Optional Mathematics — Grade IX

Grade IX

Optional Mathematics · वैकल्पिक गणित

Optional Mathematics

वैकल्पिक गणित — Grade IX

📚 8 Chapters

📝 28 Topics

🎯 TQ 33 · TM 75

🔢 K · U · A · HA

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Course Chapters

Click any chapter to expand topics · Each topic opens in a new tab

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Key Formulas

Essential results — appear directly in exams

Algebra — Quadratic

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Roots of $ax^2+bx+c=0$. Discriminant $D=b^2-4ac$ determines nature.

Algebra — Sequence

$$S_n = \frac{n}{2}\bigl[2a+(n-1)d\bigr]$$

Sum of $n$ terms of an AP with first term $a$ and common difference $d$.

Matrix — Inverse

$$A^{-1} = \frac{1}{\det A}\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$$

For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, exists only when $\det A \neq 0$.

Coordinate — Angle

$$\tan\theta = \left|\frac{m_1-m_2}{1+m_1 m_2}\right|$$

Angle between two lines with slopes $m_1$ and $m_2$.

Circle

$$(x-h)^2+(y-k)^2 = r^2$$

Centre $(h,k)$, radius $r$. General form: $x^2+y^2+2gx+2fy+c=0$.

Trigonometry — Double Angle

$$\sin 2A = 2\sin A\cos A$$

Also: $\cos 2A = \cos^2 A - \sin^2 A = 1-2\sin^2 A$.

Trigonometry — Sum to Product

$$\sin C+\sin D = 2\sin\!\frac{C+D}{2}\cos\!\frac{C-D}{2}$$

Transformation identity — used for simplification and solving equations.

Vector — Scalar Product

$$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$$

Two vectors are perpendicular when $\vec{a}\cdot\vec{b}=0$.

Statistics — Std. Deviation

$$\sigma = \sqrt{\frac{\sum f(x-\bar{x})^2}{\sum f}}$$

Coefficient of Variation $= \dfrac{\sigma}{\bar{x}}\times100\%$

Statistics — Quartile Dev.

$$QD = \frac{Q_3 - Q_1}{2}$$

Semi-interquartile range. $Q_1$ = lower quartile, $Q_3$ = upper quartile.

Cramer's Rule

$$x = \frac{D_x}{D},\quad y = \frac{D_y}{D}$$

$D$ = coefficient determinant; $D_x$, $D_y$ = determinants with RHS substituted.

Height & Distance

$$h = d\tan\alpha$$

Height $h$, horizontal distance $d$, angle of elevation $\alpha$.

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Key Concepts

Deep understanding beyond memorising formulas

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Functions

A function maps each input to exactly one output. Written $f:A\to B$. Domain, codomain, and range must be clearly identified. Composite: $(g\circ f)(x)=g(f(x))$.

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Polynomials

Remainder Theorem: $f(a)$ is the remainder when $f(x)$ is divided by $(x-a)$. Factor Theorem: $(x-a)$ is a factor iff $f(a)=0$.

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Limits & Continuity

$f$ is continuous at $x=a$ if $\lim_{x\to a}f(x)=f(a)$. A removable discontinuity can be fixed; a jump discontinuity cannot.

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Matrix Operations

Matrix multiplication is not commutative: $AB \neq BA$ in general. A matrix is invertible iff its determinant is non-zero.

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Conic Sections

A parabola, ellipse, and hyperbola are all cross-sections of a cone. The circle is the special case $r=\text{const}$ in polar form.

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Transformations

Combined transformations apply right-to-left. Inversion maps points inside a circle to outside. $P' = \dfrac{r^2}{\overrightarrow{OP}^2}\overrightarrow{OP}$.

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Vectors

Vectors have magnitude and direction. Scalar product gives a number: $\vec{a}\cdot\vec{b}=0$ implies perpendicularity. Vectors are equal iff same magnitude and direction.

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Standard Deviation

SD measures spread around the mean. A small SD means data is tightly clustered. CV$=\frac{\sigma}{\bar x}\times100\%$ allows comparison of different datasets.

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