note: If you are having difficulties accessing these formulas on your mobile, try opening the Desktop site on your mobile in your mobile’s browser settings.*

## Vectors and Three Dimensional Geometry Formulas

## Algebra Formulas

**In Vector Addition**

## Trigonometry

**Definition**

**Inve Inverse Properties**

**Double Angle and Half Angle Formulas**

**Important Maths Formulas for Entrance Exams**

### 3-Dimensional Coordinate Geometry Formulas

**1. Distance between two points**

If P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are two points, then distance between them

**2. Coordinates of division point**

Coordinates of the point dividing the line joining two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) in the ratio m_{1} : m_{2} are

(i) In case of internal division:

(b) Centroid of a Triangle:

**3. Direction cosines of a line [Dc’s]**

The cosines of the angles made by a line with coordinate axes are called Direction Cosine. If α, β, γ be the angles made by a line with coordinate axes, then direction cosine are l = cos α, m = cos β, n = cos γ and relation between dc’s: l^{2} + m^{2} + n^{2} = 1 i.e. cos^{2} α + cos^{2} β + cos^{2} γ = 1 or sin^{2} α + sin^{2} β + sin^{2} γ = 2

**4. Direction ratios of a line**

Three numbers which are proportional to the direction cosines of a line are called the direction ratios of that line. If a, b, c are such numbers then

**5. Direction cosines of a line joining two points**

Let P = (x_{1}, y_{1}, z_{1}) and Q = (x_{2}, y_{2}, z_{2}); then

(i) dr’s of PQ: (x_{2} – x_{1}), (y_{2} – y_{1}), (z_{2} – z_{1})

**6. Angle between two lines**

(i) When direction cosines of the lines are given:

If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are dc’s of given two lines, then the angle θ between them is given by

* cos θ = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}

(ii) When direction ratios of the lines are given:

If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are dr’s of given two lines, then the angle θ between them is given by

**7. Conditions of parallelism and perpendicularity of two lines**

(i) When Dc’s of two lines AB and CD say l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are known, then

AB || CD ⇔ l_{1} = l_{2}, m_{1} = m_{2}, n_{1} = n_{2}

AB ⊥ CD ⇔ l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0

(ii) When dr’s of two lines AB & CD, say a1; bj, Cj and a2, b2, c2 are known, then

AB || CD ⇔

AB ⊥ CD ⇔ a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

**8. Projection of a line segment joining two points on a line**

(i) Let PQ be a line segment where P = (x_{1}, y_{1}, z_{1}) and Q = (x_{2}, y_{2}, z_{2}); and AB be a given line with dc’s as l, m, n. Then projection of PQ is P’Q’ = l(x_{2} – x_{1}) + m (y_{2} – y_{1} + n (z_{2} – z_{1})

(ii) If a, b, c are the projections of a line segment on coordinate axes, then length of the segment =

(iii) If a, b, c are projections of a line segment on coordinate axes then its dc’s are

**9. Cartesian equation of a line passing through a given point & given direction ratios**

Cartesian equation of a straight line passing through a fixed point (x_{1}, y_{1}, z_{1}) and having direction ratios a, b, c is

**10. Cartesian equation of a line passing through two given points**

The cartesian equation of a line passing through two given points (x_{1}, y_{1}, z_{1}) and Q = (x_{2}, y_{2}, z_{2}) is given by

**11. Perpendicular distance of a point from a line**

(a) Cartesian Form:

To find the perpendicular distance of a given point (α, β, γ) from a given line

Let the coordinates of L be (x_{1} + aλ, y_{1} + bλ, + z_{1} + cλ). Then direction ratios of PL are x_{1} + aλ – α, y_{1} + bλ – β, z_{1} + cλ – γ.

Direction ratio of AB are a, b, c. Since PL is perpendicular to AB, therefore

(x_{1} + aλ – α) a + (y_{1} + bλ – β) b + (z_{1} + cλ – γ) c = 0 =

Puting this value of λ in (x_{1} + aλ, y_{1} + bλ, z_{1} + cλ), we obtain coordinates of L. Now, using distance formula we can obtain the length PL.

* Using “L” as mid point, we can find image of P with respect to given plane

(b) Co-ordinate’s of “L” let they are (p, q, r)

**12. Plane**

(i) General equation of a plane: ax + by + cz + d = 0

(ii) Equation of a plane passing through a given point:

The general equation of a plane passing through a point (x_{1}, y_{1}, z_{1}) is a(x – x_{1}) + b(y – y_{1}) + c(z – z_{1}) = 0, where a, b and c are constants.

(iii) Intercept form of a plane:

The equation of a plane intercepting lengths a, b and c with x-axis , y-axis and z-axis respectively is

(iv) Normal Form:

If l, m, n are direction cosines of the normal to a given plane which is at a distance p from the origin, then the equation of the plane is lx + my + nz = p.

(v) The reflection of the plane ax + by + cz + d = 0 on the plane, a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 is

2(aa_{1} + bb_{1} + cc_{1}) (a_{1} + b_{1}y + c_{1}z + d_{1})

= (a_{1}^{2} + a_{2}^{2} + a_{3}^{2}) × (ax + by + cz + d)

**13. Angle between two planes in Cartesian form**

The angle θ between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 is given by

**14. Distance of a point from a plane**

The length of the perpendicular from a point P(x_{1}, y_{1}, z_{1}) to the plane ax + by + cz + d = 0 is given by

**15. Equation of plane bisecting the angle between two given planes**

The equation of the planes bisecting the angles between the planes a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 are

If d_{1} and d_{2} are +ve then following table can be utilized to write acute angle bisector or obtuse angle bisector.

**16. Condition of coplanarity of two lines**

Area of a triangle:

If Ayz, Azx, Axy be the projection of an area A on the coordinate plane yz, zx and xy respectively then A =

**17. Sphere**

(i) The equation of a sphere with centre (a, b, c) and radius R is (x – a)^{2} + (y – b)^{2} + (z – c)^{2} = R^{2}

(ii) General equation of a sphere:

The equation x^{2} + y^{2} + z^{2} + 2ux + 2vy + 2wz + d = 0 represents a sphere with centre (- u, – v, – w) and radius =

(iii) Diameter form of the equation of a sphere:

If (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) are the coordinates of the extremities of a diameter of a sphere, then its equation is

(x – x_{1}) (x – x_{2}) + (y – y_{1}) (y – y_{2}) + (z – z_{1}) (z – z_{2}) = 0.

An equation of the form

ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2hxy = 0 is a homogeneous equation of 2^{nd} degree may represent pair of planes if

Let a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 be the equation of any two planes, taken together then

a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 = a_{2}x + b_{2}y + c_{2}z + d_{2}

Note:

- x
^{2}+ y^{2}+ z^{2}= r^{2}is equation of sphere with centre (0, 0, 0) and radius “r” - Equation of sphere will have coeff. of x
^{2}, y^{2}, z^{2}equal and coeff. of term xy, yz, zx must be absent. - Equation of tangent plane at any point (x
_{1}, y_{1}, z_{1}) of the sphere x^{2}+ y^{2}+ z^{2}+ 2ux + 2vy + 2wz + d = 0 is

xx_{1}+ yy_{1}+ zz_{1}+ u(x + x_{1}) + v(y + y_{1}) + w(z + z_{1}) + d = 0 - The plane lx + my + nz = p will touch the sphere x
^{2}+ y^{2}+ z^{2}+ 2ux + 2vy + 2wz + d = 0 if (ul + vm + wn + p)^{2}= (l^{2}+ m^{2}+ n^{2}) (u^{2}+ v^{2}+ w^{2}– d)

**18. Distance between the parallel planes :**

Let two parallel planes are a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0

and distance between plane

**19. Coplanarity of lines:**

(a) Let lines are

and a_{1}x + b_{1}y + c_{1}z = d_{1} = 0 = a_{2}x + b_{2}y + c_{2}z + d_{2} the condition for coplanarity is

(c) Let lines are a_{1} + b_{1}y + c_{1}z + d_{1} = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 a_{3}x + b_{3}y + c_{3}z + d_{3} = 0 = a_{4}x + b_{4}y + c_{4}z + d_{4} then condition that lines are coplanar is

**20. Family of planes:**

Equation of a plane passing through line of intersection of the planes ax + by + cz + d = 0 and a_{1}x + b_{1}y + c_{1}z + d_{1} = 0 can be represented by the equation (ax + by + cz + d) + λ(a_{1}x + b_{1}y + c_{1}z + d_{1}) = 0

**21. Skew lines:**

Two straight lines in space are called skew lines if neither j they are parallel nor they intersect each other the distance between j then may be calculated by using following formulae

**22. Volume of tetrahedron:**

If the vertices of tetrahedron are (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) and (x_{4}, y_{4}, z_{4}) then

✅ Addition of Vectors Formulas ⭐️⭐️⭐️⭐️⭐

✅ Area Under The Curve Formulas ⭐️⭐️⭐️⭐️⭐

✅ Binomial Theorem Formulas ⭐️⭐️⭐️⭐️⭐

✅ Complex Number Formulas ⭐️⭐️⭐️⭐️⭐

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