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(x1) = f(x2) ⇒ x1 = x2 ∀ x1 , x2 ∈ 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 = IX and fog = IY. 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:
![](https://tutorttd.com/wp-content/uploads/2021/10/Inverse-Trigonometric-Funct.jpg)
Formulas
![](https://tutorttd.com/wp-content/uploads/2021/10/Inverse-Trigonometric-Funct-1.jpg)
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 = [aij]m × m will be known as diagonal matrix if aij = 0, when i ≠ j.
- A = [aij]n × n is a scalar matrix if aij = 0, when i ≠ j, aij = k, (where k is some constant); and i = j.
- A = [aij]n × n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j.
- A zero matrix will contain all its element as zero.
- A = [aij] = [bij] = B if and only if:
- (i) A and B are of the same order
- (ii) aij = bij for all the certain values of i and j
Elementary Operations
- Some basic operations of matrices:
- (i) kA = k[aij]m × n = [k(aij)]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
![](https://tutorttd.com/wp-content/uploads/2021/10/Elementary-Operations.jpg)
Determinants
Definition/Theorems
- The determinant of a matrix A = [a11]1 × 1 can be given as: |a11| = a11.
- 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
![](https://tutorttd.com/wp-content/uploads/2021/10/Formulas.jpg)
![](https://tutorttd.com/wp-content/uploads/2021/10/Formulas-1.jpg)
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.
![](https://tutorttd.com/wp-content/uploads/2021/10/Continuity-And-Differentiab.jpg)
Formulas
Given below are the standard derivatives:
![](https://tutorttd.com/wp-content/uploads/2021/10/Derivative-574x1024.jpg)
Integrals
Definition/Properties
![](https://tutorttd.com/wp-content/uploads/2021/10/Integrals.jpg)
Formulas – Standard Integrals
![](https://tutorttd.com/wp-content/uploads/2021/10/Formulas-–-Standard-Integra.jpg)
Formulas – Partial Fractions
![](https://tutorttd.com/wp-content/uploads/2021/10/Partial-Fractions.jpg)
![](https://tutorttd.com/wp-content/uploads/2021/10/Integration-by-Substitution.jpg)
Formulas – Integration by Parts
![](https://tutorttd.com/wp-content/uploads/2021/10/Integration-by-Parts.jpg)
Application Of Integrals
![](https://tutorttd.com/wp-content/uploads/2021/10/Application-Of-Integrals.jpg)
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:
![](https://tutorttd.com/wp-content/uploads/2021/10/Vector-Algebra.jpg)
Formulas
![](https://tutorttd.com/wp-content/uploads/2021/10/Formulas-2.jpg)
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 l2 + m2 + n2 = 1.
Formulas
- The Direction cosines of a line joining two points P (x1 , y1 , z1) and Q (x2 , y2 , z2)
![](https://tutorttd.com/wp-content/uploads/2021/10/Geometry.jpg)
![](https://tutorttd.com/wp-content/uploads/2021/10/Geometry-1.jpg)
![](https://tutorttd.com/wp-content/uploads/2021/10/Geometry-2.jpg)
Probability
Definition/Properties
- The conditional probability of an event E holds the value of the occurrence of the event F as:
![](https://tutorttd.com/wp-content/uploads/2021/10/Probability.jpg)
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
![](https://tutorttd.com/wp-content/uploads/2021/10/Vectors.jpg)
Algebra Formulas
![](https://tutorttd.com/wp-content/uploads/2021/10/Algebra-Formulas.jpg)
In Vector Addition
![](https://tutorttd.com/wp-content/uploads/2021/10/In-Vector-Addition.jpg)
Trigonometry
Definition
![](https://tutorttd.com/wp-content/uploads/2021/10/Trigonometry.jpg)
InveInverse Properties
![](https://tutorttd.com/wp-content/uploads/2021/10/Inverse-Properties.jpg)
Double Angle and Half Angle Formulas
![](https://tutorttd.com/wp-content/uploads/2021/10/Double-Angle-and-Half-Angle.jpg)
Important Maths Formulas for Entrance Exams
3-Dimensional Coordinate Geometry Formulas
1. Distance between two points
If P(x1, y1, z1) and Q(x2, y2, z2) are two points, then distance between them
![](https://tutorttd.com/wp-content/uploads/2021/10/Distance-between-two-points.jpg)
2. Coordinates of division point
Coordinates of the point dividing the line joining two points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m1 : m2 are
(i) In case of internal division:
![](https://tutorttd.com/wp-content/uploads/2021/10/Coordinates-of-division-poi.jpg)
(b) Centroid of a Triangle:
![](https://tutorttd.com/wp-content/uploads/2021/10/Centroid-of-a-Triangle.jpg)
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: l2 + m2 + n2 = 1 i.e. cos2 α + cos2 β + cos2 γ = 1 or sin2 α + sin2 β + sin2 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
![](https://tutorttd.com/wp-content/uploads/2021/10/Direction-cosines.jpg)
5. Direction cosines of a line joining two points
Let P = (x1, y1, z1) and Q = (x2, y2, z2); then
(i) dr’s of PQ: (x2 – x1), (y2 – y1), (z2 – z1)
![](https://tutorttd.com/wp-content/uploads/2021/10/Direction-cosines-1.jpg)
6. Angle between two lines
(i) When direction cosines of the lines are given:
If l1, m1, n1 and l2, m2, n2 are dc’s of given two lines, then the angle θ between them is given by
* cos θ = l1l2 + m1m2 + n1n2
![](https://tutorttd.com/wp-content/uploads/2021/10/Angle-between-two-lines.jpg)
(ii) When direction ratios of the lines are given:
If a1, b1, c1 and a2, b2, c2 are dr’s of given two lines, then the angle θ between them is given by
![](https://tutorttd.com/wp-content/uploads/2021/10/Angle-between-two-lines-1.jpg)
7. Conditions of parallelism and perpendicularity of two lines
(i) When Dc’s of two lines AB and CD say l1, m1, n1 and l2, m2, n2 are known, then
AB || CD ⇔ l1 = l2, m1 = m2, n1 = n2
AB ⊥ CD ⇔ l1l2 + m1m2 + n1n2 = 0
(ii) When dr’s of two lines AB & CD, say a1; bj, Cj and a2, b2, c2 are known, then
AB || CD ⇔
![](https://tutorttd.com/wp-content/uploads/2021/10/Conditions-of-parallelism.jpg)
AB ⊥ CD ⇔ a1a2 + b1b2 + c1c2 = 0
8. Projection of a line segment joining two points on a line
(i) Let PQ be a line segment where P = (x1, y1, z1) and Q = (x2, y2, z2); and AB be a given line with dc’s as l, m, n. Then projection of PQ is P’Q’ = l(x2 – x1) + m (y2 – y1 + n (z2 – z1)
(ii) If a, b, c are the projections of a line segment on coordinate axes, then length of the segment =
![](https://tutorttd.com/wp-content/uploads/2021/10/Projection.jpg)
(iii) If a, b, c are projections of a line segment on coordinate axes then its dc’s are
![](https://tutorttd.com/wp-content/uploads/2021/10/projections-of-a-line.jpg)
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 (x1, y1, z1) and having direction ratios a, b, c is
![](https://tutorttd.com/wp-content/uploads/2021/10/Cartesian-equation.jpg)
10. Cartesian equation of a line passing through two given points
The cartesian equation of a line passing through two given points (x1, y1, z1) and Q = (x2, y2, z2) is given by
![](https://tutorttd.com/wp-content/uploads/2021/10/Cartesian-equation-1.jpg)
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
![](https://tutorttd.com/wp-content/uploads/2021/10/Perpendicular-distance.jpg)
Let the coordinates of L be (x1 + aλ, y1 + bλ, + z1 + cλ). Then direction ratios of PL are x1 + aλ – α, y1 + bλ – β, z1 + cλ – γ.
Direction ratio of AB are a, b, c. Since PL is perpendicular to AB, therefore
(x1 + aλ – α) a + (y1 + bλ – β) b + (z1 + cλ – γ) c = 0 =
![](https://tutorttd.com/wp-content/uploads/2021/10/Direction-ratio.jpg)
Puting this value of λ in (x1 + aλ, y1 + bλ, z1 + 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)
![](https://tutorttd.com/wp-content/uploads/2021/10/Co-ordinates.jpg)
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 (x1, y1, z1) is a(x – x1) + b(y – y1) + c(z – z1) = 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
![](https://tutorttd.com/wp-content/uploads/2021/10/Plane.jpg)
(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, a1x + b1y + c1z + d1 = 0 is
2(aa1 + bb1 + cc1) (a1 + b1y + c1z + d1)
= (a12 + a22 + a32) × (ax + by + cz + d)
13. Angle between two planes in Cartesian form
The angle θ between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by
![](https://tutorttd.com/wp-content/uploads/2021/10/Angle-between-two-planes.jpg)
14. Distance of a point from a plane
The length of the perpendicular from a point P(x1, y1, z1) to the plane ax + by + cz + d = 0 is given by
![](https://tutorttd.com/wp-content/uploads/2021/10/Distance-of-a-point.jpg)
15. Equation of plane bisecting the angle between two given planes
The equation of the planes bisecting the angles between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are
![](https://tutorttd.com/wp-content/uploads/2021/10/Equation-of-plane.jpg)
If d1 and d2 are +ve then following table can be utilized to write acute angle bisector or obtuse angle bisector.
![](https://tutorttd.com/wp-content/uploads/2021/10/If-d1-and-d2.jpg)
16. Condition of coplanarity of two lines
![](https://tutorttd.com/wp-content/uploads/2021/10/Condition-of-coplanarity.jpg)
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 =
![](https://tutorttd.com/wp-content/uploads/2021/10/Area-of-a-triangle.jpg)
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 = R2
(ii) General equation of a sphere:
The equation x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 represents a sphere with centre (- u, – v, – w) and radius =
![](https://tutorttd.com/wp-content/uploads/2021/10/The-equation.jpg)
(iii) Diameter form of the equation of a sphere:
If (x1, y1, z1) and (x2, y2, z2) are the coordinates of the extremities of a diameter of a sphere, then its equation is
(x – x1) (x – x2) + (y – y1) (y – y2) + (z – z1) (z – z2) = 0.
An equation of the form
ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = 0 is a homogeneous equation of 2nd degree may represent pair of planes if
![](https://tutorttd.com/wp-content/uploads/2021/10/An-equation.jpg)
Let a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 be the equation of any two planes, taken together then
a1x + b1y + c1z + d1 = 0 = a2x + b2y + c2z + d2
Note:
- x2 + y2 + z2 = r2 is equation of sphere with centre (0, 0, 0) and radius “r”
- Equation of sphere will have coeff. of x2, y2, z2 equal and coeff. of term xy, yz, zx must be absent.
- Equation of tangent plane at any point (x1, y1, z1) of the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 is
xx1 + yy1 + zz1 + u(x + x1) + v(y + y1) + w(z + z1) + d = 0 - The plane lx + my + nz = p will touch the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 if (ul + vm + wn + p)2 = (l2 + m2 + n2) (u2 + v2 + w2 – d)
18. Distance between the parallel planes :
Let two parallel planes are a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0
and distance between plane
![](https://tutorttd.com/wp-content/uploads/2021/10/Distance.jpg)
19. Coplanarity of lines:
(a) Let lines are
![](https://tutorttd.com/wp-content/uploads/2021/10/condition-for-coplanarity.jpg)
and a1x + b1y + c1z = d1 = 0 = a2x + b2y + c2z + d2 the condition for coplanarity is
![](https://tutorttd.com/wp-content/uploads/2021/10/condition-for-coplanarity-1.jpg)
(c) Let lines are a1 + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 a3x + b3y + c3z + d3 = 0 = a4x + b4y + c4z + d4 then condition that lines are coplanar is
![](https://tutorttd.com/wp-content/uploads/2021/10/lines-are-coplanar.jpg)
20. Family of planes:
Equation of a plane passing through line of intersection of the planes ax + by + cz + d = 0 and a1x + b1y + c1z + d1 = 0 can be represented by the equation (ax + by + cz + d) + λ(a1x + b1y + c1z + d1) = 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
![](https://tutorttd.com/wp-content/uploads/2021/10/Skew-lines.jpg)
22. Volume of tetrahedron:
If the vertices of tetrahedron are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) then
![](https://tutorttd.com/wp-content/uploads/2021/10/Volume-of-tetrahedron.jpg)
✅ Addition of Vectors Formulas ⭐️⭐️⭐️⭐️⭐
✅ Area Under The Curve Formulas ⭐️⭐️⭐️⭐️⭐
✅ Binomial Theorem Formulas ⭐️⭐️⭐️⭐️⭐
✅ Complex Number Formulas ⭐️⭐️⭐️⭐️⭐
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