Q: Modulo arithmetic and congruence

| Idea | Formula or fact | | --- | --- | | $a \bmod n$ | the remainder, one of $0, 1, \dots, n - 1$, even for negative $a$ | | Arithmetic | reduce, then add, subtract or multiply, and reduce again | | Congruence | $a \equiv b \pmod{n}$ when $n$ divides $a - b$ | | Solving $ax \equiv b$ | solvable when $\gcd(a, n)$ divides $b$; then $\gcd(a, n)$ solutions | | Residue classes | $n$ classes modulo $n$, disjoint, covering every integer |

Q: Races and games

| Idea | Formula or fact | | --- | --- | | A beats B by $x$ in $d$ | speeds $d : (d - x)$ | | Dead heat | start = the distance B loses by over the full race | | Circular track | same way: lap ÷ difference of speeds; opposite: lap ÷ sum |

Q: Rate of change, marginal cost and revenue

| Idea | Formula or fact | | --- | --- | | Related rates | $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$ | | Revenue, profit | $R = px$, $P = R - C$, average cost $\frac{C}{x}$ | | Marginals | $MC = C'(x)$, $MR = R'(x)$ | | Best output | profit: $MR = MC$; revenue: $MR = 0$; least average cost: $AC = MC$ |

Q: Consumer and producer surplus

| Idea | Formula or fact | | --- | --- | | Cost from MC | $C = \int MC\,dx + \text{fixed cost}$ | | Revenue from MR | $R = \int MR\,dx$, $R(0) = 0$; demand $p = \frac{R}{x}$ | | Consumer surplus | $\int{0}^{x0} D(x)\,dx - p0x0$ | | Producer surplus | $p0x0 - \int{0}^{x0} S(x)\,dx$ |

Q: Random variables, binomial and Poisson

| Idea | Formula or fact | | --- | --- | | Distribution | $0 \le p \le 1$, $\sum p = 1$ | | Mean, variance | $E(X) = \sum xp$, $\text{Var}(X) = \sum x^{2}p - [E(X)]^{2}$ | | Linear change | $E(aX + b) = aE(X) + b$, $\text{Var}(aX + b) = a^{2}\,\text{Var}(X)$ | | Binomial | ${}^{n}C{r}\,p^{r}q^{n - r}$; mean $np$, variance $npq$ | | Poisson | $\frac{e^{-\lambda}\lambda^{r}}{r!}$; mean = variance = $\lambda$ |

Question 1

Modulo arithmetic and congruence

Accepted Answer

| Idea | Formula or fact | | --- | --- | | $a \bmod n$ | the remainder, one of $0, 1, \dots, n - 1$, even for negative $a$ | | Arithmetic | reduce, then add, subtract or multiply, and reduce again | | Congruence | $a \equiv b \pmod{n}$ when $n$ divides $a - b$ | | Solving $ax \equiv b$ | solvable when $\gcd(a, n)$ divides $b$; then $\gcd(a, n)$ solutions | | Residue classes | $n$ classes modulo $n$, disjoint, covering every integer |

Question 2

Alligation and mixtures

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Mean price | total cost ÷ total quantity | | Alligation | cheaper : dearer $= (d - m) : (m - c)$ | | Free ingredient | water costs ₹0 | | Replacement | original left after $n$ times: $V\left(1 - \frac{x}{V}\right)^{n}$ |

Question 3

Boats and streams, pipes and cisterns

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Downstream, upstream | $b + s$, $b - s$ | | From the two speeds | $b = \frac{\text{down} + \text{up}}{2}$, $s = \frac{\text{down} - \text{up}}{2}$ | | Round trip | add the time of each leg | | Pipes | rate $= \frac{1}{\text{time}}$; rates add, outlets subtract | | Two pipes together | $\frac{ab}{a + b}$ |

Question 4

Races and games

Accepted Answer

| Idea | Formula or fact | | --- | --- | | A beats B by $x$ in $d$ | speeds $d : (d - x)$ | | Dead heat | start = the distance B loses by over the full race | | Circular track | same way: lap ÷ difference of speeds; opposite: lap ÷ sum |

Question 5

Numerical inequalities

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Negative multiplier | reverses the inequality | | Modulus | $\lvert u \rvert \le k$: $-k \le u \le k$; $\lvert u \rvert \ge k$: $u \le -k$ or $u \ge k$ | | Quadratic | $ 0$ outside them | | Words | at most $\le$, at least $\ge$ |

Question 6

Rate of change, marginal cost and revenue

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Related rates | $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$ | | Revenue, profit | $R = px$, $P = R - C$, average cost $\frac{C}{x}$ | | Marginals | $MC = C'(x)$, $MR = R'(x)$ | | Best output | profit: $MR = MC$; revenue: $MR = 0$; least average cost: $AC = MC$ |

Question 7

Consumer and producer surplus

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Cost from MC | $C = \int MC\,dx + \text{fixed cost}$ | | Revenue from MR | $R = \int MR\,dx$, $R(0) = 0$; demand $p = \frac{R}{x}$ | | Consumer surplus | $\int{0}^{x0} D(x)\,dx - p0x0$ | | Producer surplus | $p0x0 - \int{0}^{x0} S(x)\,dx$ |

Question 8

Random variables, binomial and Poisson

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Distribution | $0 \le p \le 1$, $\sum p = 1$ | | Mean, variance | $E(X) = \sum xp$, $\text{Var}(X) = \sum x^{2}p - [E(X)]^{2}$ | | Linear change | $E(aX + b) = aE(X) + b$, $\text{Var}(aX + b) = a^{2}\,\text{Var}(X)$ | | Binomial | ${}^{n}C{r}\,p^{r}q^{n - r}$; mean $np$, variance $npq$ | | Poisson | $\frac{e^{-\lambda}\lambda^{r}}{r!}$; mean = variance = $\lambda$ |

Question 9

Normal distribution

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Standard normal variate | $z = \frac{x - \mu}{\sigma}$, mean 0, standard deviation 1 | | Areas | $P(Z z) = 0.5 - A(z)$ for $z > 0$ | | Rule of thumb | 68%, 95%, 99.7% within 1, 2, 3 standard deviations |

Question 10

Time series and trend

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Components | secular trend, seasonal, cyclical, irregular | | Moving average | placed against the middle of its run; centre an even period | | Least squares, $\sum x = 0$ | $a = \frac{\sum y}{n}$, $b = \frac{\sum xy}{\sum x^{2}}$ |

Question 11

Sampling, inference and the t-test

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Systematic sampling | interval $k = \frac{N}{n}$ | | Standard error | $\frac{\sigma}{\sqrt{n}}$ | | t-statistic | $t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$, degrees of freedom $n - 1$ | | Decision | reject $H0$ when $\lvert t \rvert$ exceeds the critical value |

Question 12

Financial mathematics

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Perpetuity | $PV = \frac{R}{i}$ | | Sinking fund | $A = R\frac{(1 + i)^{n} - 1}{i}$, $R = \frac{Ai}{(1 + i)^{n} - 1}$ | | Bond | PV of the coupons + PV of the face value | | Flat-rate EMI | $\frac{P(1 + rt)}{\text{months}}$ | | Reducing-balance EMI | $\frac{Pi(1 + i)^{n}}{(1 + i)^{n} - 1}$ | | CAGR | $\left(\frac{E}{B}\right)^{1/n} - 1$ | | Linear depreciation | $\frac{\text{cost} - \text{residual value}}{\text{useful life}}$ a year |

Applied Mathematics formulas for CUET Maths