DDCET Engineering Prep

Cheatsheet Content

### Physics: Units & Measurement - **Unit Conversion:** - Length: 1 m = 100 cm = 1000 mm - Mass: 1 kg = 1000 g - Time: 1 min = 60 s, 1 hr = 60 min - **Error Calculation:** - Absolute Error: $\Delta A = |A_{mean} - A_i|$ - Relative Error: $\frac{\Delta A_{mean}}{A_{mean}}$ - Percentage Error: $\frac{\Delta A_{mean}}{A_{mean}} \times 100\%$ ### Physics: Mechanics - **Newton's Laws of Motion:** - **1st Law:** Inertia (object at rest stays at rest, object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force) - **2nd Law:** $F = ma$ (Force = mass × acceleration) - **3rd Law:** For every action, there is an equal and opposite reaction. - **Momentum:** $p = mv$ (Momentum = mass × velocity) - **Work Done:** $W = Fd\cos\theta$ (Work = Force × distance × cos of angle) - **Kinetic Energy (KE):** $KE = \frac{1}{2}mv^2$ - **Potential Energy (PE):** $PE = mgh$ (gravitational) - **Power:** $P = \frac{W}{t} = Fv$ ### Physics: Electricity - **Ohm's Law:** $V = IR$ (Voltage = Current × Resistance) - **Resistance:** - **Series:** $R_{eq} = R_1 + R_2 + ... + R_n$ - **Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$ - **Capacitance:** - **Definition:** $C = \frac{Q}{V}$ - **Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$ - **Parallel:** $C_{eq} = C_1 + C_2 + ... + C_n$ ### Physics: Heat & Thermodynamics - **Heat Transfer:** $Q = mc\Delta T$ (Heat = mass × specific heat capacity × change in temperature) - **Linear Expansion:** $L = L_0(1 + \alpha\Delta T)$ - **Area Expansion:** $A = A_0(1 + \beta\Delta T)$ where $\beta \approx 2\alpha$ - **Volume Expansion:** $V = V_0(1 + \gamma\Delta T)$ where $\gamma \approx 3\alpha$ ### Physics: Waves & Optics - **Wave Speed:** $v = f\lambda$ (Speed = frequency × wavelength) - **Snell's Law:** $n_1\sin\theta_1 = n_2\sin\theta_2$ - **Critical Angle:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) - **Lens Formula:** $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ (for lenses) - **Mirror Formula:** $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ (for mirrors) - **Magnification:** $M = \frac{h_i}{h_o} = -\frac{v}{u}$ ### Physics: Important Constants - **Acceleration due to gravity (g):** $9.8 \, \text{m/s}^2$ - **Speed of light in vacuum (c):** $3 \times 10^8 \, \text{m/s}$ - **Universal Gas Constant (R):** $8.314 \, \text{J/(mol}\cdot\text{K)}$ - **Permittivity of free space ($\epsilon_0$):** $8.854 \times 10^{-12} \, \text{F/m}$ - **Permeability of free space ($\mu_0$):** $4\pi \times 10^{-7} \, \text{H/m}$ ### Mathematics: Determinants & Matrices - **2x2 Determinant:** $$ \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc $$ - **3x3 Determinant:** $$ \det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg) $$ - **Inverse of 2x2 Matrix:** $$ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \quad \text{for } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ ### Mathematics: Trigonometry - **Basic Identities:** - $\sin^2\theta + \cos^2\theta = 1$ - $\tan\theta = \frac{\sin\theta}{\cos\theta}$ - $1 + \tan^2\theta = \sec^2\theta$ - $1 + \cot^2\theta = \csc^2\theta$ - **Compound Angle Formulas:** - $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ - $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ - $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ ### Mathematics: Vectors - **Dot Product (Scalar Product):** - $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$ - $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ - **Cross Product (Vector Product):** - $\vec{A} \times \vec{B} = |\vec{A}||\vec{B}|\sin\theta \, \hat{n}$ - $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$ ### Mathematics: Coordinate Geometry - **Equation of a Line:** - **Slope-intercept form:** $y = mx + c$ - **Point-slope form:** $y - y_1 = m(x - x_1)$ - **Two-point form:** $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$ - **Equation of a Circle:** - **Standard form:** $(x - h)^2 + (y - k)^2 = r^2$ (center $(h,k)$, radius $r$) - **General form:** $x^2 + y^2 + 2gx + 2fy + c = 0$ (center $(-g,-f)$, radius $\sqrt{g^2+f^2-c}$) ### Mathematics: Calculus - **Derivatives:** - $\frac{d}{dx}(x^n) = nx^{n-1}$ - $\frac{d}{dx}(\sin x) = \cos x$ - $\frac{d}{dx}(\cos x) = -\sin x$ - $\frac{d}{dx}(e^x) = e^x$ - $\frac{d}{dx}(\ln x) = \frac{1}{x}$ - **Chain Rule:** $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ - **Integration Formulas:** - $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ - $\int \frac{1}{x} \, dx = \ln|x| + C$ - $\int \sin x \, dx = -\cos x + C$ - $\int \cos x \, dx = \sin x + C$ - $\int e^x \, dx = e^x + C$ ### Mathematics: Logarithms - **Laws of Logarithms:** - $\log_b(MN) = \log_b M + \log_b N$ - $\log_b(\frac{M}{N}) = \log_b M - \log_b N$ - $\log_b(M^p) = p \log_b M$ - $\log_b b = 1$ - $\log_b 1 = 0$ - **Change of Base:** $\log_b M = \frac{\log_a M}{\log_a b}$ ### Mathematics: Statistics - **Mean:** $\bar{x} = \frac{\sum x_i}{n}$ - **Median:** The middle value of an ordered dataset. - **Mode:** The value that appears most frequently in a dataset.