SSLC Karnataka Math

Cheatsheet Content

### Arithmetic Progressions - **Definition:** A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference ($d$). - **General Form:** $a, a+d, a+2d, ..., a+(n-1)d$ - **$n^{th}$ term:** $a_n = a + (n-1)d$ - $a$: first term - $n$: number of terms - $d$: common difference - **Sum of first $n$ terms:** - $S_n = \frac{n}{2}[2a + (n-1)d]$ - $S_n = \frac{n}{2}(a + a_n)$, if the last term $a_n$ is known. - **Common difference:** $d = a_2 - a_1 = a_3 - a_2 = ...$ ### Triangles - **Thales Theorem (Basic Proportionality Theorem - BPT):** 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. - If $DE \parallel BC$, then $\frac{AD}{DB} = \frac{AE}{EC}$. - **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:** 1. **AAA Similarity:** If corresponding angles are equal. 2. **SSS Similarity:** If corresponding sides are in the same ratio. 3. **SAS Similarity:** If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional. - **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. - $\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = (\frac{AB}{PQ})^2 = (\frac{BC}{QR})^2 = (\frac{AC}{PR})^2$ - **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 ($c^2 = a^2 + b^2$). - **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. ### Pair of Linear Equations in Two Variables - **General Form:** - $a_1x + b_1y + c_1 = 0$ - $a_2x + b_2y + c_2 = 0$ - **Methods of Solving:** 1. **Graphical Method:** Intersection point is the solution. 2. **Substitution Method:** Express one variable in terms of the other from one equation and substitute into the second equation. 3. **Elimination Method:** Multiply equations by suitable numbers to make coefficients of one variable equal, then add or subtract. 4. **Cross-Multiplication Method:** - $x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}$ - $y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}$ - **Conditions for Solutions:** 1. **Intersecting Lines (Unique Solution):** $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ 2. **Coincident Lines (Infinitely Many Solutions):** $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ 3. **Parallel Lines (No Solution):** $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ ### Quadratic Equations - **Standard Form:** $ax^2 + bx + c = 0$, where $a \neq 0$. - **Roots of a Quadratic Equation:** Values of $x$ that satisfy the equation. - **Methods of Solving:** 1. **Factorisation Method:** Factor the quadratic polynomial into two linear factors. 2. **Completing the Square Method:** Transform the equation into $(x+k)^2 = m$. 3. **Quadratic Formula:** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - **Discriminant ($\Delta$ or $D$):** $D = b^2 - 4ac$ - **Nature of Roots:** 1. If $D > 0$: Two distinct real roots. 2. If $D = 0$: Two equal real roots. 3. If $D ### Circles - **Tangent to a Circle:** A line that intersects the circle at exactly one point. - **Properties of Tangents:** 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. 2. The lengths of tangents drawn from an external point to a circle are equal. - **Number of Tangents:** - From an interior point: 0 - From a point on the circle: 1 - From an exterior point: 2 ### Areas Related to Circles - **Circumference of a circle:** $C = 2\pi r$ - **Area of a circle:** $A = \pi r^2$ - **Area of a sector with angle $\theta$ (in degrees):** $A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2$ - **Length of an arc of a sector with angle $\theta$ (in degrees):** $L_{arc} = \frac{\theta}{360^\circ} \times 2\pi r$ - **Area of a segment:** Area of sector - Area of corresponding triangle. - For minor segment: $\frac{\theta}{360^\circ} \times \pi r^2 - \frac{1}{2}r^2 \sin\theta$ (for $\theta \le 180^\circ$) - For major segment: Area of circle - Area of minor segment. ### 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:** 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: - $x = \frac{m_1x_2 + m_2x_1}{m_1+m_2}$ - $y = \frac{m_1y_2 + m_2y_1}{m_1+m_2}$ - **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: - $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$ - **Area of a Triangle:** For a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: - Area $= \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$ - **Collinearity:** Three points are collinear if the area of the triangle formed by them is 0. ### 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 - **Degree of a Polynomial:** The highest power of the variable in a polynomial. - **Types of Polynomials:** - Linear: Degree 1 (e.g., $ax+b$) - Quadratic: Degree 2 (e.g., $ax^2+bx+c$) - Cubic: Degree 3 (e.g., $ax^3+bx^2+cx+d$) - **Zeros of a Polynomial:** The values of $x$ for which the polynomial $P(x)$ equals 0. - Geometrically, the x-coordinates where the graph intersects the x-axis. - **Relationship between Zeros and Coefficients of a Quadratic Polynomial ($ax^2+bx+c$):** - Sum of zeros ($\alpha + \beta$) $= -\frac{b}{a}$ - Product of zeros ($\alpha \beta$) $= \frac{c}{a}$ - **Relationship between Zeros and Coefficients of a Cubic Polynomial ($ax^3+bx^2+cx+d$):** - Sum of zeros ($\alpha + \beta + \gamma$) $= -\frac{b}{a}$ - Sum of products of zeros taken two at a time ($\alpha\beta + \beta\gamma + \gamma\alpha$) $= \frac{c}{a}$ - Product of zeros ($\alpha\beta\gamma$) $= -\frac{d}{a}$ - **Division Algorithm for Polynomials:** If $P(x)$ and $G(x)$ are any two polynomials with $G(x) \neq 0$, then we can find polynomials $Q(x)$ and $R(x)$ such that $P(x) = G(x) \times Q(x) + R(x)$, where $R(x) = 0$ or degree of $R(x) ### Introduction to Trigonometry - **Trigonometric Ratios (Right-Angled Triangle):** - $\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - $\tan A = \frac{\text{Opposite}}{\text{Adjacent}}$ - $\csc A = \frac{1}{\sin A}$ - $\sec A = \frac{1}{\cos A}$ - $\cot A = \frac{1}{\tan A}$ - **Trigonometric Ratios of Some 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 | - **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 Identities:** 1. $\sin^2 A + \cos^2 A = 1$ 2. $1 + \tan^2 A = \sec^2 A$ 3. $1 + \cot^2 A = \csc^2 A$ ### Some Applications of Trigonometry - **Line of Sight:** The line drawn from the eye of an observer to the object viewed. - **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. - **Solving Problems:** Use appropriate trigonometric ratios ($\sin, \cos, \tan$) based on the given information (angle, side lengths) to find unknown heights or distances. ### Surface Areas and Volumes - **Cuboid:** - Volume: $V = l \times b \times h$ - Lateral Surface Area: $2h(l+b)$ - Total Surface Area: $2(lb + bh + hl)$ - **Cube:** - Volume: $V = a^3$ - Lateral Surface Area: $4a^2$ - Total Surface Area: $6a^2$ - **Cylinder:** - Volume: $V = \pi r^2 h$ - Curved Surface Area: $2\pi r h$ - Total Surface Area: $2\pi r (r+h)$ - **Cone:** - Volume: $V = \frac{1}{3}\pi r^2 h$ - Curved Surface Area: $\pi r l$ (where $l = \sqrt{r^2+h^2}$ is slant height) - Total Surface Area: $\pi r (l+r)$ - **Sphere:** - Volume: $V = \frac{4}{3}\pi r^3$ - Surface Area: $A = 4\pi r^2$ - **Hemisphere:** - Volume: $V = \frac{2}{3}\pi r^3$ - Curved Surface Area: $2\pi r^2$ - Total Surface Area: $3\pi r^2$ - **Frustum of a Cone:** (If covered, typically for higher levels, but might be in some syllabi) - Volume: $V = \frac{1}{3}\pi h (R^2 + r^2 + Rr)$ - Curved Surface Area: $A = \pi l (R+r)$ (where $l = \sqrt{h^2+(R-r)^2}$) - Total Surface Area: $\pi l (R+r) + \pi R^2 + \pi r^2$ ### Statistics - **Measures of Central Tendency:** 1. **Mean ($\bar{x}$):** - 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 $u_i = \frac{x_i - A}{h}$ 2. **Median:** The middle-most value when data is arranged in ascending/descending order. - For grouped data: Median $= L + \left(\frac{\frac{n}{2} - cf}{f}\right)h$ - $L$: lower limit of median class - $n$: total frequency - $cf$: cumulative frequency of class preceding median class - $f$: frequency of median class - $h$: class size 3. **Mode:** The value that appears most frequently. - 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:** $3 \text{ Median } = \text{ Mode } + 2 \text{ Mean }$ - **Ogive (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 intersection point of less than and more than ogives gives the Median. ### Probability - **Definition:** Probability of an event $E$, denoted $P(E)$, is a measure of the likelihood of the event occurring. - $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$ - **Range of Probability:** $0 \le P(E) \le 1$ - **Impossible Event:** An event that cannot occur ($P(E) = 0$). - **Sure Event (Certain Event):** An event that is certain to occur ($P(E) = 1$). - **Complementary Events:** The event 'not E' (denoted $\bar{E}$ or $E^c$). - $P(E) + P(\bar{E}) = 1$ - $P(\bar{E}) = 1 - P(E)$ - **Elementary Event:** An event having only one outcome of the experiment. The sum of probabilities of all elementary events of an experiment is 1.