Junior Intermediate Mathematics Comprehensive Study Guide (NARAYANA GROUP CDF MATERIAL)

Functions and Core Geometrical Concepts Related to Sets A set is defined as a well-defined collection of objects. If A and B are non-empty sets, their Cartesian product, denoted as $A \times B$ (read as A cross B), consists of the set of all ordered pairs $(a,b)$ such that $a \in A$ and $b \in B$. A relation is any subset of this Cartesian product. A function $f: A \rightarrow B$ is a specific type of relation that associates every element in the domain A to a unique element in the co-domain B. If $f(a) = b$, then $b$ is the image of $a$, and $a$ is the pre-image of $b$. The range of as function is the subset of co-domain B containing all images of elements in A, expressed as $Range = \{f(a) : a \in A\}$. A function is an injection (one-one) if $f(a_1) = f(a_2)$ implies $a_1 = a_2$. It is a surjection (onto) if every element in the co-domain B has at least one pre-image in A (meaning the range equals the co-domain). A bijection is a function that is both one-one and onto. Two functions $f$ and $g$ are equal ($f = g$) if they share the same domain and $f(x) = g(x)$ for all $x$ in that domain. A constant function is defined as $f(x) = c$ for all $x$, while an identity function $I_A$ maps every element to itself ($f(x) = x$). If a function is a bijection, its inverse $f^{-1}: B \rightarrow A$ exists such that $f^{-1}(b) = a$ if and only if $f(a) = b$. The composition of functions $f: A \rightarrow B$ and $g: B \rightarrow C$ is denoted $g \circ f: A \rightarrow C$ where $(g \circ f)(a) = g(f(a))$. A function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$. Special algebraic functions include the greatest integer function $[x]$ (the largest integer less than or equal to $x$), the modulus function $|x|$ (defined as $x$ for $x \ge 0$ and $-x$ for $x < 0$), and the signum function $sgn(x)$ (which returns $1$ for $x > 0$, $0$ for $x = 0$, and $-1$ for $x < 0$). Algebraically, $(f \pm g)(x) = f(x) \pm g(x)$, $(fg)(x) = f(x)g(x)$, and $(f/g)(x) = f(x)/g(x)$ provided $g(x) \neq 0$. # Mathematical Induction and Series Summations The principle of mathematical induction involves three stages: the Basis of Induction (showing $P(1)$ is true), the Inductive Hypothesis (assuming $P(k)$ is true for $k \ge 1$), and the Inductive Step (proving $P(k+1)$ is true). Fundamental summation formulas include the sum of the first $n$ natural numbers $\sum n = \frac{n(n+1)}{2}$, the sum of their squares $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$, and the sum of their cubes $\sum n^3 = \frac{n^2(n+1)^2}{4}$. The sum of the first $n$ odd integers is $1+3+5+...+(2n-1) = n^2$, while the sum of the first $n$ even integers is $2+4+6+...+2n = n(n+1)$. In an Arithmetic Progression (A.P.) with first term $a$ and common difference $d$, the $n^{th}$ term is $t_n = a + (n-1)d$ and the sum of the first $n$ terms is $S_n = \frac{n}{2}[2a + (n-1)d]$. In a Geometric Progression (G.P.) with first term $a$ and common ratio $r$, the $n^{th}$ term is $t_n = a.r^{n-1}$. The sum $S_n = \frac{a(r^n - 1)}{r - 1}$ for $r > 1$, and the sum to infinity for $|r| < 1$ is $S_{\infty} = \frac{a}{1 - r}$. # Matrices and Determinants A matrix is an ordered rectangular array of elements. A matrix with $m$ rows and $n$ columns has order $m \times n$. The trace of a square matrix, $Tr(A)$, is the sum of its principal diagonal elements: $a_{11} + a_{22} + a_{33}$. Types of matrices include the diagonal matrix (non-diagonal elements are zero), scalar matrix (diagonal matrix with equal diagonal elements), and unit/identity matrix (diagonal elements are all $1$). Triangular matrices are classified as upper (elements below the diagonal are zero, $a_{ij} = 0$ for $i > j$) or lower (elements above the diagonal are zero, $a_{ij} = 0$ for $i < j$). Two matrices are equal if they have the same order and identical corresponding elements. Matrix transpose $A^T$ is formed by swapping rows and columns. A matrix is symmetric if $A^T = A$ and skew-symmetric if $A^T = -A$ (where diagonal elements must be zero). Determinants distinguish singular matrices ($|A| = 0$) from non-singular ones ($|A| \neq 0$). The rank of a matrix is the maximum order of its non-singular square sub-matrices. Linear systems can be solved using Cramer's Rule where $x = \frac{\Delta_1}{\Delta}$, $y = \frac{\Delta_2}{\Delta}$, and $z = \frac{\Delta_3}{\Delta}$, or via the Matrix Inversion Method ($X = A^{-1}D$). The Adjoint of a matrix is the transpose of its cofactor matrix, and the inverse is calculated as $A^{-1} = \frac{Adj A}{|A|}$. A system is consistent if it has solutions and inconsistent otherwise. # Vector Algebra and Product of Vectors A vector is a directed line segment with magnitude and direction. Two vectors are collinear if $a = tb$ for some scalar $t$. The modulus of a vector $r = xi + yj + zk$ is $|r| = \sqrt{x^2 + y^2 + z^2}$. Direction cosines are defined as $\cos(\alpha) = \frac{x}{|r|}$, $\cos(\beta) = \frac{y}{|r|}$, and $\cos(\gamma) = \frac{z}{|r|}$. The internal section formula for a point P dividing the segment AB in ratio $m:n$ is $P = \frac{mb + na}{m + n}$. The vector equation of a line through point $a$ parallel to $b$ is $r = a + tb$. The scalar (dot) product is defined as $a \cdot b = |a||b|\cos(\theta)$, and vectors are perpendicular if $a \cdot b = 0$. The vector (cross) product is $a \times b = |a||b|\sin(\theta)n$, where $n$ is a unit vector perpendicular to both. The area of a triangle with adjacent sides $AB$ and $AC$ is $\frac{1}{2}|AB \times AC|$. Scalar triple product $[a b c]$ results in the volume of a parallelepiped, while the volume of a tetrahedron is $\frac{1}{6}|[a b c]|$ for co-terminous edges. The shortest distance between skew lines is given by $\frac{|(a-c) \cdot (b \times d)|}{|b \times d|}$. # Trigonometric Ratios and Transformations Standard trigonometric ratios for an angle $\theta$ are $\sin(\theta) = \frac{Opp}{Hyp}$, $\cos(\theta) = \frac{Adj}{Hyp}$, and $\tan(\theta) = \frac{Opp}{Adj}$. Fundamental identities include $\sin^2(\theta) + \cos^2(\theta) = 1$, $\sec^2(\theta) - \tan^2(\theta) = 1$, and $\csc^2(\theta) - \cot^2(\theta) = 1$. Trigonometric values undergo sign changes based on quadrants ($Q_1, Q_2, Q_3, Q_4$): all are positive in $Q_1$, sine/cosecant in $Q_2$, tangent/cotangent in $Q_3$, and cosine/secant in $Q_4$. For angles involving multiples of $90^{\circ}$, functions change (sine to cosine, etc.), whereas they remain the same for $180^{\circ}$ multiples. The range of $a \cos(x) + b \sin(x) + c$ is $[c - \sqrt{a^2+b^2}, c + \sqrt{a^2+b^2}]$. Compound angle formulas include $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$ and $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$. Multiple angle formulas such as $\sin(2A) = 2\sin(A)\cos(A)$ and $\cos(2A) = \cos^2(A) - \sin^2(A) = 2\cos^2(A) - 1$ are central. Transformation formulas convert sums to products: $\sin(C) + \sin(D) = 2\sin(\frac{C+D}{2})\cos(\frac{C-D}{2})$ and $\cos(C) - \cos(D) = -2\sin(\frac{C+D}{2})\sin(\frac{C-D}{2})$. General solutions for equations like $\sin(\theta) = \sin(\alpha)$ are $\theta = n\pi + (-1)^n\alpha$. # Hyperbolic and Inverse Trigonometric Functions Hyperbolic functions are defined using exponentials: $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$. They follow identities similar to circular functions, such as $\cosh^2(x) - \sinh^2(x) = 1$. Inverse hyperbolic functions are expressed logarithmically: $\sinh^{-1}(x) = \log(x + \sqrt{x^2 + 1})$ and $\cosh^{-1}(x) = \log(x + \sqrt{x^2 - 1})$. Inverse trigonometric functions like $\sin^{-1}(x)$ have specific domains and ranges: $\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$ for $x \in [-1, 1]$. Sum and difference formulas for $\tan^{-1}$ vary depending on the product of variables (e.g., $\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}(\frac{x+y}{1-xy})$ if $xy < 1$). # Properties of Triangles In any triangle ABC with sides $a, b, c$ and semi-perimeter $s = \frac{a+b+c}{2}$, the Sine Rule states $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R$, where $R$ is the circumradius. The Cosine Rule provides expressions like $a^2 = b^2 + c^2 - 2bc \cos(A)$. Projection rules state $a = b \cos(C) + c \cos(B)$. The area $\Delta$ of a triangle can be calculated via $\sqrt{s(s-a)(s-b)(s-c)}$, $\frac{abc}{4R}$, or $rs$ (where $r$ is in-radius). Half-angle formulas provide $\sin(\frac{A}{2}) = \sqrt{\frac{(s-b)(s-c)}{bc}}$ and $\tan(\frac{A}{2}) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$. The in-radius $r = 4R \sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2})$. Length of the median through A is $\frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$. # Coordinate Geometry (2D and 3D) In 2D plane coordinate geometry, distance between points is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. The centroid of a triangle is $(\frac{\sum x_i}{3}, \frac{\sum y_i}{3})$. Important points of concurrency include the In-center (distance $r$ from sides), Orthocenter (intersection of altitudes), and Circumcenter (equidistant from vertices). A locus is a set of points satisfying specific geometric conditions. Shifting the origin to $(h,k)$ is a translation where $x = X+h$; rotation through angle $\theta$ transforms coordinates as $X = x\cos(\theta) + y\sin(\theta)$ and $Y = -x\sin(\theta) + y\cos(\theta)$. Straight lines can be represented in various forms: Slope-intercept ($y = mx+c$), Intercept ($\frac{x}{a} + \frac{y}{b} = 1$), and Normal ($x\cos(\alpha) + y\sin(\alpha) = p$). The distance from a point to a line $ax+by+c=0$ is $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$. For 3D geometry, the distance of point P from axes involves square roots of sums of squares (e.g., from x-axis: $\sqrt{y^2+z^2}$). Any line in 3D has direction cosines $(l, m, n)$ where $l^2+m^2+n^2=1$. The equation of a plane through $(x_1, y_1, z_1)$ with normal d.r.'s $(a, b, c)$ is $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$. # Calculus: Limits, Continuity, and Differentiation The limit of function $f(x)$ as $x \rightarrow a$ exists if the left-hand and right-hand limits are equal. Standard limits include $\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$ and $\lim_{x \rightarrow a} \frac{x^n - a^n}{x - a} = n.a^{n-1}$. A function is continuous at $a$ if the limit at $a$ equals $f(a)$. Differentiation follows the First Principle: $f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$. Basic derivatives include $\frac{d}{dx}(x^n) = nx^{n-1}$, $\frac{d}{dx}(\sin(x)) = \cos(x)$, $\frac{d}{dx}(e^x) = e^x$. Composite functions are differentiated via the chain rule: $(f \circ g)'(x) = f'(g(x))g'(x)$. Applications include finding equations of tangents ($y-y_1 = m(x-x_1)$) and normals ($y-y_1 = -\frac{1}{m}(x-x_1)$). Lengths of tangent, normal, sub-tangent, and sub-normal are specifically defined using the slope $m$. Rolle's Theorem and Lagrange's Mean Value Theorem (LMVT) provide existence conditions for points where a derivative equals zero or an average slope. Functions are strictly increasing if $f'(x) > 0$ and decreasing if $f'(x) < 0$. Extremum values (maxima/minima) are identified through stationary points ($f'(x) = 0$) and checked using the second derivative test ($f''(c) < 0$ for maximum, $f''(c) > 0$ for minimum).