Class 10 Maths Formulas (CBSE)

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Chapter 1: Real Numbers Euclid's Division Lemma: Given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where $0 \le r Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. HCF and LCM relation for two positive integers $a, b$: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. Rational Numbers: Numbers that can be expressed in the form $p/q$, where $p, q$ are integers and $q \ne 0$. Irrational Numbers: Numbers that cannot be expressed in the form $p/q$. Examples: $\sqrt{2}, \sqrt{3}, \pi$. Decimal Expansion of Rational Numbers: Terminating: If the prime factorization of the denominator $q$ (in $p/q$ form, where $p,q$ are coprime) is of the form $2^m 5^n$, where $m, n$ are non-negative integers. Non-terminating Repeating: If the prime factorization of the denominator $q$ is NOT of the form $2^m 5^n$. Chapter 2: Polynomials General Form of a Linear Polynomial: $ax + b$, where $a \ne 0$. Degree 1. General Form of a Quadratic Polynomial: $ax^2 + bx + c$, where $a \ne 0$. Degree 2. General Form of a Cubic Polynomial: $ax^3 + bx^2 + cx + d$, where $a \ne 0$. Degree 3. Zeros of a Polynomial: The values of the variable for which the polynomial evaluates to zero. Graphically, these are the x-intercepts. Relationship between Zeros and Coefficients (Quadratic Polynomial $ax^2 + bx + c$, zeros $\alpha, \beta$): Sum of zeros: $\alpha + \beta = -b/a$ Product of zeros: $\alpha \beta = c/a$ Relationship between Zeros and Coefficients (Cubic Polynomial $ax^3 + bx^2 + cx + d$, zeros $\alpha, \beta, \gamma$): Sum of zeros: $\alpha + \beta + \gamma = -b/a$ Sum of products of zeros taken two at a time: $\alpha \beta + \beta \gamma + \gamma \alpha = c/a$ Product of zeros: $\alpha \beta \gamma = -d/a$ Division Algorithm for Polynomials: If $p(x)$ and $g(x)$ are any two polynomials with $g(x) \ne 0$, then we can find polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that $p(x) = g(x) \cdot q(x) + r(x)$, where $r(x) = 0$ or $\text{deg } r(x) Chapter 3: Pair of Linear Equations in Two Variables General Form: $a_1x + b_1y + c_1 = 0$ $a_2x + b_2y + c_2 = 0$ Graphical Representation and Conditions for Solutions: Ratio Comparison Graphical Representation Algebraic Interpretation $a_1/a_2 \ne b_1/b_2$ Intersecting Lines Unique Solution (Consistent) $a_1/a_2 = b_1/b_2 = c_1/c_2$ Coincident Lines Infinitely Many Solutions (Consistent & Dependent) $a_1/a_2 = b_1/b_2 \ne c_1/c_2$ Parallel Lines No Solution (Inconsistent) Methods of Solving: Substitution Method Elimination Method Cross-Multiplication Method: $x / (b_1c_2 - b_2c_1) = y / (c_1a_2 - c_2a_1) = 1 / (a_1b_2 - a_2b_1)$ Chapter 4: Quadratic Equations Standard Form: $ax^2 + bx + c = 0$, where $a \ne 0$. Quadratic Formula: Solutions (roots) are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Discriminant ($D$): $D = b^2 - 4ac$ Nature of Roots: If $D > 0$: Two distinct real roots. If $D = 0$: Two equal real roots. If $D Relationship between Roots and Coefficients (for roots $\alpha, \beta$): Sum of Roots: $\alpha + \beta = -b/a$ Product of Roots: $\alpha \beta = c/a$ Forming a Quadratic Equation (given roots $\alpha, \beta$): $x^2 - (\alpha + \beta)x + (\alpha \beta) = 0$ Chapter 5: Arithmetic Progressions (AP) General Form: $a, a+d, a+2d, a+3d, \dots$ First Term: $a$ Common Difference: $d = a_2 - a_1 = a_3 - a_2$, etc. $n^{\text{th}}$ Term ($a_n$ or $l$): $a_n = a + (n-1)d$ Sum of First $n$ Terms ($S_n$): $S_n = \frac{n}{2} [2a + (n-1)d]$ $S_n = \frac{n}{2} (a + l)$, where $l$ is the last term ($a_n$). Chapter 6: Triangles Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. If $DE \parallel BC$ in $\triangle ABC$, then $AD/DB = AE/EC$. Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Criteria for Similarity of Triangles: AA (Angle-Angle) Similarity: If two angles of one triangle are respectively equal to two angles of another triangle. SSS (Side-Side-Side) Similarity: If the corresponding sides of two triangles are in the same ratio. SAS (Side-Angle-Side) Similarity: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are in the same ratio. Area of Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If $\triangle ABC \sim \triangle PQR$, then $\text{Area}(\triangle ABC) / \text{Area}(\triangle PQR) = (AB/PQ)^2 = (BC/QR)^2 = (AC/PR)^2$. Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In $\triangle ABC$ right-angled at $B$, $AC^2 = AB^2 + BC^2$. Converse of Pythagoras Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Chapter 7: Coordinate Geometry Distance Formula: Distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Section Formula (Internal Division): Coordinates of point $P(x, y)$ that divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$ are: $x = (mx_2 + nx_1) / (m+n)$ $y = (my_2 + ny_1) / (m+n)$ Mid-point Formula: Coordinates of the mid-point of the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ are: $x = (x_1 + x_2) / 2$ $y = (y_1 + y_2) / 2$ Area of a Triangle: Area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is: $\text{Area} = 1/2 |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ If the area is 0, the three points are collinear. Chapter 8: Introduction to Trigonometry Trigonometric Ratios (for acute angle $A$ in a right-angled $\triangle ABC$, right-angled at $B$): $\sin A = \text{Opposite Side} / \text{Hypotenuse} = BC/AC$ $\cos A = \text{Adjacent Side} / \text{Hypotenuse} = AB/AC$ $\tan A = \text{Opposite Side} / \text{Adjacent Side} = BC/AB$ $\csc A = 1/\sin A = \text{Hypotenuse} / \text{Opposite Side}$ $\sec A = 1/\cos A = \text{Hypotenuse} / \text{Adjacent Side}$ $\cot A = 1/\tan A = \text{Adjacent Side} / \text{Opposite Side}$ Reciprocal Identities: $\csc A = 1/\sin A$ $\sec A = 1/\cos A$ $\cot A = 1/\tan A$ Quotient Identities: $\tan A = \sin A / \cos A$ $\cot A = \cos A / \sin A$ Pythagorean Identities: $\sin^2 A + \cos^2 A = 1$ $1 + \tan^2 A = \sec^2 A$ (for $0^\circ \le A $1 + \cot^2 A = \csc^2 A$ (for $0^\circ Trigonometric Ratios of Complementary Angles: $\sin (90^\circ - A) = \cos A$ $\cos (90^\circ - A) = \sin A$ $\tan (90^\circ - A) = \cot A$ $\cot (90^\circ - A) = \tan A$ $\sec (90^\circ - A) = \csc A$ $\csc (90^\circ - A) = \sec A$ Trigonometric Values for Specific Angles: Angle ($\theta$) $0^\circ$ $30^\circ$ $45^\circ$ $60^\circ$ $90^\circ$ $\sin \theta$ $0$ $1/2$ $1/\sqrt{2}$ $\sqrt{3}/2$ $1$ $\cos \theta$ $1$ $\sqrt{3}/2$ $1/\sqrt{2}$ $1/2$ $0$ $\tan \theta$ $0$ $1/\sqrt{3}$ $1$ $\sqrt{3}$ Undefined Chapter 9: Some Applications of Trigonometry (Heights and Distances) Angle of Elevation: The angle formed by the line of sight with the horizontal when the object is above the horizontal level. Angle of Depression: The angle formed by the line of sight with the horizontal when the object is below the horizontal level. Problems involve using trigonometric ratios ($\sin, \cos, \tan$) and their values for $30^\circ, 45^\circ, 60^\circ$ to find unknown heights or distances. Chapter 10: Circles Tangent to a Circle: A line that intersects the circle at exactly one point. Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. ($\angle OPA = 90^\circ$ if $OP$ is radius and $PA$ is tangent). Theorem 2: The lengths of tangents drawn from an external point to a circle are equal. ($PA = PB$ for tangents from external point $P$). Number of Tangents from a Point: From a point inside the circle: 0 tangents. From a point on the circle: 1 tangent. From a point outside the circle: 2 tangents. Chapter 11: Areas Related to Circles Circumference of a Circle: $C = 2\pi r$ or $C = \pi d$ (where $d=2r$ is diameter). Area of a Circle: $A = \pi r^2$. Area of a Sector of Angle $\theta$ (in degrees): $A_{sector} = (\theta/360^\circ) \times \pi r^2$. Length of an Arc of Angle $\theta$ (in degrees): $L_{arc} = (\theta/360^\circ) \times 2\pi r$. Area of a Segment: Area of Minor Segment = Area of Sector - Area of corresponding $\triangle$ formed by radii and chord. Area of Minor Segment = $(\theta/360^\circ) \pi r^2 - 1/2 r^2 \sin \theta$ (for angle $\theta$ in degrees). Area of Major Segment = Area of Circle - Area of Minor Segment. Chapter 12: Surface Areas and Volumes Cuboid (length $l$, breadth $b$, height $h$): Volume ($V$) = $l \times b \times h$ Lateral Surface Area ($LSA$) = $2h(l+b)$ (Area of 4 walls) Total Surface Area ($TSA$) = $2(lb + bh + hl)$ Cube (side $a$): Volume ($V$) = $a^3$ Lateral Surface Area ($LSA$) = $4a^2$ Total Surface Area ($TSA$) = $6a^2$ Cylinder (radius $r$, height $h$): Volume ($V$) = $\pi r^2 h$ Curved Surface Area ($CSA$) = $2\pi r h$ Total Surface Area ($TSA$) = $2\pi r(r+h)$ Cone (radius $r$, height $h$, slant height $l = \sqrt{r^2 + h^2}$): Volume ($V$) = $1/3 \pi r^2 h$ Curved Surface Area ($CSA$) = $\pi r l$ Total Surface Area ($TSA$) = $\pi r(r+l)$ Sphere (radius $r$): Volume ($V$) = $4/3 \pi r^3$ Surface Area ($SA$) = $4\pi r^2$ Hemisphere (radius $r$): Volume ($V$) = $2/3 \pi r^3$ Curved Surface Area ($CSA$) = $2\pi r^2$ Total Surface Area ($TSA$) = $3\pi r^2$ Frustum of a Cone (radii $r_1, r_2$, height $h$, slant height $l=\sqrt{h^2+(r_1-r_2)^2}$): Volume ($V$) = $1/3 \pi h (r_1^2 + r_2^2 + r_1 r_2)$ Curved Surface Area ($CSA$) = $\pi (r_1 + r_2) l$ Total Surface Area ($TSA$) = $\pi (r_1 + r_2) l + \pi r_1^2 + \pi r_2^2$ (Area of CSA + Area of both circular bases) Chapter 13: Statistics Mean (for ungrouped data): $\bar{x} = (\sum x_i) / n$ Mean (for grouped data): Direct Method: $\bar{x} = (\sum f_i x_i) / (\sum f_i)$ Assumed Mean Method: $\bar{x} = a + (\sum f_i d_i) / (\sum f_i)$, where $d_i = x_i - a$ Step-Deviation Method: $\bar{x} = a + h \cdot [(\sum f_i u_i) / (\sum f_i)]$, where $u_i = (x_i - a) / h$ Median (for grouped data): $\text{Median} = L + \left[\frac{n/2 - cf}{f}\right] \times h$ $L$: lower limit of median class $n$: total frequency $cf$: cumulative frequency of class preceding median class $f$: frequency of median class $h$: class size Mode (for grouped data): $\text{Mode} = L + \left[\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right] \times h$ $L$: lower limit of modal class $f_1$: frequency of modal class $f_0$: frequency of class preceding modal class $f_2$: frequency of class succeeding modal class $h$: class size Empirical Relationship between Mean, Median, Mode: $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$ Ogive: A cumulative frequency curve. "Less than" ogive: Plots upper class limits vs. less than cumulative frequencies. "More than" ogive: Plots lower class limits vs. more than cumulative frequencies. The x-coordinate of the intersection point of the "less than" and "more than" ogives gives the median. Chapter 14: Probability Probability of an Event ($E$): $P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of possible outcomes}}$ Range of Probability: $0 \le P(E) \le 1$ Complementary Events: For any event $E$, $P(E) + P(\text{not } E) = 1$. $P(\text{not } E)$ is often denoted as $P(\bar{E})$ or $P(E')$. Impossible Event: An event that has no chance of occurring. $P(\text{Impossible Event}) = 0$. Sure Event (Certain Event): An event that is certain to occur. $P(\text{Sure Event}) = 1$. Elementary Event: An event having only one outcome of the experiment. The sum of the probabilities of all elementary events of an experiment is 1.