Chapter 1 Relations and Functions

1. Definition and representations of a relation:

- A relation R is defined as a set of ordered pairs (x, y) where x and y belong to sets A and B respectivel y.

- Representation: R = {(x, y) | x belongs to A and y belongs to B}

2. Types of Relations:

a) Reflexive relation: A relation R is reflexive if (a, a) exists for every element a in set A. Example: R = {(a, a), (b, b), (c, c)}

b) Symmetric relation: A relation R is symmetric if (a, b) exists implies (b, a) also exists for every (a, b) i n set A.

Example: R = {(a, b), (b, a), (c, c)}

c) Transitive relation: A relation R is transitive if (a, b) exists and (b, c) exists implies (a, c) also exists for every (a, b, c) in set A.

Example: R = {(a, b), (b, c), (a, c)}

d) Equivalence relation: A relation R is an equivalence relation if it is reflexive, symmetric, and transitive .

Example: R = {(a, a), (b, b), (c, c), (a, b), (b, a)}

e) Partial order relation: A relation R is a partial order relation if it is reflexive, anti-symmetric, and transit ive.

Example: R = {(a, a), (b, b), (c, c), (a, b), (a, c)}

3. Functions:

- A function f from set A to set B is a relation that maps every element in A to a unique element in B.

- Representation: f: A → B, (x, f(x))

- Domain: Set of all possible input values (x) of a function.

- Co-domain: Set of all possible output values of a function.

- Range: Set of all actual output values of a function.

4. Types of Functions:

a) One-to-One (Injective) function: A function f is one-to-one if each element in the domain maps to a un ique element in the co-domain. Example: f(x) = x + 1

b) Onto (Surjective) function: A function f is onto if every element in the co-domain has a corresponding element in the domain.

Example: f(x) = x^2

c) One-to-One Correspondence (Bijective) function: A function f is a one-to-one correspondence if it is b oth one-to-one and onto.

Example: f(x) = 2x

d) Identity function: A function f(x) = x, where the output is the same as the input. Example: f(x) = x

e) Constant function: A function f(x) = c, where the output is a constant value, regardless of the input. Example: f(x) = 3

Chapter 2 inverse trigonometric functions

1. Inverse Sine function: sin^(-1)(x)

2. Inverse Cosine function: cos^(-1)(x)

3. Inverse Tangent function: tan^(-1)(x)

4. Inverse Cotangent function: cot^(-1)(x)

5. Inverse Secant function: sec^(-1)(x)

6. Inverse Cosecant function: csc^(-1)(x)

The caret (^) symbol is used to denote the power or exponent of a function.

Chapter 3 matrices

1. Addition of Matrices:

If A and B are two matrices of the same order m x n, then their sum A + B is a matrix of the same order, where each element (i, j) of the sum matrix is obtained by adding the corresponding elements (i, j) of the matrices A and B.

2. Scalar Multiplication:

If A is a matrix of order m x n and k is a scalar value, then the scalar multiplication kA is a matrix of the s ame order m x n, where each element (i, j) of the resulting matrix is obtained by multiplying the element (i,

j) of the matrix A by scalar k.

3. Subtraction of Matrices:

If A and B are two matrices of the same order m x n, then their difference A - B is a matrix of the same o rder, where each element (i, j) of the difference matrix is obtained by subtracting the corresponding eleme nts (i, j) of the matrices A and B.

4. Multiplication of Matrices:

If A is a matrix of order m x n and B is a matrix of order n x p, then their product AB is a matrix of order m x p, where each element (i, j) of the resulting matrix is obtained by multiplying the elements of the ith ro w of matrix A with the corresponding elements of the jth column of matrix B and summing them.

5. Transpose of a Matrix:

If A is a matrix of order m x n, then its transpose A^T is a matrix of order n x m, where each element (i, j) of the transpose matrix is obtained by swapping the rows and columns of the element (j, i) of the original matrix.

6. Determinant of a Matrix:

If A is a square matrix of order n x n, then the determinant |A| of matrix A can be calculated using variou s methods such as expansion by minors, cofactor expansion, or row reduction.

7. Inverse of a Matrix:

If A is a square matrix of order n x n and |A| ≠ 0, then its inverse A^(-1) exists. The inverse of a matrix A can be found using methods like the adjoint method or elementary row operations.

Chapter 4 determinants

1. Determinants of 2x2 matrices:

a. Given a 2x2 matrix A = [[a, b], [c, d]], the determinant is given by: det(A) = ad - bc

2. Determinants of 3x3 matrices:

a. Given a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]], the determinant is given by: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

3. Properties of determinants:

a. If A is a square matrix and B is obtained from A by interchanging any two rows (or two columns), then det(B) = -det(A).

b. If A is a square matrix and B is obtained from A by multiplying any row (or column) by a scalar k, then det(B) = k * det(A).

c. If A is a square matrix and B is obtained from A by adding a scalar multiple of one row (or column) to another row (or column), then det(B) = det(A).

4. Cramer's Rule:

a. Cramer's Rule states that given a system of linear equations Ax = b, where A is a square matrix and x , b are column vectors, the solution can be expressed as: x = (A^(-1)) * b where A^(-1) is the inverse of matrix A.

5. Adjoint and Inverse of a matrix:

a. The adjoint of a matrix A is denoted as adj(A) and is obtained by taking the transpose of the cofactor matrix of A. b. The inverse of a matrix A, denoted as A^(-1), is given by: A^(-1) = (1 / det(A)) * adj(A)

6. Finding the inverse of a matrix:

a. To find the inverse of a 2x2 matrix A = [[a, b], [c, d]], the formula is:

A^(-1) = (1 / (ad - bc)) * [[d, -b], [-c, a]]

b. To find the inverse of a 3x3 matrix A, the formula involves finding the adjoint matrix and dividing it by t he determinant of A.

Chapter 5 continuity and differentiability

1. Definition of Continuity:

A function f(x) is said to be continuous at a point x = c if the following three conditions are satisfied:

- f(c) is defined.

- \lim_{x \to c} f(x) exists.

- \lim_{x \to c} f(x) = f(c)

2. Continuity on an Interval:

A function f(x) is said to be continuous on an interval (a, b) if it is continuous at every point in the interval. Similarly, f(x) is said to be continuous on a closed interval [a, b] if it is continuous at every point in the inter val as well as at the endpoints a and b.

3. Algebra of Continuous Functions:

If f(x) and g(x) are both continuous functions at x = c, then the following functions are also continuous at x = c:

- (f(x) + g(x))

- (f(x) - g(x))

- (f(x) * g(x))

- (f(x)/g(x)) (if g(c) ≠ 0)

4. Composite Functions and Continuity:

If f(x) is continuous at x = c and g(x) is continuous at x = f(c), then the composite function (g f)(x) = g(f(x)) is continuous at x = c. 5. Differentiability: A function f(x) is said to be differentiable at x = c if the following limit exists:

- \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}

6. Differentiability and Continuity:

If a function f(x) is differentiable at x = c, then it must be continuous at x = c. However, if a function is conti nuous at x = c, it may or may not be differentiable at x = c.

7. Differentiation Rules:

- Constant Rule: \frac{d}{dx} (c) = 0, where c is a constant.

- Power Rule: \frac{d}{dx} (x^n) = nx^{n-1}, where n is a constant.

- Sum/Difference Rule: \frac{d}{dx} (f(x) ± g(x)) = \frac{d}{dx} (f(x)) ± \frac{d}{dx} (g(x))

- Product Rule: \frac{d}{dx} (f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)

- Quotient Rule: \frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) * g(x) - f(x) * g'(x)}{(g(x))^2} (if g(x) ≠ 0)

8. Chain Rule:

If y = f(g(x)), then \frac{dy}{dx} = \frac{dy}{dg} * \frac{dg}{dx}

9. Rolle's Theorem:

If a function f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0.

10. Mean Value Theorem:

If a function f(x) is continuous on the interval [a, b], differentiable on the interval (a, b), then there exists at least one point c in (a, b) such that \frac{f(b) - f(a)}{b - a} = f'(c).

Chapter 6 application of derivatives

1. Rate of change: If y is a function of x, then the rate of change of y with respect to x is given by dy/dx.

2. Differentiation of basic functions:

a) d/dx (k) = 0 (where k is a constant)

b) d/dx (x^n) = nx^(n-1) (for any real number n)

c) d/dx (e^x) = e^x

d) d/dx (ln(x)) = 1/x

e) d/dx (sin(x)) = cos(x)

f) d/dx (cos(x)) = -sin(x)

3. Differentiation rules:

a) Sum/Difference Rule: d/dx (f(x) ± g(x)) = f'(x) ± g'(x)

b) Product Rule: d/dx (f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)

c) Quotient Rule: d/dx (f(x) / g(x)) = (f'(x) * g(x) - f(x) * g'(x)) / g(x)^2

d) Chain Rule: d/dx (f(g(x))) = f'(g(x)) * g'(x)

4. Tangents, normals, and derivatives:

a) Equation of a tangent at point x=a: y - f(a) = f'(a) * (x - a)

b) Equation of a normal at point x=a: y - f(a) = -1/f'(a) * (x - a)

5. Increasing and decreasing functions:

a) If f'(x)>0 for all x in an interval, then f(x) is increasing in that interval.

6. Points of inflection:

a) A point of inflection occurs at x=c if the concavity of a function changes at x=c.

b) If f''(c) = 0 and f''(x) changes sign at x=c, then x=c is a point of inflection.

Chapter 7 integrals

1. Fundamental theorem of calculus: ∫(a to b) f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).

2. Integration by parts: ∫ u dv = uv - ∫ v du

3. Integration of algebraic functions:

- ∫ x^n dx = (x^(n+1))/(n+1) + C, (n ≠ -1)

- ∫ 1/x dx = ln|x| + C

- ∫ a^x dx = (a^x)/(ln(a)) + C, (a > 0, a ≠ 1)

- ∫ e^x dx = e^x + C

- ∫ sin(x) dx = -cos(x) + C

- ∫ cos(x) dx = sin(x) + C

- ∫ sec^2(x) dx = tan(x) + C

- ∫ csc^2(x) dx = -cot(x) + C

4. Integration of trigonometric functions:

- ∫ sin^n(x) cos^m(x) dx = (-1)^m (cos^(m-1)(x))/(m-1) sin^(n+1)(x) + ∫ sin^(n+2)(x) cos^(m-2)(x) dx, (n, m > 0)

- ∫ cos^n(x) sin^m(x) dx = (cos^(n+1)(x))/(n+1) (-1)^m sin^(m-1)(x) + ∫ cos^(n-2)(x) sin^(m+2)(x) dx, (n, m > 0)

5. Integration of exponential and logarithmic functions:

- ∫ (1/x) dx = ln|x| + C

- ∫ e^x dx = e^x + C

- ∫ a^x dx = (a^x)/(ln(a)) + C, (a > 0, a ≠ 1)

- ∫ log_a(x) dx = (x ln(x) - x)/(ln(a)) + C, (a > 0, a ≠ 1)

6. Integration of inverse trigonometric functions:

- ∫(1/(a^2-x^2)) dx = (1/a) arcsin(x/a) + C, (a ≠ 0)

- ∫(1/(x^2-a^2)) dx = (1/2a) ln|(x-a)/(x+a)| + C, (a ≠ 0)

- ∫(1/(x^2+a^2)) dx = (1/a) arctan(x/a) + C, (a ≠ 0)

Chapter 8 application of integrals

1. Area between curves: If y=f(x) and y=g(x) have continuous derivatives on [a, b] and g(x) ≤ f(x) for all x i n [a, b], then the area A between the curves is given by:

A = ∫[a, b] (f(x) - g(x)) dx

2. Area bounded by a curve, x-axis, and two ordinates: If x=a and x=b are two ordinates (where b > a) an d y=f(x) is a continuous function on [a, b], then the area A bounded by the curve, x-axis, and the ordinates is given by:

A = ∫[a, b] |f(x)| dx

3. Area bounded by a curve, y-axis, and two abscissas: If y=c and y=d are two abscissas (where d > c) an d x=g(y) is a continuous function on [c, d], then the area A bounded by the curve, y-axis, and the abscissa s is given by:

A = ∫[c, d] |g(y)| dy

4. Length of a curve: If a curve is represented by y=f(x), where f'(x) exists and is continuous on [a, b], then the length L of the curve on [a, b] is given by:

L = ∫[a, b] √[1 + (f'(x))^2] dx

5. Volume of solid of revolution: If a curve y=f(x) is rotated about the x-axis or y-axis between [a, b], then t he volume V of the resulting solid is given by:

- For rotation about x-axis:

V = π ∫[a, b] (f(x))^2 dx

- For rotation about y-axis:

V = π ∫[c, d] (g(y))^2 dy

6. Length of a plane curve: If a curve is represented by x=g(t) and y=h(t), where g'(t) and h'(t) exist and ar e continuous on [a, b], then the length L of the curve on [a, b] is given by:

L = ∫[a, b] √[(g'(t))^2 + (h'(t))^2] dt

7. Centre of mass/centroid: If a lamina has a density ρ(x, y) at each point (x, y) and is bounded by a curve y=f(x), the x-axis, and two ordinates x=a and x=b, then the coordinates of the centre of mass (x , ) are giv en by:

xx = (1/A) ∫[a, b] x ρ(x, f(x)) √[1 + (f'(x))^2] dx = (1/A) ∫[a, b] (0.5(y)^2) ρ(x, f(x)) √[1 + (f'(x))^2] dx where A is the area of the lamina.

Chapter 9 differential equations

1. Ordinary Differential Equations:

a) First Order Differential Equations:

- Linear Differential Equation: \(\frac{dy}{dx} + P(x)y = Q(x)\)

- Bernoulli’s Differential Equation: \(\frac{dy}{dx} + P(x)y = Q(x)y^n\)

- Exact Differential Equation: \(M(x,y)dx + N(x,y)dy = 0\) where \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\)

- Homogeneous Differential Equation: \(\frac{dy}{dx} = F\left(\frac{y}{x}\right)\)

- Separable Differential Equation: \(\frac{dy}{dx} = f(x)g(y)\)

b) Second Order Linear Differential Equations:

- Homogeneous Equation: \(ay'' + by' + cy = 0\)

- Non-Homogeneous Equation: \(ay'' + by' + cy = g(x)\)

2. Linear Differential Equations:

- Variable Separable: \(y = e^{\int P(x)dx}\left(\int Q(x)e^{-\int P(x)dx}dx + C\right)\)

- Homogeneous Equation: \(y = e^{\int P(x)dx}\int Q(x)e^{-\int P(x)dx}dx\)

- Exact Differential Equation: \(M(x,y)dx + N(x,y)dy = 0\) where \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\)

3. Homogeneous Linear Differential Equations:

- Auxiliary Equation: \(ar^2 + br + c = 0\)

- General Solution: \(y = C_1e^{r_1x} + C_2e^{r_2x}\)

4. Linear Differential Equations with Constant Coefficients:

- Auxiliary Equation: \(ar^2 + br + c = 0\)

- Case 1: \(r_1 \neq r_2\) (Distinct Real Roots): \(y = C_1e^{r_1x} + C_2e^{r_2x}\)

- Case 2: \(r_1 = r_2\) (Repeated Real Roots): \(y = (C_1 + C_2x)e^{r_1x}\)

- Case 3: \(r_1, r_2 = \alpha \pm i \beta\) (Complex Roots): \(y = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin( \beta x))\)

5. Bernoulli’s Differential Equations:

- Convert to Linear Form: \(v = y^{1-n}\)

- Substitute: \(y = v^{1/(1-n)}\)

- Solve the resulting linear equation

Chapter 10 vector algebra

1. Magnitude of a Vector:

|u| = √(u ² + u ² + u ²) where u , u , and u are the components of the vector u.

2. Unit Vector:

If u is a non-zero vector, then the unit vector along u is given by: u = u / |u|

3. Addition and Subtraction of Vectors:

u + v = (u + v ) i + (u + v ) j + (u + v ) k u - v = (u - v ) i + (u - v ) j + (u - v ) k

4. Scalar Multiplication:

au = (a u ) i + (a u ) j + (a u ) k where a is a scalar and u is a vector.

5. Dot Product:

u ∙ v = u v + u v + u v

6. Cross Product:

u x v = (u v - u v ) i - (u v - u v ) j + (u v - u v ) k

7. Angle between Vectors:

cos θ = (u ∙ v) / (|u| |v|)

8. Projection of a Vector:

proj u = (u ∙ x ) xx proj u = (u ∙ ) proj u = (u ∙ ) where xx, , and are unit vectors in the direction of x, , and respectively.

9. Scalar Triple Product:

u ∙ (v x w) = (u x v) ∙ w = (w x u) ∙ v

10. Vector Triple Product:

u x (v x w) = (u ∙ w) v - (u ∙ v) w

Chapter 11 three-dimensional geometry

1. Distance between two points in 3D space:

- If P(x , y , z ) and Q(x , y , z ) are two points, then the distance between them is given by:

d(PQ) = √((x - x )² + (y - y )² + (z - z )²)

2. Section Formula:

- If P(x , y , z ) and Q(x , y , z ) divide a line segment PQ in the ratio m : m internally, then the coordinates of the point of division R(x, y, z) are given by:

x = (m x + m x )/(m + m ), y = (m y + m y )/(m + m ), z = (m z + m z )/(m + m )

- If P(x , y , z ) and Q(x , y , z ) divide a line segment PQ in the ratio m : m externally, then the coordinates o f the point of division R(x, y, z) are given by:

x = (m x - m x )/(m - m ), y = (m y - m y )/(m - m ), z = (m z - m z )/(m

- m )

3. Direction Cosines of a Line:

- If a line makes angles α, β, and γ with the positive X, Y, and Z-axes respectively, then its direction cosi nes are given by:

l = cosα, m = cosβ, n = cosγ

4. Direction Ratios of a Line:

- If a line makes angles α, β, and γ with the positive X, Y, and Z-axes respectively, then its direction ratio s are given by:

l = sinβ sinγ, m = sinγ sinα, n = sinα sinβ

5. Equation of a Line:

- A line passing through a point P(x , y , z ) in the direction of the vector a = (l, m, n) can be expressed as:

(x - x )/l = (y - y )/m = (z - z )/n

6. Distance between a point and a line:

- If P(x , y , z ) is a point and ax + by + cz + d = 0 is the equation of a line, then the distance between the p oint and the line is given by:

dist(P, line) = |ax + by + cz + d|/√(a² + b² + c²)

Chapter 12 linear programming

1. Objective function:

Z = c x + c x + ... + c x

2. Constraints:

a x + a x + ... + a nx ≤ b

a x + a x + ... + a nx ≤ b

... a x + a x + ... + a nx ≤ b

3. Non-negativity constraints:

x ≥ 0, x ≥ 0, ..., x ≥ 0

4. Feasible region:

The set of all possible solutions that satisfy the given constraints.

5. Corner point:

A point on the boundary of the feasible region where two or more constraints intersect.

6. Optimal solution:

The point(s) on the feasible region that maximizes or minimizes the objective function.

7. Graphical method:

A method of solving linear programming problems by graphing the constraints and identifying the feasible region.

8. Simplex method:

An algorithm used to solve linear programming problems by iteratively improving a feasible solution until a n optimal solution is reached.

9. Basic variables:

Variables corresponding to nonzero values in the optimal solution.

10. Slack variables:

Variables introduced to convert inequality constraints into equality constraints during the formulation of lin ear programming problems.

Chapter 13 probability

1. Probability of an event A: P(A) = (number of favorable outcomes for A) / (total number of possible outco mes)

2. Probability of the complementary event A': P(A') = 1 - P(A)

3. Addition Rule: For two mutually exclusive events A and B, P(A or B) = P(A) + P(B)

4. Multiplication Rule for Independent Events: For two independent events A and B, P(A and B) = P(A) * P (B)

5. Conditional Probability: The probability of event A occurring given that event B has already occurred is denoted as P(A|B) and is calculated as P(A|B) = P(A and B) / P(B)

6. Bayes' Theorem: P(A|B) = (P(B|A) * P(A)) / P(B)

7. Law of Total Probability: If B1, B2, ..., Bn are mutually exclusive and exhaustive events, then for any ev ent A, P(A) = P(A|B1) * P(B1) + P(A|B2) * P(B2) + ... + P(A|Bn) * P(Bn)

8. Permutation Formula: P(n, r) = n! / (n - r)!

9. Combination Formula: C(n, r) = n! / (r! * (n - r)!)

10. Independent Events: Two events A and B are independent if and only if P(A and B) = P(A) * P(B)

11. Mutually Exclusive Events: Two events A and B are mutually exclusive if and only if P(A and B) = 0

12. Expected Value: E(X) = Σ (x * P(X=x)) where x represents each possible outcome and P(X=x) represe nts the probability of that outcome occurring

13. Variance: Var(X) = Σ ((x - )^2 * P(X=x)) where x represents each possible outcome, P(X