Mathematics Formulas PDF

Mathematics 󰗊 Click Below to Join Telegram 󰗊 Join HP Gurukul :- Click Here  Matrices & Determinant Types of Matrices...

Mathematics 󰗊 Click Below to Join Telegram 󰗊 Join HP Gurukul :- Click Here  Matrices & Determinant Types of Matrices 1. Row Matrix : A matrix having only one row is called a row matrix or a Definition - Matrices row vector. 2. Column Matrix: A matrix having only one column is called a column A set of (m × n) numbers arranged in the form of an matrix or a column vector. ordered set of m rs and n columns is called a matrix of 3. Square matrix: A matrix in which the number of rows is equal to the order m × n. number of columns, say (n × n) is called a square matrix of order n. 4. Diagonal Matrix: A square matrix is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero. A = [aij]n×n , aij = 0 for all i ≠ j 5. Scalar Matrix: A diagonal matrix in which all the diagonal elements are equal is called the scalar matrix. A square matrix A = [aij]n×n is called a scalar matrix if. (i) aij = 0 for all i ≠ j and (ii) aii = C for all i ∈ {1, 2,..., n} 6. Identity or Unit Matrix: A square matrix each of whose diagonal element is unity and each of whose non diagonal element is equal to zero is called an identity or unit matrix. Equality of Matrices 7. Null Matrix: A matrix whose all elements are zero is called a null matrix or a zero matrix, represented by O. Two matrices A = [aij]m×n and B = [bij]r×s are equal if 8. Upper Triangular Matrix: A square matrix A = [aij] is called an upper (i) m = r, i.e., the number of rows in A equals the number of rows in B. triangular matrix if aij = 0 ∀ i > j. (ii) n = s, i.e., the number of columns in A equals the number of columns in 9. Lower Triangular Matrix: A square matrix A = [aij] is called lower B. triangular if aij = 0 ∀ i < j. (iii) aij = bij for i = 1, 2,..., m and j = 1, 2,..., n. 10. Singular Matrix: A square matrix with zero determinant is called a If two matrices A and B are equal, we write A = B, singular matrix. otherwise we write A ≠ B.  Matrices & Determinant Scalar Multiplication of Matrices If A be a given matrix and k is any scalar number real or Sum of Matrices Complex. Let A = [aij], B = [bij] be matrices of the same order m×n. Then matrix kA is a matrix of same order, where all the elements of kA Then C = A + B = [cij], is a matrix of order m×n. are k times of the corresponding elements of A Where, [cij] = [aij + bij] Properties of Multiplication by a Scalar Properties of Matrix Addition If A = [aij] and B = [bij] are matrix of the same order and a and (i) Matrix addition is commutative b are any scalars, then A+B=B+A (i) ⍺ (A + B) = ⍺A + ⍺B (ii) Matrix addition is associative (ii) (⍺ + β)A = ⍺A + βA A + (B + C) = (A + B) + C. (iii) ⍺(βA) = (⍺β) A. (iv) If A is a square matrix of order ‘n’ Then |kA| = kn |A| Transpose of a Matrix If A be a given matrix of the order m × n then the matrix obtained by changing the rows of A into columns and columns of A into rows is Multiplication of Matrices called Transpose of matrix A and is denoted by A' or AT. Hence the matrix A' is of order n × m. If A = [aij]m×p and B = [bjk]p×n Properties of Transpose (i) (A')' = A. Then Am×p × Bp×n = (AB)m×n (ii) (kA)' = kA'. k being a scalar. (iii) (A + B)' = A' + B'. (iv) (AB)' = B'A'. (v) (ABC)' = C'B'A'.  Matrices & Determinant Properties of Matrix Multiplication (i) Multiplication of matrices is distributive with respect to a addition of matrices. Symmetric and Skew-symmetric A (B + C) = AB + AC and (A + B) C = AC + BC Matrix (i) A square matrix A = [aij] will be called symmetric if AT = A. (ii) Matrix multiplication is associative if conformability is assured. i.e. A (BC) = (AB) C. i.e. every ijth element = jith element. (iii) The multiplication of matrices is not always commutative. i.e. AB is not (ii) A square matrix A = [aij] will be called skew symmetric if AT = –A. i.e. every ijth element = -(jith element). always equal to BA. Let A and B be symmetric matrices of the same order. (iv) Multiplication of a matrix A by a null matrix conformable with A for Then the following hold: multiplication is a null matrix i.e. AO = O. 1. Aₙ is symmetric for all positive integers n. In particular if A be a square matrix and O be square null matrix of the 2. AB is symmetric if and only if AB = BA. same order, then OA = AO = O. 3. AB + BA is symmetric. (v) If AB = O then it does not necessarily mean that A = O or B = O. 4. AB – BA is skew - symmetric Adjoint of Matrix If A is a square matrix, then transpose of a matrix made from None of the matrices on the left is a null matrix whereas their products is a cofactors of elements of A is called adjoint matrix of A. It’s null matrix. denoted by adj A. (vi) Multiplication of matrix A by a unit matrix I : Properties of Adjoint Matrix Let A be a m × n matrix. Then AIn = A and Im A = A.. (i) A. (Adj A) = | A | In = (adj A). A (vii) If A and B are square matrices of order ‘n’. Then |AB| = |A| |B| (ii) |adj A | = | A |n–1 (iii) adj (adj A) = |A|n–2 A (iv) (adj A)T = adj (AT) (v) adj (AB) = (adj B). (adj A) (iv) Adj (A–1) = (adj A)–1 (vii) |(adj (adj (A)) | =  Matrices & Determinant Determinants Definition Inverse of Matrix A square matrix A of order n is said to be invertible or nonsingular if there exists a square matrix B of order n such that AB = In= BA where In is the identity matrix of order n, B is called inverse of A and is denoted by A–1. Properties of Inverse Matrices (i) (AT)–1 = (A–1)T (ii) (AB)–1 = B–1 A–1 (iii) (A–1)–1 = A (iv) (v)  Matrices & Determinant Properties of Determinants (iii) if a determinant has any two rows (or columns) identical or proportional, then its values is zero. Properties of Determinants If a determinant has all the elements zero in any row (or column) then its values is zero. (iv) If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number. (i) The value of a determinant remains unaltered; if the rows and columns are interchanged. Then D’ = KD (iv) Determinant of a skew-symmetric matrix of odd order is (ii) If any two rows (or column) of a determinant be zero. interchanged, the value of determinant changes in sign only. (v) Determinant of a skew-symmetric matrix of even order is always a perfect square. (vi) If a determinant has all the elements zero in any row (or column) then its values is zero. (vii) If a determinant has any two rows (or columns) identical or proportional, then its values is zero.  Matrices & Determinant System of Equation Minors and Cofactors Definition Consistency of a System of Equations (i) If D ≠ 0 then the given system of equations are consistent and have unique solution. (ii) If D = 0 but at least one of Dx, Dy, Dz is not zero then the equations are inconsistent and have no solution. (iii) If D = Dx = Dy = Dz = 0 then the given system of equations are consistent and have infinite solution except the case of parallel planes when there is no solution. (iv) If d1 = d2 = d3 = 0 then system of equation is called Homogeneous system of equations. (v) Solution of Homogeneous Equations is always consistent, as x = 0 = y = z is always a solution. This is known as TRIVIAL solution. (vi) For Homogenous Equations, if D ≠ 0. Then x = 0 = y = z is the only solution. (vii) For Homogenous Equations, if D = 0, then there exists non zero solutions [NON TRIVIAL SOLUTiONS] also Maths : Important topic with PYQ Types of Problems Type 1 Adjoint of Matrix  Maths : Important topic with PYQ JEE Main 2024 | 4t April S1 1 Let ⍺ ∈ (0, ∞) and If det (adj (2A - AT) MCQ Type adj(A - 2AT)) = 28, then (det(A))2 is equal to: A 16 B 36 C 49 D 1  Maths : Important topic with PYQ JEE Main 2024 | 4t April S1 1 Let ⍺ ∈ (0, ∞) and If det (adj (2A - AT) MCQ Type adj(A - 2AT)) = 28, then (det(A))2 is equal to: A 16 B 36 C 49 D 1  Maths : Important topic with PYQ Solution:  Maths : Important topic with PYQ JEE Main 2023 | 24th Jan S2 2 Let A be a 3 × 3 matrix such that |adj(adj(adj A))| = 124. Then |A-1 adj A| is equal to MCQ Type A 12 B 2√3 C 1 D √6  Maths : Important topic with PYQ JEE Main 2023 | 24th Jan S2 2 Let A be a 3 × 3 matrix such that |adj(adj(adj A))| = 124. Then |A-1 adj A| is equal to MCQ Type A 12 B 2√3 C 1 D √6  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 2 System of Equation  Maths : Important topic with PYQ JEE Main 2023 | 30th Jan S2 3 10 For ⍺, β ∈ R, suppose the system of linear equations x - y + z = 5, 2x + 2y + ⍺z = 8, 3x - y + 4z = β MCQ Type has infinitely many solutions. Then ⍺ and β are the roots of A x2 + 18x + 56 = 0 B x2 - 10x + 16 = 0 C x2 + 14x + 24 = 0 D x2 - 18x + 56 = 0  Maths : Important topic with PYQ JEE Main 2023 | 30th Jan S2 3 10 For ⍺, β ∈ R, suppose the system of linear equations x - y + z = 5, 2x + 2y + ⍺z = 8, 3x - y + 4z = β MCQ Type has infinitely many solutions. Then ⍺ and β are the roots of A x2 + 18x + 56 = 0 B x2 - 10x + 16 = 0 C x2 + 14x + 24 = 0 D x2 - 18x + 56 = 0  Maths : Important topic with PYQ Solution: 1. Vector Quantity Vector Algebra A quantity which has magnitude & also a direction in space is called a vector quantity. 2. Position Vector Let O be the origin & P be a point in space having coordinates (x, y, z) with respect to the origin O. Then the vector is called the position vector of the point P with respect to O. The angles made by with positive direction of x, y & z-axes (say ⍺, β & γ respectively) are called its direction angles, and the cosine value of these angles i.e., cos ⍺, cos β & cos γ are called direction cosines of , denoted by l, m & n respectively. 3. Types of Vectors Vector Algebra 1. Zero Vector: A vector whose initial and terminal 4. Addition of Vector points coincide, is called a zero vector (or null vector) denoted as It has zero magnitude 2. Unit Vector: A vector whose magnitude is unity (i.e., 1 unit) is called unit vector. The unit vector in the direction of is denoted as. 3. Coinitial Vectors: Two or more vectors having the same initial point are called coinitial vectors. 4. Collinear Vectors: Two or more vectors are called collinear, if they are parallel to the same line, irrespective of their magnitude. 5. Equal Vectors: Two vectors are said to be equal, if they have same magnitude & direction regardless of the position of their initial points. 6. Negative of a vector: A vector whose magnitude is the same as that of the given vector, but the direction is opposite to that of t, is called negative of the given vector. 5. Multiplication of a Vector by a Scalar Vector Algebra 6. Component of Vector 7. Vector Joining two Points Vector Algebra 8. Section Formulae 9. Scalar (or dot) Product of two Vectors 11. Vector (or Cross) Product of two Vectors Vector Algebra 10. Projection of Vector Along a Directed Line 12. Properties Regarding Scalar and Vector Product Maths : Important topic with PYQ Types of Problems Type 1 Product of Vectors  Maths : Important topic with PYQ JEE Main 2024 | 1st Feb S1 4 MCQ Type A -12 B -10 C -13 D -15  Maths : Important topic with PYQ JEE Main 2024 | 1st Feb S1 4 MCQ Type A -12 B -10 C -13 D -15  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 2 Angle between two vectors  Maths : Important topic with PYQ JEE Main 2024 | 30th Jan S2 5 8 MCQ Type A 85 B 90 C 75 D 95  Maths : Important topic with PYQ JEE Main 2024 | 30th Jan S2 5 8 MCQ Type A 85 B 90 C 75 D 95  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 3 Projection of Vector  Maths : Important topic with PYQ JEE Main 2022 | July 6 13 Let If the projection of on the vector is 30, then MCQ Type 𝛼 is equal to: A 15/2 B 8 C 13/2 D 7  Maths : Important topic with PYQ JEE Main 2022 | July 6 13 Let If the projection of on the vector is 30, then MCQ Type 𝛼 is equal to: A 15/2 B 8 C 13/2 D 7  Maths : Important topic with PYQ Solution:  1. Direction Cosines of a Line (DC’s) 3D - Geometry The direction cosines are generally denoted by l, m, n. Hence, l = cos ⍺, m = cos β, n = cos γ Note that l2 + m2 + n2 = 1 3. Equation of a Line 2. Direction Ratios of a Line (DR’s) 1. Equation of a line through a given point with Any three numbers a, b and c proportional to the position vector and parallel to a given vector : direction cosines l, m and n, respectively are called In vector form, direction ratios of the line. In cartesian form, The direction ratios of a line passing through two points P(x1, y1, z1) and Q(x2, y2, z2) are (x2 - x1), (y2 - y1), (z2 - z1) where, Here, a, b, c are also the direction ratios of the line 2. Equation of a line passing through two given points with position vectors In vector form, In cartesian form,  4. Angle between Two Lines 3D - Geometry 5. Shortest Distance between Two Lines In vector form, 1. Distance between Parallel Lines The angle between two lines The shortest distance parallel lines 2. Distance between Two Skew Lines In cartesian form, In vector form, The angle between two lines: The distance between two skew lines In cartesian form, The distance between two skew lines cos θ = |l1l2 + m1m2 + n1n2| If two lines are perpendicular, then If two lines are parallel, then Maths : Important topic with PYQ Types of Problems Type 1 Shortest distance between lines  Maths : Important topic with PYQ JEE Main 2024 | 8th April S2 7 If the shortest distance between the lines MCQ Type then a value of λ is : A 13/25 B 1 C -1 D -13/25  Maths : Important topic with PYQ JEE Main 2024 | 8th April S2 7 If the shortest distance between the lines MCQ Type then a value of λ is : A 13/25 B 1 C -1 D -13/25  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 2 Image of Point  Maths : Important topic with PYQ JEE Main 2021 | 24th Feb S2 8 Let a, b ∈ R. If the mirror image of the point P(a, 6, 9) with respect to the line MCQ Type is (20, b, -a-9), then |a + b|, is equal to: A 88 B 90 C 86 D 84  Maths : Important topic with PYQ JEE Main 2021 | 24th Feb S2 8 Let a, b ∈ R. If the mirror image of the point P(a, 6, 9) with respect to the line MCQ Type is (20, b, -a-9), then |a + b|, is equal to: A 88 B 90 C 86 D 84  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 3 Foot of the perpendicular from a point  Maths : Important topic with PYQ JEE Main 2024 | 31st Jan S1 9 Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x = y, z = 1 and x = -1, z = -1 Integer Type respectively. If ∠QPR is a right angle, then 12a2 is equal to ______.  Maths : Important topic with PYQ JEE Main 2024 | 31st Jan S1 9 Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x = y, z = 1 and x = -1, z = -1 Integer Type respectively. If ∠QPR is a right angle, then 12a2 is equal to ______. Correct Answer : 12  Maths : Important topic with PYQ Solution:  Definition Null Set A set is a well-defined A set which doesn’t contain collection of objects any element is called an empty or null or void set Finite Set & Infinite Set Equal Sets Types of A set which is empty or consists of a definite Sets Two sets A and B are said number of elements is to be equal if they have called finite otherwise, the exactly the same elements set is called infinite and we write A = B Power Sets Singleton Set Subset A set which is said to be a The set of all subsets of a set A is If a set A has only one element, subset of a set B if every element of A is also an called the power set of A. It is we call it a singleton set. Set of element of B denoted by P(A). In P(A), every all even prime numbers-(2) is a element is a set singleton set  De-Morgan’s Law Complement of a Set Intersection of Sets Let U be the universal set and A a subset De-Morgan’s Laws: of U. Then the complement of A is the set (i) (A ∪ B)’ = A’ ∩ B’ of all elements of U which are noth the The intersection of two sets (ii) (A ∩ B”’ = A’ ∪ B’ elements of A. symbolically, we write A’ A and B is the set of all to denote the complement of A with those elements which respect to U. Thus, belong to both A and B. A’ = {x : x ∈ U and x ∉ A} Symbolically, we write So A’ = U - A A ∩ B = {x : x ∈ A and x ∈ B} Operations on Sets Difference Union of Sets of Sets Let A and B any two sets. The union of A The difference of the sets A and B is the set which consists of all the and B in this order is the set elements of A and all the elements of B, of elements which belong to A the common elements being taken only but not to B. Symbolically, we once. The symbol ‘U’ is used to denote write A-B and read as “A the union. Symbolically, we write A to B animus B” and usually read as ‘A and B’  Types of Relations ANTI SYMMETRIC RELATION If (a, b) ∈ R and (b, a) ∈ R, for all a, b ∈ A, then a=b EMPTY RELATION TRANSITIVE RELATION Relation R in Set A is Empty if no element of A is related to any element of A ; R = ϕ ⊂ A × A (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A UNIVERSAL RELATION EQUIVALENCE RELATION When each element of A is related to every element of A in a relation R ; R = A × A A relation R in A when R is reflexive, Symmetric & Transitive SYMMETRIC RELATION EQUIVALENCE CLASS If (a, b) ∈ R, then (b, a) ∈ R, for all a, b ∈ A All subsets of equivalent relation All elements of Ai are related to each REFLEXIVE RELATION other, for all i No element of Ai is related to any element (a, a) ∈ R, for every a ∈ A of Aj, for all i ≠ J ϕ U Ai = A and Ai ∩ Aj = ϕ ; i ≠ j  Types of Relations IRREFLEXIVE R is irreflexive iff ∀ a ∈ ((a, a) ∉ R) IDENTITY RELATION PARTIAL ORDER RELATION A relation in which each element is related to itself only. I = {(a, a), a ∈ A} R is a partial order, if R is Reflexive, Antisymmetric and Transitive. INVERSE RELATION ASYMMETRIC RELATION Inverse relation of R from A to B, denoted by R-1, is a relation from B to A is defined by (x, y) ∈ R ⇒ (y, x) ∉ R R-1 = {(b, a), : (a, b) ∈ R}.  Properties Counting of Relation 1. R is not reflexive does not imply R is irreflexive. Number of relations from set A to B = 2mn, where |A| = m, Counter example: |B| = n A = {1, 2, 3}, R = {(1, 1)} Number of identity relation on a set with ‘n’ elements = 1 2. R is asymmetric implies that R is irreflexive. By definition, for all a, b ∈ A, Number of reflexive relation set on a set with ‘n’ elements = (a, b), ∈ R and (b, a) ∉ R. This implies that for all 2n(n-1) (a, b) ∈ R, a ≠ b Thus, for all a ∈ A, Number of symmetric relation set on a set with ‘n’ elements = (a, a) ∉ R Therefore, R is irreflexive. 2n(n+1)/2 3. R is not symmetric does not imply r is The number of antisymmetric binary relations possible on A antisymmetric counter example: is A = {1, 2, 3}, R = {(1, 2), (2, 3), (3, 2)} The number of binary relation on A which are both symmetric 4. R is not symmetric does not imply R is and antisymmetric is 2n. asymmetric. Counter example: The number of binary relation on A which are both symmetric A = {1, 2, 3}, R = {(1, 2), (2, 2)} and asymmetric is 1. 5. R is not antisymmetric does not imply R is The number of binary relation which are both relation which symmetric. Counter example: are both reflexive and antisymmetric on the set Aa is A = {1, 2, 3}, R = {(1, 2), (2, 3), (3, 2)} The number of asymmetric binary relation possible on the set 6. R is reflexive implies that R is not asymmetric. By A is definition, for all a ∈ A, (a, a) ∈ R. There are at least 2n transitive relations (lower bound) and at This implies that, both (a, b) and (b, a) are in R most (upper bound) when a = b. Thus, R is not asymmetric.  Operation on Relations 3. R1 ∪ R2 = {(a, b) | (a, b) ∈ R1 and (a, b) ∈ 1. R1 - R2 = {(a, b) | (a, b) ∈ R1 and (a, b) ∉ R2} R2} 2. R2 - R1 = {(a, b) | (a, b) ∈ R2 and (a, b) ∉ R1} 4. R1 ∩ R2 = {(a, b) | (a, b) ∈ R1 and (a, b) ∈ R2} Properties 1) If R1 and R2 are reflexive and symmetric, then R1 ∪ R2 4) If R1 and R2 are equivalence relations on A, is reflexive, and symmetric. R1 - R2is not an equivalence relation (reflexivity fails). 2) If R1 is transitive and R2 is transitive, then R1 ∪ R2 R1 - R2 is not a partial order (since R1 - R2 is not need not be transitive. reflexive). Counter example: Let A = {1, 2} such that R1 ⨁ R2 = R1 ∪ R2 - (R1 ∩ R2) is neither R1 = {(1, 2)} and R2 = {(2, 1)}. R1 ∪ R2 = {(1, 2), (2, 1)} equivalence relation nor partial order (reflexivity and (1, 1) ∉ R1 ∪ R2 implies that R1 ∪ R2 is not fails) Transitive. 5) The union of two equivalence relation on a set is not 3) If R1 and R2 are equivalence relations, then R1 ∩ R2 is necessarily an equivalence relation on the set. an equivalence relation. 6) The inverse of a equivalence relation R is an equivalence relation.  Composition of Relations Equivalence Class Let R1 ⊆ A × B and R2 ⊆ B × C, Composition of R2 on Equivalence class of a ∈ A is defined as [a] = {x | (x, a) ∈ R}, that R1, denoted as R1 R2 or simply R1 R2 is is all the elements related to a under the relation R. R1 R2 = {(a, c) | a ∈ A, c ∈ C ^ ∃ b ∈ B such that Example ((a, b) ∈ R1, (b, c) ∈ R2)} E = Even integers, O = Odd integers NOTE (i) All elements of E are related to each other and all elements of R1 (R2 ∩ R3) ⊂ R1 R2 ∩ R1 R3 O are related to each other. R1 (R2 ∪ R3) = R1 R2 ∪ R1 R3 (ii) No element of E is related to any element of O and vice-versa. R1 ⊆ A × B, R2 ⊆ B × C, R3 ⊆ C × D.(R1R2)R3 = R1 (R2R3) (iii) E and O are disjoint and Z = E ∪ O -1 (R1oR2) = R2-1 oR1-1 The subset E is called the equivalence class containing zero and is denoted by. Properties: Consider an equivalence relation R defined on a set A. 4. For any two equivalence class [a] and [b], either [a] = [b] or [a] ∩ [b] = ϕ 2. For every a, b ∈ A such that a ∈ [b], a ∉ b it follows 5. For all a, b ∈ A, if a ∈ [b] then b ∈ [a] that [a] = [b] 6. For all a, b, c ∈ A, if a ∈ [b] and b ∈ [c], then a ∈ [c] 7. For all a ∈ A, [a] ≠ ϕ  Classification of Functions 1. Constant Function Functions f(x) = k, k is a constant. 2. Identity Function The function y = f(x) = x, ∀ x ∈ R Logarithmic Exponential Here domain & Range both R function Function 3. Polynomial function y = f(x) = a0 xn + a1 xn-1 + … + an n is non negative integer, a1 are real constants. Given a0 ≠ 0, n is the degree of polynomial f(x) = logax[a > 0, a ≠ 1] f(x) = ax, a > 0, a ≠ 1. function There are two polynomial functions, f(x) = 1 + xn & f(x) = 1 - xn satisfying the relation: f(x) ⋅ f(1/x) - f(x) + f(1/x) where ‘n’ is a positive integer. 4. Rational Function It is defined as the ratio of two polynomials. Domain = (0, ∞), Range = R Domain = R, Rang = (0, ∞) f(x) = P(x)/Q(x) provided Q(x) ≠ 0 Dom {f(x)} is all real numbers except when denominator is zero [i.e., Q(x) ≠ 0]  Properties of Log Functions Functions Logarithmic function 5. 6. 1. loga(xy) = loga|x| + loga|y|, where a > 0, a ≠ 1 and xy > 0 7. If a > 1, then the values of f(x) = logax increase with the 2. increase in x. I.e. x < y ⇔ loga x < loga y 3. Also, 4. logn(xn) = n loga |x|, where a > 0, a ≠ 1 and xn > 0 Functions Sine function Tangent function Cosecant function Trigonometric Functions f(x) = sin x f(x) = tan x f(x) = cosec x Dom (f) = R Dom (f) = R - {(2n + I)π/2, n∈ z Dom (f) = R - {mR, n e z} Ran (f) = [-1, 1] Ran (f) = R Ran (f) = R - (-1, 1) Cosine function Secant function Cotangent function f(x) = cos x f(x) = sec x f(x) = cot x Dom (f) = R Dom (s) = R - {(2n + 1)π/2|n∈Z} Dom (f) = R - {nπ | n e z} Ran (f) = [-1, 1] Ran (f) = R - (-1, 1) Ran (f) = R  Absolute Value Functions Function 1. |x|2 = x2 2. √x2 = |x| 3. |x| = max {-x, x} 4. -|x| = min {-x, x} 5. 6. 7. |x + y| ≤ |x| + |y| 8. |x + y| = |x| + |y| if xy > 0 9. |x - y| = |x| + |y| if xy ≤ 0 10. |x| ≥ a (is -ve) x ∈ R 11. a < |x|< b ⇒ b ≤ x ≤ -a or a ≤ x ≤ b. x ∈ [-b, -a] ∪ [a, b] Functions Signum Function Greatest Integer f(x) = [x] the integral part of x, which Function is nearest & smaller integer 1. [x] ≤ x < [x] + 1 2. x - 1 < [x] < x 6. [x] ≤ n ⇔ x < n + 1, n ∈ 7. [x] < n ⇔ x < n 8. I 3. I ≤ x < I + I ⇒ [x] = I 9. 10. [x] + [y] ≤ [x + y] ≤ [x] + [y] + 1 4. 11. 5.  Fractional Part Odd and Even Functions Function Function 1. if f(-x) =-f(x) ∀ x ∈R then f is an odd function, odd y = {x} fractional past of x. functions are symmetrical in opposite quadrants or y = {x} = x - [x] about origin. 2. If f(-x) = f(x). then even. It is symmetric about y axis. 1. {x} = x, 0 ≤ x 0 and directrix x = -a is The equation of ellipse with 'foci' on the x-axis is x²/a2 + y²/b2 = 1 y² = 4ax, where 4a is the length of the Length of the latus rectum of the ellipse x /a + y /b = 1 is 2b2/a 2 2 2 2 latus rectum The eccentricity of an ellipse is the ratio of distances from centre of ellipse to one of foci and to one of the vertices of ellipse i.e., e = c/a Maths : Important topic with PYQ Types of Problems Type 1 Standard Equations of Circle  Maths : Important topic with PYQ JEE Main 2024 | 6th April S2 46 If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, -2), B(8, 3) and C(h, k). Then the point C MCQ Type lies on the circle: A x2 + y2 - 74 = 0 B x2 + y2 - 65 = 0 C x2 + y2 - 61 = 0 D x2 + y2 - 52 = 0  Maths : Important topic with PYQ JEE Main 2024 | 6th April S2 46 If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, -2), B(8, 3) and C(h, k). Then the point C MCQ Type lies on the circle: A x2 + y2 - 74 = 0 B x2 + y2 - 65 = 0 C x2 + y2 - 61 = 0 D x2 + y2 - 52 = 0  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 2 Positions of Two Circles  Maths : Important topic with PYQ JEE Main 2024 | 8th April S1 47 Let the circles C1 (x - ⍺)2 + (y - β)2 = r12 and C2 : (x - 8) 2 + (y - 15/2)2 = r22 touch each other externally MCQ Type at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles C₁ and C2 internally in the ratio 2 : 1, then (⍺ + β) + 4(r12 + r22) equals A 130 B 110 C 145 D 125  Maths : Important topic with PYQ JEE Main 2024 | 8th April S1 47 Let the circles C1 (x - ⍺)2 + (y - β)2 = r12 and C2 : (x - 8) 2 + (y - 15/2)2 = r22 touch each other externally MCQ Type at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles C₁ and C2 internally in the ratio 2 : 1, then (⍺ + β) + 4(r12 + r22) equals A 130 B 110 C 145 D 125  Maths : Important topic with PYQ Solution:  Maths : Important topic with PYQ Solution: Maths : Important topic with PYQ Types of Problems Type 3 Intersection of Circle & Line Problems  Maths : Important topic with PYQ JEE Main 2024 | 4th April S2 48 Let C be a circle with radius √10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and MCQ Type slope -1. Then, a distance (in units) between the chord PQ and the chord MN is A 3 - √2 B 2 - √3 C √2 - 1 D √2 + 1  Maths : Important topic with PYQ JEE Main 2024 | 4th April S2 48 Let C be a circle with radius √10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and MCQ Type slope -1. Then, a distance (in units) between the chord PQ and the chord MN is A 3 - √2 B 2 - √3 C √2 - 1 D √2 + 1  Maths : Important topic with PYQ Solution: Let the line by x + y = λ  Properties Permutations Restricted Permutations The number of ways in which r objects can be arranged from n dissimilar objects if k particular objects are Always included (or never excluded) = n-kCr-k r! = rPk n-kPr-k. Always excluded (never included) = n-kCr r! = n-kPr Distributions into Groups Distribution of n distinct things into r groups G1, G2, …….., Gr containing P1, P2, ….., Pr elements respectively. Permutations Arranging r objects out of n difference things Groups are distinct: When repetition is not allowed = Groups are identical: where 0 ≤ r ≤ n When repetition is allowed = nr Factorial Notation Product of first n natural numbers is denoted by n! i.e., n! = n(n - 1)(n - 2) … 3⋅2⋅1 De-arrangements Any change in the existing order of things is called Combinations Combinations De-arrangement. If m things are arranged in a row, the Selecting r objects out of n number of ways in which they can be deranged so that difference things given by none of them occupies its original place (no one of them Properties occupies the place assigned to it) n Pr = nCr r!, 0 ≤ r ≤ n For 0 ≤ r ≤ n, nCr = nCn-r For 1 ≤ r ≤ n, nCr + nCr-1 = n+1Cr n Ca = nCb ⇒ a = b or n = a + b Fundamental Principle of Counting n In an operation A can be performed C0 + nC1 + … + nCn = 2n in m difference ways and another operation B can be performed in n difference ways. Then Circular Permutations Both the operations can be Restricted combinations (i) Arrangement of an difference things The number of ways in which r objects can be selected performed in m × n ways. taken all at a time in form of circle is from n dissimilar objects if k particular objects are Either of the two operations ➔ (n - 1)!, if sense matter. can be performed in (m + n) Always included = n-kCr-k = n-kCn--r ➔ ½(n - 1)!, if sense doesn’t matter ways. n-k Never included (Always excluded) = Cr (ii) Number of circular permutations of n dissimilar things taken r at a time = nPr/r if clockwise and anticlockwise orders are considered as difference. = nPr/2r if clockwise and anticlockwise order is considered as same. Maths : Important topic with PYQ Types of Problems Type 1 Rank/Dictionary Problem  Maths : Important topic with PYQ JEE Main 2024 | 29th Jan S1 49 All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words Integer Type are written as in a dictionary. The serial number of the word "GTWENTY" is _________.  Maths : Important topic with PYQ JEE Main 2024 | 29th Jan S1 49 All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words Integer Type are written as in a dictionary. The serial number of the word "GTWENTY" is _________. Correct Answer: 553  Maths : Important topic with PYQ Solution: Words starting with E = 360 Words starting with GE = 60 Words starting with GN = 60 Words starting with GTE = 24 Words starting with GTN = 24 Words starting with GTT = 24 GTWENTY =1 Total 553 Maths : Important topic with PYQ Types of Problems Type 4 Selection-Arrangement Problems  Maths : Important topic with PYQ JEE Main 2024 | 30t Jan S2 50 In an examination of Mathematics paper, there are 20 Integer Type questions of equal marks and the question paper is divided into three sections : A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is __________.  Maths : Important topic with PYQ JEE Main 2024 | 30t Jan S2 50 In an examination of Mathematics paper, there are 20 Integer Type questions of equal marks and the question paper is divided into three sections : A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is __________. Correct Answer : 11376  Maths : Important topic with PYQ Solution: If 4 questions from each section are selected Remaining 3 questions can be selected either in (1, 1, 1) or (3, 0, 0) or (2, 1, 0) ∴ Total ways = 8C5 ⋅ 6C5 ⋅ 6C5 + 8C6 6C5 ⋅ 6C4 × 2 + 8 C5 ⋅ 6C6 ⋅ 6C4 × 2 +8C4 ⋅ 6C6 ⋅ 6C5 × 2 + 8C7 ⋅ 6C4 ⋅ 6C4 = 56. 6. 6 + 28. 6. 15. 2 + 56. 15. 2 + 70. 6. 2 + 8. 15. 15 = 2016 + 5040 + 1680 + 840 + 1800 = 11376 Join HP Gurukul :- Click Here