Class 10 NCERT Maths

Cheatsheet Content

### Real Numbers - **Euclid's Division Lemma:** $a = bq + r$, where $0 \le r ### Polynomials - **Degree of a Polynomial:** The highest power of the variable in a polynomial. - **Types of Polynomials:** - Linear: $ax+b$, degree 1 - Quadratic: $ax^2+bx+c$, degree 2 - Cubic: $ax^3+bx^2+cx+d$, degree 3 - **Zeros of a Polynomial:** Values of $x$ for which $P(x) = 0$. - For a quadratic polynomial $ax^2+bx+c$: - Sum of zeros ($\alpha + \beta$) = $-b/a$ - Product of zeros ($\alpha \beta$) = $c/a$ - If $\alpha, \beta$ are zeros, then $P(x) = k[x^2 - (\alpha+\beta)x + \alpha\beta]$ - **Division Algorithm for Polynomials:** $P(x) = G(x) \cdot Q(x) + R(x)$, where $R(x)=0$ or degree $R(x) ### Pair of Linear Equations in Two Variables - **General Form:** $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ - **Graphical Method:** - **Intersecting Lines:** $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$ (Unique solution) - **Coincident Lines:** $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Infinitely many solutions) - **Parallel Lines:** $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$ (No solution) - **Algebraic Methods:** - **Substitution Method:** Express one variable in terms of the other, substitute into the second equation. - **Elimination Method:** Make coefficients of one variable equal, then add or subtract equations. - **Cross-Multiplication Method:** $$\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$$ ### Quadratic Equations - **General Form:** $ax^2+bx+c=0$, where $a \ne 0$. - **Methods to Solve:** - **Factorisation:** Split the middle term. - **Completing the Square:** $x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 = - \frac{c}{a} + (\frac{b}{2a})^2$ - **Quadratic Formula:** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - **Discriminant ($D$):** $D = b^2 - 4ac$ - If $D > 0$: Two distinct real roots. - If $D = 0$: Two equal real roots. - If $D ### Arithmetic Progressions (AP) - **Definition:** A sequence where the difference between consecutive terms is constant. - **nth Term:** $a_n = a + (n-1)d$ - $a$: first term, $d$: common difference, $n$: number of terms. - **Sum of First n Terms:** - $S_n = \frac{n}{2}[2a + (n-1)d]$ - $S_n = \frac{n}{2}[a + l]$, where $l$ is the last term ($a_n$). ### Triangles - **Similar Triangles:** - Two triangles are similar if: - Their corresponding angles are equal (AAA similarity). - Their corresponding sides are in the same ratio (SSS similarity). - Two corresponding sides are in the same ratio and the included angles are equal (SAS similarity). - **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, the other two sides are divided in the same ratio. - **Areas of Similar Triangles:** The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. - $\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = (\frac{AB}{PQ})^2 = (\frac{BC}{QR})^2 = (\frac{CA}{RP})^2$ - **Pythagoras Theorem:** In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($h^2 = b^2 + p^2$). - Converse: If 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. ### Coordinate Geometry - **Distance Formula:** $D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ - **Section Formula:** - **Internal Division:** $P(x,y) = (\frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2 + m_2y_1}{m_1+m_2})$ - **Mid-point:** $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$ - **Area of a Triangle:** - $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$ - If area is 0, points are collinear. ### Introduction to Trigonometry - **Trigonometric Ratios (Right Triangle):** - $\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{P}{H}$ - $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{B}{H}$ - $\tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{P}{B}$ - $\csc A = \frac{1}{\sin A}$, $\sec A = \frac{1}{\cos A}$, $\cot A = \frac{1}{\tan A}$ - **Trigonometric Identities:** - $\sin^2 A + \cos^2 A = 1$ - $1 + \tan^2 A = \sec^2 A$ (for $A \ne 90^\circ$) - $1 + \cot^2 A = \csc^2 A$ (for $A \ne 0^\circ$) - **Complementary Angles:** - $\sin(90^\circ - A) = \cos A$ - $\cos(90^\circ - A) = \sin A$ - $\tan(90^\circ - A) = \cot A$ - $\cot(90^\circ - A) = \tan A$ - $\sec(90^\circ - A) = \csc A$ - $\csc(90^\circ - A) = \sec A$ - **Specific Angle Values:** | 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 | ### Applications of Trigonometry (Heights and Distances) - **Line of Sight:** The line drawn from the eye of an observer to the object being viewed. - **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. - Use $\sin$, $\cos$, $\tan$ to find unknown heights/distances. ### Circles - **Tangent to a Circle:** A line that intersects the circle at exactly one point. - The tangent at any point of a circle is perpendicular to the radius through the point of contact. - The lengths of tangents drawn from an external point to a circle are equal. - **Number of Tangents from a Point:** - Inside the circle: 0 - On the circle: 1 - Outside the circle: 2 ### Areas Related to Circles - **Circumference of a Circle:** $2\pi r$ - **Area of a Circle:** $\pi r^2$ - **Area of a Sector (angle $\theta$ in degrees):** $\frac{\theta}{360^\circ} \times \pi r^2$ - **Length of an Arc (angle $\theta$ in degrees):** $\frac{\theta}{360^\circ} \times 2\pi r$ - **Area of a Segment:** Area of sector - Area of corresponding triangle. - For minor segment: $\frac{\theta}{360^\circ} \times \pi r^2 - \frac{1}{2}r^2 \sin\theta$ ### Surface Areas and Volumes - **Cuboid:** - TSA = $2(lb+bh+hl)$ - Volume = $lbh$ - **Cube:** - TSA = $6a^2$ - Volume = $a^3$ - **Cylinder:** - CSA = $2\pi rh$ - TSA = $2\pi r(r+h)$ - Volume = $\pi r^2h$ - **Cone:** - CSA = $\pi rl$ (where $l = \sqrt{r^2+h^2}$) - TSA = $\pi r(r+l)$ - Volume = $\frac{1}{3}\pi r^2h$ - **Sphere:** - SA = $4\pi r^2$ - Volume = $\frac{4}{3}\pi r^3$ - **Hemisphere:** - CSA = $2\pi r^2$ - TSA = $3\pi r^2$ - Volume = $\frac{2}{3}\pi r^3$ - **Frustum of a Cone:** (radii $r_1, r_2$, height $h$, slant height $l = \sqrt{h^2+(r_1-r_2)^2}$) - CSA = $\pi (r_1+r_2)l$ - TSA = $\pi (r_1+r_2)l + \pi r_1^2 + \pi r_2^2$ - Volume = $\frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1r_2)$ ### Statistics - **Mean:** - **Direct Method:** $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ - **Assumed Mean Method:** $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$, where $d_i = x_i - a$ - **Step-Deviation Method:** $\bar{x} = a + (\frac{\sum f_i u_i}{\sum f_i})h$, where $u_i = \frac{x_i - a}{h}$ - **Median (for grouped data):** - Median $= L + (\frac{N/2 - CF}{f})h$ - $L$: 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):** - Mode $= L + (\frac{f_1 - f_0}{2f_1 - f_0 - f_2})h$ - $L$: 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 Median = Mode + 2 Mean - **Ogive:** Cumulative frequency curve. Intersection of 'less than' and 'more than' ogives gives the median. ### Probability - **Definition:** $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$ - **Properties:** - $0 \le P(E) \le 1$ - $P(E) + P(\text{not } E) = 1$ - $P(\text{sure event}) = 1$ - $P(\text{impossible event}) = 0$