Q: Relations and functions

| Idea | Formula or fact | | --- | --- | | Reflexive | $(a, a) \in R$ for every $a$ | | Symmetric | $(a, b) \in R \Rightarrow (b, a) \in R$ | | Transitive | $(a, b), (b, c) \in R \Rightarrow (a, c) \in R$ | | Equivalence | all three; its classes split the set | | Counts on $n$ elements | $2^{n^{2}}$ relations, $2^{n^{2} - n}$ reflexive | | One-one | $f(a1) = f(a2) \Rightarrow a1 = a2$ | | Onto | range = codomain | | Counts from $m$ to $n$ elements | $n^{m}$ functions, ${}^{n}P{m}$ one-one |

Q: Continuity and differentiability

| Idea | Formula or fact | | --- | --- | | Continuity at $a$ | left limit = right limit = $f(a)$ | | Standard limits | $\frac{\sin ax}{x} \to a$, $\frac{e^{ax} - 1}{x} \to a$, $\frac{1 - \cos ax}{x^{2}} \to \frac{a^{2}}{2}$ | | Differentiability at a join | values match and slopes match | | Link | differentiable ⇒ continuous, not the converse |

Q: Chain rule, log differentiation and rates

| Idea | Formula or fact | | --- | --- | | Chain rule | $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ | | $\sin^{-1} x$, $\cos^{-1} x$ | $\pm\frac{1}{\sqrt{1 - x^{2}}}$ | | $\tan^{-1} x$, $\cot^{-1} x$ | $\pm\frac{1}{1 + x^{2}}$ | | $e^{u}$, $\log u$ | $e^{u}u'$, $\frac{u'}{u}$ | | $x^{x}$ | $x^{x}(1 + \log x)$ | | Related rates | $\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$ |

Q: Area of bounded regions

| Region | Area | | --- | --- | | Circle $x^{2} + y^{2} = a^{2}$ | $\pi a^{2}$ | | Ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ | $\pi ab$ | | $y^{2} = 4ax$ up to $x = a$ | $\frac{8}{3}a^{2}$ | | Key integral | $\int0^{a} \sqrt{a^{2} - x^{2}}\,dx = \frac{\pi a^{2}}{4}$ |

Q: Vectors

| Idea | Formula or fact | | --- | --- | | Magnitude | $\sqrt{a1^{2} + a2^{2} + a3^{2}}$ | | Section formula | $\frac{m\vec{b} + n\vec{a}}{m + n}$ inside, $\frac{m\vec{b} - n\vec{a}}{m - n}$ outside | | Dot product | $a1b1 + a2b2 + a3b3 = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos \theta$ | | Projection on $\vec{b}$ | $\frac{\vec{a} \cdot \vec{b}}{\lvert \vec{b} \rvert}$ | | Cross product | magnitude $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin \theta$, perpendicular to both | | Areas | parallelogram $\lvert \vec{a} \times \vec{b} \rvert$; triangle half |

Q: Three-dimensional geometry

| Idea | Formula or fact | | --- | --- | | Line | $\vec{r} = \vec{a} + \lambda\vec{b}$; $\frac{x - x1}{a} = \frac{y - y1}{b} = \frac{z - z1}{c}$ | | Angle | $\cos \theta = \frac{\lvert a1a2 + b1b2 + c1c2 \rvert}{\sqrt{a1^{2} + b1^{2} + c1^{2}}\sqrt{a2^{2} + b2^{2} + c2^{2}}}$ | | Skew lines | $d = \frac{\lvert (\vec{a}2 - \vec{a}1) \cdot (\vec{b}1 \times \vec{b}2) \rvert}{\lvert \vec{b}1 \times \vec{b}2 \rvert}$ | | Parallel lines | $d = \frac{\lvert \vec{b} \times (\vec{a}2 - \vec{a}1) \rvert}{\lvert \vec{b} \rvert}$ |

Q: Conditional probability and Bayes' theorem

| Idea | Formula or fact | | --- | --- | | Conditional | $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ | | Multiplication | $P(A \cap B) = P(A)P(B \mid A)$ | | Independent | $P(A \cap B) = P(A)P(B)$ | | Total probability | $P(A) = \sum P(Ei)P(A \mid Ei)$ | | Bayes | $P(Ei \mid A) = \frac{P(Ei)P(A \mid Ei)}{P(A)}$ |

Question 1

Relations and functions

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Reflexive | $(a, a) \in R$ for every $a$ | | Symmetric | $(a, b) \in R \Rightarrow (b, a) \in R$ | | Transitive | $(a, b), (b, c) \in R \Rightarrow (a, c) \in R$ | | Equivalence | all three; its classes split the set | | Counts on $n$ elements | $2^{n^{2}}$ relations, $2^{n^{2} - n}$ reflexive | | One-one | $f(a1) = f(a2) \Rightarrow a1 = a2$ | | Onto | range = codomain | | Counts from $m$ to $n$ elements | $n^{m}$ functions, ${}^{n}P{m}$ one-one |

Question 2

Inverse trigonometric functions

Accepted Answer

| Function | Domain | Principal branch | | --- | --- | --- | | $\sin^{-1} x$ | $[-1, 1]$ | $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ | | $\cos^{-1} x$ | $[-1, 1]$ | $[0, \pi]$ | | $\tan^{-1} x$ | $\mathbb{R}$ | $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ | | $\cot^{-1} x$ | $\mathbb{R}$ | $(0, \pi)$ | | $\sec^{-1} x$ | $\lvert x \rvert \ge 1$ | $[0, \pi] - \left\{\frac{\pi}{2}\right\}$ | | $\operatorname{cosec}^{-1} x$ | $\lvert x \rvert \ge 1$ | $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$ | | Negative inputs | $\sin^{-1}(-x) = -\sin^{-1} x$; $\cos^{-1}(-x) = \pi - \cos^{-1} x$ | |

Question 3

Continuity and differentiability

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Continuity at $a$ | left limit = right limit = $f(a)$ | | Standard limits | $\frac{\sin ax}{x} \to a$, $\frac{e^{ax} - 1}{x} \to a$, $\frac{1 - \cos ax}{x^{2}} \to \frac{a^{2}}{2}$ | | Differentiability at a join | values match and slopes match | | Link | differentiable ⇒ continuous, not the converse |

Question 4

Chain rule, log differentiation and rates

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Chain rule | $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ | | $\sin^{-1} x$, $\cos^{-1} x$ | $\pm\frac{1}{\sqrt{1 - x^{2}}}$ | | $\tan^{-1} x$, $\cot^{-1} x$ | $\pm\frac{1}{1 + x^{2}}$ | | $e^{u}$, $\log u$ | $e^{u}u'$, $\frac{u'}{u}$ | | $x^{x}$ | $x^{x}(1 + \log x)$ | | Related rates | $\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$ |

Question 5

Area of bounded regions

Accepted Answer

| Region | Area | | --- | --- | | Circle $x^{2} + y^{2} = a^{2}$ | $\pi a^{2}$ | | Ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ | $\pi ab$ | | $y^{2} = 4ax$ up to $x = a$ | $\frac{8}{3}a^{2}$ | | Key integral | $\int0^{a} \sqrt{a^{2} - x^{2}}\,dx = \frac{\pi a^{2}}{4}$ |

Question 6

Homogeneous and linear differential equations

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Homogeneous of degree $n$ | $F(\lambda x, \lambda y) = \lambda^{n} F(x, y)$ | | Homogeneous equation | put $y = vx$, $\frac{dy}{dx} = v + x\frac{dv}{dx}$ | | Linear | $\frac{dy}{dx} + Py = Q$; I.F. $= e^{\int P\,dx}$ | | Solution | $y \cdot \text{I.F.} = \int Q \cdot \text{I.F.}\,dx + C$ |

Question 7

Vectors

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Magnitude | $\sqrt{a1^{2} + a2^{2} + a3^{2}}$ | | Section formula | $\frac{m\vec{b} + n\vec{a}}{m + n}$ inside, $\frac{m\vec{b} - n\vec{a}}{m - n}$ outside | | Dot product | $a1b1 + a2b2 + a3b3 = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos \theta$ | | Projection on $\vec{b}$ | $\frac{\vec{a} \cdot \vec{b}}{\lvert \vec{b} \rvert}$ | | Cross product | magnitude $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin \theta$, perpendicular to both | | Areas | parallelogram $\lvert \vec{a} \times \vec{b} \rvert$; triangle half |

Question 8

Three-dimensional geometry

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Line | $\vec{r} = \vec{a} + \lambda\vec{b}$; $\frac{x - x1}{a} = \frac{y - y1}{b} = \frac{z - z1}{c}$ | | Angle | $\cos \theta = \frac{\lvert a1a2 + b1b2 + c1c2 \rvert}{\sqrt{a1^{2} + b1^{2} + c1^{2}}\sqrt{a2^{2} + b2^{2} + c2^{2}}}$ | | Skew lines | $d = \frac{\lvert (\vec{a}2 - \vec{a}1) \cdot (\vec{b}1 \times \vec{b}2) \rvert}{\lvert \vec{b}1 \times \vec{b}2 \rvert}$ | | Parallel lines | $d = \frac{\lvert \vec{b} \times (\vec{a}2 - \vec{a}1) \rvert}{\lvert \vec{b} \rvert}$ |

Question 9

Conditional probability and Bayes' theorem

Accepted Answer