Class 10 Mathematics - Formulas & Key Points (Chapters 1–15)

Comprehensive review flashcards covering definitions, formulas, properties, and key concepts across Chapters 1 to 15 of Class 10 Mathematics.

Euclid's Division Lemma

A lemma stating that for any given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where $0 \le r < b$.

Co-primes

Two numbers $p$ and $q$ such that if $p$ is prime, then $\text{HCF}(p, q) = 1$.

Decimal Expansion of a Rational Number

The decimal representation of a rational number $\frac{p}{q}$ ($q \ne 0$), which is always either terminating or non-terminating recurring.

Division Algorithm for Polynomials

The relation $p(x) = g(x)q(x) + r(x)$, where $\text{deg } r(x) < \text{deg } g(x)$.

Factor Theorem

A theorem stating that $p(a) = 0 \Leftrightarrow (x - a)$ is a factor of the polynomial $p(x)$.

Remainder Theorem

A theorem stating that when a polynomial $p(x)$ is divided by $(x - a)$, the remainder is $R = p(a)$.

Cramer's Rule for Linear Systems

A method to solve a pair of linear equations where $x = \frac{b_1 c_2 - b_2 c_1}{\Delta}$, $y = \frac{a_2 c_1 - a_1 c_2}{\Delta}$, and $\Delta = a_1 b_2 - a_2 b_1$.

Conditions for Solutions in Two Variables

Criteria for linear systems: unique solution occurs when $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$; infinitely many solutions occur when $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$; and no solution occurs when $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$.

Discriminant ($D$)

The expression $D = b^2 - 4ac$ in a quadratic equation $ax^2 + bx + c = 0$, used to determine the nature of its roots.

Nature of Roots

For $ax^2 + bx + c = 0$: if $D > 0$, roots are two real & distinct; if $D = 0$, roots are real & equal with $\alpha = \beta = -\frac{b}{2a}$; and if $D < 0$, there are no real roots.

$n^{\text{th}}$ Term of an Arithmetic Progression

The general term of an AP given by the formula $a_n = a + (n - 1)d$ or $a_n = a_m + (n - m)d$.

Sum of First $n$ Odd Numbers

The total sum of the first $n$ positive odd integers, equal to $n^2$.

Sum of First $n$ Even Numbers

The total sum of the first $n$ positive even integers, equal to $n(n + 1)$.

Pythagoras Theorem

In a right-angled triangle with sides $a$, $b$, and hypotenuse $c$, the relationship is $c^2 = a^2 + b^2$.

Heron's Formula

A formula to find the area of a triangle: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$, where $s = \frac{a + b + c}{2}$.

Section Formula (Internal Division)

The coordinates of a point dividing a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m : n$, given by $\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$.

Distance of Point from a Line

The perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $ax + by + c = 0$, given by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.

Perpendicular Lines Condition

Two lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $m_1 m_2 = -1$.

Fundamental Trigonometric Identities

The three standard relations: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, and $1 + \cot^2(\theta) = \csc^2(\theta)$.

Angle of Elevation

The angle formed by the line of sight with the horizontal plane when an observer looks at an object above the horizontal level.

Angle of Depression

The angle formed by the line of sight with the horizontal plane when an observer looks at an object below the horizontal level.

Central Angle Theorem

A circle theorem stating that the angle subtended by an arc at the centre is twice the angle subtended by it at the circumference ($\text{Angle at centre} = 2 \times \text{angle at circumference}$).

Cyclic Quadrilateral

A quadrilateral inscribed in a circle whose opposite angles are supplementary ($\text{sum} = 180^\circ$).

Length of Tangent

The length of a tangent line segment from an external point $P$ to a circle with radius $r$ and centre-to-point distance $PO$, given by $PA = \sqrt{PO^2 - r^2}$.

Area of Sector

The portion of a circle's area enclosed by two radii and an arc, calculated as $A = \frac{\theta}{360^\circ} \times \pi r^2$.

Arc Length

The distance along the curved line of a circle sector, given by $l = \frac{\theta}{360^\circ} \times 2\pi r = r\theta\text{ (rad)}$.

Frustum of a Cone

The part of a cone remaining when its top is sliced off parallel to the base; it has $\text{CSA} = \pi (r_1 + r_2)l$, $\text{TSA} = \pi (r_1 + r_2)l + \pi (r_1^2 + r_2^2)$, and $V = \frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1 r_2)$.

Class Mark

The midpoint of a class interval in statistics, calculated as $\frac{l + u}{2}$, where $l$ is the lower limit and $u$ is the upper limit.

Empirical Relation in Statistics

The approximate relationship connecting central tendency measures: $\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})$.

Mode for Grouped Data

The value calculated using the formula $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$, where $l$ is the lower limit, $f_1$ is the modal frequency, $f_0$ and $f_2$ are frequencies of the preceding and succeeding classes, and $h$ is class width.

Conditional Probability

The probability of an event $A$ occurring given that event $B$ has already occurred, formulated as $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) \ne 0$.

Independent Events

Two events $A$ and $B$ whose joint probability satisfies $P(A\text{ and }B) = P(A) \times P(B)$.

Euclid's Division Lemma

A fundamental result stating that for any given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ satisfying $a = bq + r$, where $0 \le r < b$.

Fundamental Relation of HCF and LCM

For any two positive numbers, the relationship is given by $\text{HCF} \times \text{LCM} = \text{product of two numbers}$.

Irrational Number

A real number with a non-terminating, non-recurring decimal expansion, such as $\sqrt{2} = 1.414213\dots$.

Zeroes and Coefficients of a Quadratic Polynomial

For a quadratic polynomial $p(x) = ax^2 + bx + c$, the sum of zeroes is $\alpha + \beta = -\frac{b}{a}$ and the product of zeroes is $\alpha\beta = \frac{c}{a}$.

Division Algorithm for Polynomials

Given polynomials $p(x)$ and $g(x)$, there exist unique polynomials $q(x)$ and $r(x)$ such that $p(x) = g(x)q(x) + r(x)$, where $\deg(r) < \deg(g)$.

Factor Theorem

A theorem stating that for a polynomial $p(x)$, $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$.

Discriminant ($D$)

The algebraic expression $D = b^2 - 4ac$ from $ax^2 + bx + c = 0$ used to determine root nature: $D > 0$ gives two real and distinct roots, $D = 0$ gives real and equal roots, and $D < 0$ indicates no real roots.

Quadratic Formula

The formula yielding the solutions to any quadratic equation $ax^2 + bx + c = 0$, given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Consistency Conditions for Linear Systems

For two linear equations $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$, a unique solution exists when $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$, infinitely many solutions exist when $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, and no solution exists when $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$.

Cramer's Rule (Two Variables)

A method to solve linear systems using determinants: $x = \frac{b_1 c_2 - b_2 c_1}{\Delta}$ and $y = \frac{a_2 c_1 - a_1 c_2}{\Delta}$, where $\Delta = a_1 b_2 - a_2 b_1$.

$n^{\text{th}}$ Term of an Arithmetic Progression ($a_n$)

The formula to find any specific term in an AP: $a_n = a + (n - 1)d$, which can also be expressed relative to the $m^{\text{th}}$ term as $a_n = a_m + (n - m)d$.

Sum of First $n$ Terms of an AP ($S_n$)

The total sum of the first $n$ terms of an arithmetic progression, given by $S_n = \frac{n}{2}[2a + (n - 1)d]$.

Sum of First $n$ Odd and Even Numbers

The sum of the first $n$ positive odd numbers equals $n^2$, while the sum of the first $n$ positive even numbers equals $n(n + 1)$.

Heron's Formula

A geometric formula for the area of a triangle given side lengths $a$, $b$, and $c$: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$, where semi-perimeter $s = \frac{a + b + c}{2}$.

Section Formula (Internal Division)

Formula giving the coordinates of a point dividing the line joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$ internally: $\left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$.

Perpendicular Lines Condition

The algebraic condition for two straight lines with slopes $m_1$ and $m_2$ to be perpendicular: $m_1 m_2 = -1$.

Distance of a Point from a Line

The perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $ax + by + c = 0$, defined by $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$.

Fundamental Trigonometric Identities

The core Pythagorean identities in trigonometry: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, and $1 + \cot^2(\theta) = \csc^2(\theta)$.

Complementary Angles in Trigonometry

Identities relating angles that sum to $90^\circ$: $\sin(90^\circ - \theta) = \cos(\theta)$, $\cos(90^\circ - \theta) = \sin(\theta)$, and $\tan(90^\circ - \theta) = \cot(\theta)$.

Angle of Elevation

The acute angle formed between the horizontal line of sight and the line directed upward toward an object situated above the observer.

Angle of Depression

The acute angle formed between the horizontal line of sight and the line directed downward toward an object situated below the observer.

Cyclic Quadrilateral Angle Property

A theorem stating that the opposite angles of any cyclic quadrilateral are supplementary, meaning their sum is $180^\circ$.

Tangent-Radius Theorem

A geometric rule stating that a tangent to a circle is perpendicular to the radius at the point of contact ($\text{Tangent} \perp \text{radius}$).

Length of Tangents from an External Point

Tangents drawn to a circle from an external point $P$ have equal lengths ($PA = PB$), each calculated as $PA = \sqrt{PO^2 - r^2}$.

Area of a Sector of a Circle

The area bounded by two radii and an arc with central angle $\theta$, given by $A = \frac{\theta}{360^\circ} \times \pi r^2$.

Arc Length of a Sector ($l$)

The length of an arc subtending a central angle $\theta$, defined as $l = \frac{\theta}{360^\circ} \times 2\pi r = r\theta\text{ (rad)}$.

Total Surface Area and Volume of a Frustum of a Cone

For radii $r_1, r_2$ and slant height $l$, the total surface area is $\text{TSA} = \pi(r_1 + r_2)l + \pi(r_1^2 + r_2^2)$, and its volume is $V = \frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1 r_2)$.

Empirical Relation of Central Tendency

The statistical formula connecting the three common averages: $\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})$.

Mode for Grouped Data

The formula to determine the modal value in grouped frequencies: $\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$, where $l$ is the lower limit of the modal class and $h$ is the class width.

Median for Grouped Data

The formula to calculate the median of grouped data: $\text{Median} = l + \left(\frac{\frac{N}{2} - c.f.}{f}\right) \times h$.

Class Mark

The midpoint of a class interval with lower limit $l$ and upper limit $u$, computed as $\frac{l + u}{2}$.

Conditional Probability ($P(A|B)$)

The probability that event $A$ occurs given that event $B$ has already occurred, formulated as $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) \ne 0$.

Independent Events (Probability)

Events $A$ and $B$ whose occurrence does not affect each other, defined mathematically by the multiplication rule: $P(A \cap B) = P(A) \times P(B)$.