Q: Determinants and the inverse

| Idea | Formula or fact | | --- | --- | | 2 × 2 | $ad - bc$ | | Scalar multiple, order $n$ | $\lvert kA \rvert = k^n\lvert A \rvert$ | | Product | $\lvert AB \rvert = \lvert A \rvert\lvert B \rvert$ | | Adjoint | $A(\operatorname{adj} A) = \lvert A \rvert I$, $\lvert \operatorname{adj} A \rvert = \lvert A \rvert^{n-1}$ | | Inverse | $A^{-1} = \frac{1}{\lvert A \rvert}\operatorname{adj} A$, only when $\lvert A \rvert \ne 0$ | | Area of a triangle | $\frac{1}{2}\lvert \Delta \rvert$; collinear when $\Delta = 0$ | | $AX = B$ | $X = A^{-1}B$; if $\lvert A \rvert = 0$, look at $(\operatorname{adj} A)B$ |

Question 1

Matrices

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Order | rows × columns; $a{ij}$ is row $i$, column $j$ | | Counting | $m \times n$ matrices with entries from $r$ values: $r^{mn}$ | | Types | identity ⇒ scalar ⇒ diagonal ⇒ square | | Equal | same order and equal corresponding elements | | Product defined | columns of $A$ = rows of $B$; $AB$ takes the outer numbers | | Product rules | $(AB)C = A(BC)$, $A(B + C) = AB + AC$; $AB \ne BA$ in general | | $2 \times 2$ polynomial | $A^2 - (a + d)A + (ad - bc)I = O$ | | Transpose | $(AB)' = B'A'$ | | Symmetric, skew-symmetric | $A' = A$; $A' = -A$, with a zero diagonal | | Splitting $A$ | $\frac{1}{2}(A + A') + \frac{1}{2}(A - A')$ | | Inverse | $AB = BA = I$, square only, unique; $(AB)^{-1} = B^{-1}A^{-1}$ |

Question 2

Determinants and the inverse

Accepted Answer

| Idea | Formula or fact | | --- | --- | | 2 × 2 | $ad - bc$ | | Scalar multiple, order $n$ | $\lvert kA \rvert = k^n\lvert A \rvert$ | | Product | $\lvert AB \rvert = \lvert A \rvert\lvert B \rvert$ | | Adjoint | $A(\operatorname{adj} A) = \lvert A \rvert I$, $\lvert \operatorname{adj} A \rvert = \lvert A \rvert^{n-1}$ | | Inverse | $A^{-1} = \frac{1}{\lvert A \rvert}\operatorname{adj} A$, only when $\lvert A \rvert \ne 0$ | | Area of a triangle | $\frac{1}{2}\lvert \Delta \rvert$; collinear when $\Delta = 0$ | | $AX = B$ | $X = A^{-1}B$; if $\lvert A \rvert = 0$, look at $(\operatorname{adj} A)B$ |

Question 3

Derivatives

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Product, quotient | $(uv)' = u'v + uv'$; $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ | | Chain | outside's derivative × inside's derivative | | Implicit | each $y$ term picks up $\frac{dy}{dx}$ | | Parametric | $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ | | Second derivative, parametric | $\frac{d}{dt}\left(\frac{dy}{dx}\right) \div \frac{dx}{dt}$ |

Question 4

Increasing functions and extrema

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Rising, falling | $f' > 0$; $f' 0$ minimum; $f'' = 0$ no verdict | | Closed interval | compare critical points inside and both ends |

Question 5

Integrals and area

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Power, reciprocal | $\frac{x^{n+1}}{n+1}$; $\log\lvert x \rvert$ | | $\frac{f'}{f}$ | $\log\lvert f \rvert$ | | By parts | $\int u\,dv = uv - \int v\,du$ | | $\frac{1}{x^2 + a^2}$, $\frac{1}{\sqrt{a^2 - x^2}}$ | $\frac{1}{a}\tan^{-1}\frac{x}{a}$; $\sin^{-1}\frac{x}{a}$ | | Definite | $F(b) - F(a)$ | | Odd, even on $[-a, a]$ | 0; $2\int0^a$ | | $a - x$ property | $\int0^a f(x)\,dx = \int0^a f(a - x)\,dx$ | | Area under a curve | $\inta^b f(x)\,dx$; below the axis, its absolute value | | Ellipse | area $\pi ab$ |

Question 6

Differential equations

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Order, degree | highest derivative; its power, if polynomial in derivatives | | Constants | $n$ arbitrary constants give order $n$ | | Separable | $\int \frac{dy}{g(y)} = \int f(x)\,dx$ | | Growth, decay | $P = P0e^{kt}$ |

Question 7

Probability

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Classical | favourable ÷ total, outcomes equally likely | | Addition rule | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | | Complement | $P(E') = 1 - P(E)$ | | Odds $a : b$ in favour | $\frac{a}{a + b}$ |

Question 8

Linear programming

Accepted Answer

| Idea | Formula or fact | | --- | --- | | Bounded region | optimum at a corner point | | Unbounded region | check the half-plane beyond the corner value | | Empty region | no feasible solution |

Section A formulas for CUET Maths