Maths Formulas for Class 12
Relations And Functions
- 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.
- 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.
The domain and range of inverse trigonometric functions are given below:
- 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
- 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
- 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.
Continuity And Differentiability
- 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.
Given below are the standard derivatives:
Formulas – Standard Integrals
Formulas – Partial Fractions
Formulas – Integration by Parts
Application Of Integrals
- 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:
- 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.
- The Direction cosines of a line joining two points P (x1 , y1 , z1) and Q (x2 , y2 , z2)
- The conditional probability of an event E holds the value of the occurrence of the event F as:
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Vectors and Three Dimensional Geometry Formulas
In Vector Addition
Double Angle and Half Angle Formulas
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
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:
(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: 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
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)
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
(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
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 ⇔
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 =
(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 (x1, y1, z1) 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 (x1, y1, z1) and Q = (x2, y2, z2) 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 (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 =
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)
(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
(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
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
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
If d1 and d2 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 =
(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 =
(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
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
- 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
19. Coplanarity of lines:
(a) Let lines are
and a1x + b1y + c1z = d1 = 0 = a2x + b2y + c2z + d2 the condition for coplanarity is
(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
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
22. Volume of tetrahedron:
If the vertices of tetrahedron are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) then
✅ Addition of Vectors Formulas ⭐️⭐️⭐️⭐️⭐
✅ Area Under The Curve Formulas ⭐️⭐️⭐️⭐️⭐
✅ Binomial Theorem Formulas ⭐️⭐️⭐️⭐️⭐
✅ Complex Number Formulas ⭐️⭐️⭐️⭐️⭐