Mathematics Syllabus | Ioe Entrance

📘 ENGINEERING / ENTRANCE MATHEMATICS NOTES ________________________________________ 1. SET, LOGIC AND FUNCTIONS 1.1 Set, Real Number System, Intervals, Absolute Value, Logic Set A set is a well-defined collection of objects. Real Number System The set of rational and irrational numbers together forms the real number system. Types of Numbers • Natural Numbers • Whole Numbers • Integers • Rational Numbers • Irrational Numbers • Real Numbers ________________________________________ Intervals Interval Type Representation Open Interval $(a,b)$ Closed Interval $[a,b]$ Half Open Interval $(a,b]$, $[a,b)$ ________________________________________ Absolute Value The absolute value represents distance from zero. $$ |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases} $$ ________________________________________ Logic Logic deals with truth values. Connectives Connective Symbol AND (\land) OR $\lor$ NOT $\lnot$ Implication $\to$ Biconditional $\leftrightarrow$ ________________________________________ Laws of Logic Commutative Laws $$p \lor q = q \lor p$$ $$p \land q = q \land p$$ De Morgan’s Laws $$\lnot(p \lor q) = \lnot p \land \lnot q$$ $$\lnot(p \land q) = \lnot p \lor \lnot q$$ ________________________________________ 1.2 FUNCTIONS Function A function maps each input to exactly one output. ________________________________________ Types of Functions Injective Function (One-One) Different inputs have different outputs. Surjective Function (Onto) Every element in codomain has at least one pre-image. Bijective Function Both injective and surjective. ________________________________________ Other Functions • Algebraic Functions • Trigonometric Functions • Exponential Functions • Logarithmic Functions ________________________________________ Composite Function $$(f \circ g)(x) = f(g(x))$$ ________________________________________ Inverse Function $$f^{-1}(x)$$ If: $$f(x) = y$$ then: $$f^{-1}(y) = x$$ ________________________________________ 2. ALGEBRA ________________________________________ 2.1 MATRICES AND DETERMINANTS Matrix A rectangular arrangement of numbers. ________________________________________ Types of Matrices • Row Matrix • Column Matrix • Square Matrix • Identity Matrix • Diagonal Matrix ________________________________________ Determinant $$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$ ________________________________________ Inverse of Matrix For matrix: $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ Inverse: $$ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $$ ________________________________________ 2.2 COMPLEX NUMBERS AND POLYNOMIAL EQUATIONS Complex Number $$z = a + ib$$ where: • $a$ = real part • $b$ = imaginary part ________________________________________ Properties $$i^2 = -1$$ ________________________________________ Polynomial Equation General form: $$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$$ ________________________________________ 2.3 SEQUENCE AND SERIES Arithmetic Progression (AP) $$a_n = a + (n-1)d$$ ________________________________________ Geometric Progression (GP) $$a_n = ar^{n-1}$$ ________________________________________ PERMUTATION AND COMBINATION Permutation $$^n P_r = \frac{n!}{(n-r)!}$$ ________________________________________ Combination $$^n C_r = \frac{n!}{r!(n-r)!}$$ ________________________________________ 2.4 BINOMIAL THEOREM Formula $$ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k $$ ________________________________________ Exponential Series $$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $$ ________________________________________ Logarithmic Series $$ \log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots $$ ________________________________________ 3. TRIGONOMETRY ________________________________________ 3.1 TRIGONOMETRIC EQUATIONS Basic Ratios $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$ ________________________________________ General Solution For: $$\sin x = 0$$ General solution: $$x = n\pi$$ ________________________________________ 3.2 INVERSE TRIGONOMETRIC FUNCTIONS Principal Value Function Principal Value $$(\sin^{-1}x) ([-\pi/2,\pi/2]) (\cos^{-1}x) ([0,\pi]) (\tan^{-1}x) ((-\pi/2,\pi/2)) $$ ________________________________________ 3.3 PROPERTIES OF TRIANGLES Important Centers • In-centre • Circum-centre • Ortho-centre ________________________________________ Law of Sines $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$ ________________________________________ 4. COORDINATE GEOMETRY ________________________________________ 4.1 STRAIGHT LINES Slope Formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ ________________________________________ Equation of Line $$y = mx + c$$ ________________________________________ 4.2 CIRCLES Standard Equation $$x^2 + y^2 + 2gx + 2fy + c = 0$$ ________________________________________ Tangent Equation $$xx_1 + yy_1 = r^2$$ ________________________________________ 4.3 CONIC SECTIONS Parabola $$y^2 = 4ax$$ ________________________________________ Ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ ________________________________________ Hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ ________________________________________ 4.4 COORDINATES IN SPACE Distance Formula $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $$ ________________________________________ 5. CALCULUS ________________________________________ 5.1 LIMIT AND CONTINUITY Limit $$\lim_{x \to a} f(x)$$ ________________________________________ L’Hospital’s Rule If: $$\frac{0}{0} \text{ or } \frac{\infty}{\infty}$$ then: $$ \lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)} $$ ________________________________________ 5.2 DERIVATIVES Basic Formula $$ \frac{d}{dx}(x^n) = nx^{n-1} $$ ________________________________________ Applications • Tangent and Normal • Rate of Change • Maxima and Minima ________________________________________ 5.3 INTEGRATION Standard Integral $$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $$ ________________________________________ Applications • Area under curve • Area between curves ________________________________________ 5.4 DIFFERENTIAL EQUATIONS Definition An equation involving derivatives. ________________________________________ Types • Variable Separable • Homogeneous • Linear Differential Equation • Exact Differential Equation ________________________________________ 6. VECTORS AND THEIR PRODUCTS ________________________________________ 6.1 VECTORS Magnitude $$ |\vec{a}| = \sqrt{x^2 + y^2 + z^2} $$ ________________________________________ Types • Linearly Dependent • Linearly Independent ________________________________________ 6.2 PRODUCT OF VECTORS Scalar Product $$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta $$ ________________________________________ Vector Product $$ \vec{a} \times \vec{b} $$ ________________________________________ Scalar Triple Product $$ \vec{a} \cdot (\vec{b} \times \vec{c}) $$ ________________________________________ 7. STATISTICS AND PROBABILITY ________________________________________ 7.1 MEASURES OF LOCATION Mean $$ \bar{x} = \frac{\sum x}{n} $$ ________________________________________ Dispersion • Variance • Standard Deviation ________________________________________ 7.2 CORRELATION AND REGRESSION Correlation Coefficient $$ r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} $$ ________________________________________ 7.3 PROBABILITY ________________________________________ Probability Probability measures the chance of occurrence of an event. Probability Formula $$ P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} $$ Important Points • Probability always lies between 0 and 1 • Probability of impossible event = 0 • Probability of certain event = 1 Example If a coin is tossed: $$ P(\text{Head}) = \frac{1}{2} $$ ________________________________________ Bayes’ Theorem Formula $$ P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)} $$ Where • $P(A \mid B)$ = Probability of A given B • $P(B \mid A)$ = Probability of B given A • $P(A)$ = Probability of A • $P(B)$ = Probability of B Applications • Medical diagnosis • Statistical prediction • Machine learning • Data analysis ________________________________________ Binomial Distribution Formula $$ P(X = r) = \binom{n}{r} p^r q^{\,n-r} $$ Where • $n$ = Total number of trials • $r$ = Number of successful outcomes • $p$ = Probability of success • $q = 1-p$ = Probability of failure Example For tossing a coin 3 times: $$ P(X = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 $$ ________________________________________ EXAM TIPS • Practice formulas daily • Revise derivations regularly • Solve previous year questions • Focus on weak chapters • Make short revision notes ________________________________________ QUICK REVISION FORMULAS Arithmetic Progression (AP) $$a_n = a + (n-1)d$$ Geometric Progression (GP) $$a_n = ar^{n-1}$$ Derivative Formula $$\frac{d}{dx}(x^n) = nx^{n-1}$$ Integration Formula $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ Probability Formula $$P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$$ Binomial Theorem $$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

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