HSC Higher Math Short Syllabus for Priva

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Higher Math 1st Paper - Key Formulas for UIU CSE Admission 1. Matrices and Determinants Determinant of $2 \times 2$: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc$ Inverse of a Matrix: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ (Focus on $2 \times 2$ and $3 \times 3$) Cramer's Rule for $AX=B$: $x_i = \frac{\det(A_i)}{\det(A)}$ 2. Vector Magnitude of Vector: If $\vec{v} = (v_1, v_2, v_3)$, then $|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$ Unit Vector: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$ Dot Product: $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y + A_z B_z$ Angle between Vectors: $\cos\theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}$ Cross Product: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$ Area of Parallelogram: $|\vec{A} \times \vec{B}|$ Area of Triangle: $\frac{1}{2}|\vec{A} \times \vec{B}|$ 3. Straight Line Slope: $m = \frac{y_2-y_1}{x_2-x_1}$ Equation (Point-slope): $y - y_1 = m(x - x_1)$ Equation (Intercept form): $\frac{x}{a} + \frac{y}{b} = 1$ Distance from $(x_1, y_1)$ to $Ax+By+C=0$: $D = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$ Condition for Parallel Lines: $m_1 = m_2$ Condition for Perpendicular Lines: $m_1m_2 = -1$ 4. Circle Standard Equation: $(x-h)^2 + (y-k)^2 = r^2$ General Equation: $x^2+y^2+2gx+2fy+c=0$ (center $(-g,-f)$, radius $\sqrt{g^2+f^2-c}$) 5. Permutations and Combinations Permutation: $P(n,r) = \frac{n!}{(n-r)!}$ Combination: $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ 6. Binomial Expansion General Term ($T_{r+1}$): $\binom{n}{r} a^{n-r} x^r$ in $(a+x)^n$ 7. Differential Calculus Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ Trigonometric Derivatives: $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ Exponential/Logarithmic Derivatives: $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(\ln x) = \frac{1}{x}$ Maxima/Minima: $f'(x)=0$ for critical points. $f''(x)>0 \implies \text{min}$, $f''(x) Higher Math 2nd Paper - Key Formulas for UIU CSE Admission 1. Polynomial Equations Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ Sum of Roots ($\alpha+\beta$): $-\frac{b}{a}$ Product of Roots ($\alpha\beta$): $\frac{c}{a}$ 2. Partial Fractions Example for distinct linear factors: $\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$ Example for repeated linear factors: $\frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$ 3. Inverse Trigonometric Functions and Trigonometric Equations Identities: $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$ General Solutions: $\sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n\alpha$ $\cos\theta = \cos\alpha \implies \theta = 2n\pi \pm \alpha$ $\tan\theta = \tan\alpha \implies \theta = n\pi + \alpha$ 4. Complex Numbers Complex Number: $z = x+iy$ Modulus: $|z| = \sqrt{x^2+y^2}$ Argument: $\arg(z) = \theta = \tan^{-1}\left(\frac{y}{x}\right)$ (considering quadrant) Polar Form: $z = r(\cos\theta + i\sin\theta)$ De Moivre's Theorem: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta)$ 5. Conics (Parabola, Ellipse, Hyperbola) Parabola ($y^2=4ax$): Focus: $(a,0)$, Directrix: $x=-a$ Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$): Foci: $(\pm ae, 0)$, Eccentricity: $e = \sqrt{1-\frac{b^2}{a^2}}$ Hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$): Foci: $(\pm ae, 0)$, Eccentricity: $e = \sqrt{1+\frac{b^2}{a^2}}$ 6. Integral Calculus Standard Integrals: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \ne -1$) $\int \frac{1}{x} dx = \ln|x| + C$ $\int e^x dx = e^x + C$ $\int \sin x dx = -\cos x + C$ $\int \cos x dx = \sin x + C$ $\int \sec^2 x dx = \tan x + C$ $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$ $\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$ Integration by Parts: $\int u dv = uv - \int v du$ Definite Integral: $\int_a^b f(x) dx = F(b) - F(a)$ Area under curve: $\int_a^b y dx$ 7. Differential Equations Variable Separable: $\int \frac{dy}{g(y)} = \int f(x) dx$ for $\frac{dy}{dx} = f(x)g(y)$ Linear Equation: $\frac{dy}{dx} + Py = Q$, Solution: $y \cdot e^{\int P dx} = \int Q \cdot e^{\int P dx} dx + C$