Class 10 CBSE Math Cheatsheet

Cheatsheet Content

### Real Numbers - **Euclid's Division Lemma:** Given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ satisfying $a = bq + r$, where $0 \le r ### Polynomials - **General Form of a Quadratic Polynomial:** $ax^2 + bx + c$, where $a \ne 0$. - **Zeros of a Polynomial:** Values of $x$ for which $P(x) = 0$. - **Relationship between Zeros and Coefficients:** - For $ax^2 + bx + c$: - Sum of zeros ($\alpha + \beta$) = $-b/a$ - Product of zeros ($\alpha\beta$) = $c/a$ - For $ax^3 + bx^2 + cx + d$: - 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 $Q(x)$ and $R(x)$ such that $P(x) = G(x) \cdot Q(x) + R(x)$, 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 Solvability:** - **Intersecting Lines (Unique Solution):** $a_1/a_2 \ne b_1/b_2$ (Consistent) - **Coincident Lines (Infinitely Many Solutions):** $a_1/a_2 = b_1/b_2 = c_1/c_2$ (Consistent and Dependent) - **Parallel Lines (No Solution):** $a_1/a_2 = b_1/b_2 \ne c_1/c_2$ (Inconsistent) - **Methods of Solving:** - Substitution Method - Elimination Method - Cross-Multiplication Method: $x / (b_1c_2 - b_2c_1) = y / (c_1a_2 - c_2a_1) = 1 / (a_1b_2 - a_2b_1)$ ### Quadratic Equations - **Standard Form:** $ax^2 + bx + c = 0$, where $a \ne 0$. - **Quadratic Formula:** $x = [-b \pm \sqrt{b^2 - 4ac}] / 2a$ - **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) - **General Form:** $a, a+d, a+2d, ...$ - **$n^{th}$ term:** $a_n = a + (n-1)d$ - $a$: first term - $d$: common difference - $n$: number of terms - **Sum of first $n$ terms:** - $S_n = n/2 [2a + (n-1)d]$ - $S_n = n/2 (a + a_n)$ (when the last term $a_n$ is known) ### 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 || BC, 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. - **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 ($h^2 = b^2 + p^2$). - **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. - **Area of Similar Triangles Theorem:** The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. - Area($\triangle ABC$) / Area($\triangle DEF$) = $(AB/DE)^2 = (BC/EF)^2 = (AC/DF)^2$. ### Coordinate Geometry - **Distance Formula:** Distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. - **Section Formula:** - **Internal Division:** Coordinates of point $P(x,y)$ dividing 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:** $x = (x_1+x_2)/2$, $y = (y_1+y_2)/2$ (when $m=n=1$) - **Area of a Triangle:** With vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: Area $= 1/2 |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$ - If area is 0, points are collinear. ### Introduction to Trigonometry - **Trigonometric Ratios:** - $\sin A = \text{Opposite} / \text{Hypotenuse}$ - $\cos A = \text{Adjacent} / \text{Hypotenuse}$ - $\tan A = \text{Opposite} / \text{Adjacent} = \sin A / \cos A$ - $\csc A = 1/\sin A$ - $\sec A = 1/\cos A$ - $\cot A = 1/\tan A = \cos A / \sin A$ - **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:** - $\sin(90^\circ - A) = \cos A$ - $\cos(90^\circ - A) = \sin A$ - $\tan(90^\circ - A) = \cot A$ - $\cot(90^\circ - A) = \tan A$ - $\sec(90^\circ - A) = \csc A$ - $\csc(90^\circ - A) = \sec A$ - **Table of 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 | ### Applications of Trigonometry (Heights & 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. - Use $\sin, \cos, \tan$ ratios to solve problems involving heights and distances. ### Circles - **Tangent to a Circle:** A line that intersects the circle at only one point. - **Theorem 1:** The tangent at any point of a circle is perpendicular to the radius through the point of contact. - **Theorem 2:** The lengths of tangents drawn from an external point to a circle are equal. - **Number of Tangents:** - No tangent from an interior point. - One tangent from a point on the circle. - Two tangents from an exterior point. ### Areas Related to Circles - **Circumference of a circle:** $2\pi r$ - **Area of a circle:** $\pi r^2$ - **Area of a sector of angle $\theta$:** $(\theta/360^\circ) \times \pi r^2$ - **Length of an arc of a sector of angle $\theta$:** $(\theta/360^\circ) \times 2\pi r$ - **Area of a segment:** Area of sector - Area of corresponding triangle. - For minor segment: Area $= (\theta/360^\circ) \times \pi r^2 - 1/2 r^2 \sin \theta$ ### Surface Areas and Volumes - **Cuboid:** - Volume: $l \times b \times h$ - Lateral Surface Area: $2h(l+b)$ - Total Surface Area: $2(lb + bh + hl)$ - **Cube:** - Volume: $a^3$ - Lateral Surface Area: $4a^2$ - Total Surface Area: $6a^2$ - **Cylinder:** - Volume: $\pi r^2 h$ - Curved Surface Area: $2\pi r h$ - Total Surface Area: $2\pi r(r+h)$ - **Cone:** - Volume: $1/3 \pi r^2 h$ - Curved Surface Area: $\pi r l$, where $l = \sqrt{r^2+h^2}$ (slant height) - Total Surface Area: $\pi r(l+r)$ - **Sphere:** - Volume: $4/3 \pi r^3$ - Surface Area: $4\pi r^2$ - **Hemisphere:** - Volume: $2/3 \pi r^3$ - Curved Surface Area: $2\pi r^2$ - Total Surface Area: $3\pi r^2$ - **Frustum of a Cone:** (If included in syllabus) - Volume: $1/3 \pi h (R^2 + r^2 + Rr)$ - Curved Surface Area: $\pi l (R+r)$, where $l = \sqrt{h^2 + (R-r)^2}$ - Total Surface Area: $\pi l (R+r) + \pi R^2 + \pi r^2$ ### Statistics - **Mean:** - **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:** $\bar{x} = A + (\sum f_i u_i / \sum f_i) \times h$, where $u_i = (x_i - A) / h$ - **Median:** - For grouped data: Median $= L + [(N/2 - cf) / f] \times h$ - $L$: 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 $= L + [(f_1 - f_0) / (2f_1 - f_0 - f_2)] \times h$ - $L$: 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:** $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$ - **Ogives:** Cumulative frequency curves (less than type and more than type). Intersection gives the Median. ### Probability - **Definition:** $P(E) = \text{(Number of favorable outcomes)} / \text{(Total number of possible outcomes)}$ - **Properties:** - $0 \le P(E) \le 1$ - $P(E) + P(\text{not } E) = 1$ - $P(\text{impossible event}) = 0$ - $P(\text{sure event}) = 1$