Class 10 Maths Formula Sheet
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Chapter 1: Real Numbers
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Euclid's Division Lemma:
For $a$ and $b$, $a = bq + r$, where $0 \le r < b$.
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Product of HCF and LCM (Two Numbers):
$HCF(a, b) \times LCM(a, b) = a \times b$
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Condition for Terminating Decimal $\frac{p}{q}$:
$q$ must be of the form $2^m \times 5^n$.
Chapter 2: Polynomials
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Sum of Zeroes ($\alpha + \beta$) for $ax^2 + bx + c$:
$\alpha + \beta = -\frac{b}{a}$
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Product of Zeroes ($\alpha\beta$) for $ax^2 + bx + c$:
$\alpha\beta = \frac{c}{a}$
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Division Algorithm:
$p(x) = g(x) \times q(x) + r(x)$
Chapter 3 & 4: Linear & Quadratic Equations
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Quadratic Formula ($ax^2 + bx + c = 0$):
$x = \frac{-b \pm \sqrt{D}}{2a}$, where $D = b^2 - 4ac$ (Discriminant)
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Nature of Roots:
D > 0: Real & Distinct
D = 0: Real & Equal
D < 0: No Real Roots -
Cross Multiplication Method:
$x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}$ and $y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}$
Chapter 5: Arithmetic Progression (AP)
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$n^{\text{th}}$ Term ($T_n$):
$T_n = a + (n-1)d$
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Sum of $n$ Terms ($S_n$):
$S_n = \frac{n}{2}[2a + (n-1)d] \quad \text{or} \quad S_n = \frac{n}{2}[a + l]$
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Arithmetic Mean ($a, b, c$ in AP):
$2b = a + c$
Chapter 6: Triangles
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Basic Proportionality Theorem (BPT/Thales):
If $\text{DE} \parallel \text{BC}$, then $\frac{AD}{BD} = \frac{AE}{CE}$
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Ratio of Areas (Similar $\triangle$s):
$\frac{\text{ar}(\Delta ABC)}{\text{ar}(\Delta PQR)} = \frac{AB^2}{PQ^2}$
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Pythagoras Theorem ($\angle B = 90^\circ$):
$AB^2 + BC^2 = AC^2$
Chapter 7: Coordinate Geometry
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Distance Formula:
$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
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Section Formula ($m:n$):
$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$
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Area of Triangle:
$\frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
Chapter 8 & 9: Trigonometry
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Trigonometric Identity 1:
$\sin^2 \theta + \cos^2 \theta = 1$
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Trigonometric Identity 2:
$1 + \tan^2 \theta = \sec^2 \theta$
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Trigonometric Identity 3:
$1 + \cot^2 \theta = \csc^2 \theta$
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Complementary Angle:
$\sin(90^\circ - \theta) = \cos \theta$ and $\tan(90^\circ - \theta) = \cot \theta$
Chapter 10: Circles
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Tangent-Radius Theorem:
Tangent at any point is $\perp$ to the radius through point of contact.
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Tangents from External Point:
Lengths of tangents drawn from an external point are equal ($PQ=PR$).
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Cyclic Quadrilateral:
Sum of opposite angles is $180^\circ$ ($\angle A + \angle C = 180^\circ$).
Chapter 12: Area Related to Circles
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Area of Circle:
$\pi r^2$
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Arc Length (Angle $\theta$):
Arc Length $= \frac{\theta}{360} \times 2\pi r$
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Area of Sector (Angle $\theta$):
Area of Sector $= \frac{\theta}{360} \times \pi r^2$
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Area of Minor Segment:
Area of Sector $-\frac{1}{2} r^2 \sin \theta$
Chapter 13: Surface Area & Volume
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Volume of Cuboid:
$L \times B \times H$
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Volume of Cylinder:
$\pi r^2 h$
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CSA of Cone:
$\pi r l$, where $l = \sqrt{r^2+h^2}$
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Volume of Sphere:
$\frac{4}{3} \pi r^3$
Chapter 14: Statistics
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Mean (Direct Method):
$\overline{X} = \frac{\sum f_i x_i}{\sum f_i}$
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Median (Grouped Data):
$L + \left(\frac{\frac{n}{2} - \text{pcf}}{f}\right) \times h$
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Mode (Grouped Data):
$L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$
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Empirical Relation:
Mode = 3 Median - 2 Mean
Chapter 15: Probability
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Probability of an Event $E$ ($P(E)$):
$P(E) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}$
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Complementary Event ($\overline{E}$):
$P(E) + P(\overline{E}) = 1$
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Range of Probability:
$0 \le P(E) \le 1$