Class 10 Maths Formula Sheet

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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: For any two positive integers $a$ and $b$, $HCF(a, b) \times LCM(a, b) = a \times b$. Decimal Expansion of Rational Numbers: Terminating: Denominator is of the form $2^n 5^m$. Non-terminating Repeating: Denominator is not of the form $2^n 5^m$. Polynomials Degree of a Polynomial: The highest power of the variable in a polynomial. Linear Polynomial: $ax + b$, degree 1. Quadratic Polynomial: $ax^2 + bx + c$, degree 2 ($a \ne 0$). Cubic Polynomial: $ax^3 + bx^2 + cx + d$, degree 3 ($a \ne 0$). Zeros of a Polynomial: Values of $x$ for which $P(x) = 0$. For quadratic $ax^2+bx+c$: If $\alpha, \beta$ are zeros, then: Sum of zeros: $\alpha + \beta = -b/a$ Product of zeros: $\alpha \beta = c/a$ For cubic $ax^3+bx^2+cx+d$: If $\alpha, \beta, \gamma$ are zeros, then: 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: $P(x) = G(x) \cdot Q(x) + R(x)$, where $R(x)=0$ or degree of $R(x) Pair of Linear Equations in Two Variables General Form: $a_1x + b_1y + c_1 = 0$ $a_2x + b_2y + c_2 = 0$ Conditions for Solutions: Unique Solution (Intersecting Lines): $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$ No Solution (Parallel Lines): $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$ Infinitely Many Solutions (Coincident Lines): $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ Methods of Solving: Substitution, Elimination, Cross-multiplication. Cross-multiplication Method: $$ \frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1} $$ Provided $a_1b_2 - a_2b_1 \ne 0$. Quadratic Equations General Form: $ax^2 + bx + c = 0$, where $a \ne 0$. Methods of Solving: Factorisation by splitting the middle term. Completing the square. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ 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 Arithmetic Progressions (AP) General Form: $a, a+d, a+2d, \dots$ $n$-th Term ($a_n$): $a_n = a + (n-1)d$ $a$: first term $d$: common difference $n$: number of terms Sum of First $n$ Terms ($S_n$): $S_n = \frac{n}{2}[2a + (n-1)d]$ $S_n = \frac{n}{2}[a + l]$, where $l = a_n$ (last term) Coordinate Geometry Distance Formula: Distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ is $$ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$ Section Formula: Coordinates of a point $P(x, y)$ that divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m_1:m_2$ are: $$ \left( \frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2 + m_2y_1}{m_1+m_2} \right) $$ Mid-point Formula: Mid-point of the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ is: $$ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) $$ Area of a Triangle: Area of triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is: $$ \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| $$ If the area is 0, the points are collinear. Introduction to Trigonometry Trigonometric Ratios (Right-angled triangle): $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$ $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$ $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$ $\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}}$ Relations: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$ Trigonometric Identities: $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ Complementary Angles: $\sin (90^\circ - \theta) = \cos \theta$ $\cos (90^\circ - \theta) = \sin \theta$ $\tan (90^\circ - \theta) = \cot \theta$ $\cot (90^\circ - \theta) = \tan \theta$ $\sec (90^\circ - \theta) = \csc \theta$ $\csc (90^\circ - \theta) = \sec \theta$ Trigonometric Ratios Table: Angle $\theta$ $0^\circ$ $30^\circ$ $45^\circ$ $60^\circ$ $90^\circ$ $\sin \theta$ $0$ $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ $1$ $\cos \theta$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ $0$ $\tan \theta$ $0$ $\frac{1}{\sqrt{3}}$ $1$ $\sqrt{3}$ Undefined $\csc \theta$ Undefined $2$ $\sqrt{2}$ $\frac{2}{\sqrt{3}}$ $1$ $\sec \theta$ $1$ $\frac{2}{\sqrt{3}}$ $\sqrt{2}$ $2$ Undefined $\cot \theta$ Undefined $\sqrt{3}$ $1$ $\frac{1}{\sqrt{3}}$ $0$ Some Applications of Trigonometry Angle of Elevation: Angle formed by the line of sight with the horizontal when the object is above the horizontal level. Angle of Depression: Angle formed by the line of sight with the horizontal when the object is below the horizontal level. Circles Tangent to a Circle: A line that intersects the circle at exactly one point. Radius and Tangent: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Length of Tangents from an External Point: The lengths of tangents drawn from an external point to a circle are equal. Areas Related to Circles Circumference of a Circle: $2\pi r$ Area of a Circle: $\pi r^2$ Area of a Sector of Angle $\theta$: $\frac{\theta}{360^\circ} \times \pi r^2$ Length of an Arc of Angle $\theta$: $\frac{\theta}{360^\circ} \times 2\pi r$ Area of a Segment: Area of sector - Area of corresponding triangle. Surface Areas and Volumes Cuboid: Lateral Surface Area: $2h(l+b)$ Total Surface Area: $2(lb+bh+hl)$ Volume: $lbh$ Cube: Lateral Surface Area: $4a^2$ Total Surface Area: $6a^2$ Volume: $a^3$ Cylinder: Curved Surface Area: $2\pi rh$ Total Surface Area: $2\pi r(r+h)$ Volume: $\pi r^2 h$ Cone: Curved Surface Area: $\pi rl$ (where $l = \sqrt{r^2+h^2}$ is slant height) Total Surface Area: $\pi r(l+r)$ Volume: $\frac{1}{3}\pi r^2 h$ Sphere: Surface Area: $4\pi r^2$ Volume: $\frac{4}{3}\pi r^3$ Hemisphere: Curved Surface Area: $2\pi r^2$ Total Surface Area: $3\pi r^2$ Volume: $\frac{2}{3}\pi r^3$ Frustum of a Cone: (where $r_1, r_2$ are radii of bases, $h$ height, $l$ slant height) Slant Height $l = \sqrt{h^2 + (r_1-r_2)^2}$ Curved Surface Area: $\pi (r_1+r_2)l$ Total Surface Area: $\pi (r_1+r_2)l + \pi r_1^2 + \pi r_2^2$ Volume: $\frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1r_2)$ Statistics Mean: Direct Method: $\bar{x} = \frac{\sum f_ix_i}{\sum f_i}$ Assumed Mean Method: $\bar{x} = a + \frac{\sum f_id_i}{\sum f_i}$, where $d_i = x_i - a$ Step-Deviation Method: $\bar{x} = a + \left(\frac{\sum f_iu_i}{\sum f_i}\right)h$, where $u_i = \frac{x_i - a}{h}$ Median: For grouped data, Median $= L + \left(\frac{N/2 - cf}{f}\right)h$ $L$: lower limit of median class $N$: sum of frequencies $cf$: cumulative frequency of class preceding median class $f$: frequency of median class $h$: class size Mode: For grouped data, Mode $= L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right)h$ $L$: 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: $3 \text{ Median } = \text{ Mode } + 2 \text{ Mean}$ Probability Probability of an Event $E$: $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$ Range of Probability: $0 \le P(E) \le 1$ Sum of Probabilities: $P(E) + P(\text{not } E) = 1$ $P(\text{certain event}) = 1$ $P(\text{impossible event}) = 0$