Class 10 CBSE Math (Standard)

Cheatsheet Content

Real Numbers Euclid's Division Lemma: Given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, $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 & LCM: For any two positive integers $a$ and $b$, $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. Rational Numbers: Numbers of the form $p/q$, where $p, q$ are integers and $q \ne 0$. Terminating or non-terminating repeating decimal expansion. Irrational Numbers: Non-terminating, non-repeating decimal expansion. Examples: $\sqrt{2}, \sqrt{3}, \pi$. Polynomials Degree: Highest power of the variable. Linear: Degree 1 ($ax+b$) Quadratic: Degree 2 ($ax^2+bx+c$) Cubic: Degree 3 ($ax^3+bx^2+cx+d$) 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: $\alpha + \beta + \gamma = -b/a$ $\alpha\beta + \beta\gamma + \gamma\alpha = c/a$ $\alpha\beta\gamma = -d/a$ Division Algorithm: $P(x) = G(x) \cdot Q(x) + R(x)$, where $R(x)=0$ or $\text{deg}(R(x)) Pair of Linear Equations in Two Variables Standard Form: $a_1x + b_1y + c_1 = 0$, $a_2x + b_2y + c_2 = 0$. Graphical Representation: Ratio Graphical Algebraic $a_1/a_2 \ne b_1/b_2$ Intersecting lines Unique solution (consistent) $a_1/a_2 = b_1/b_2 = c_1/c_2$ Coincident lines Infinitely many solutions (consistent) $a_1/a_2 = b_1/b_2 \ne c_1/c_2$ Parallel lines No solution (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 $a \ne 0$. Methods of Solving: Factorisation by splitting the middle term. Completing the Square. Quadratic Formula: $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$. Nature of Roots (Discriminant $D = b^2 - 4ac$): $D > 0$: Two distinct real roots. $D = 0$: Two equal real roots. $D Arithmetic Progressions (AP) General Form: $a, a+d, a+2d, ...$ $n$th Term: $a_n = a + (n-1)d$ Sum of first $n$ terms: $S_n = n/2 [2a + (n-1)d]$ or $S_n = n/2 [a + a_n]$ $d = a_2 - a_1$ (common difference) Triangles Similar Triangles: Corresponding angles are equal. Corresponding sides are in the same ratio. Similarity Criteria: AAA, SSS, SAS. 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. 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. Areas of Similar Triangles: Ratio of areas is equal to the square of the ratio of their corresponding sides. $\text{Area}(\triangle ABC) / \text{Area}(\triangle PQR) = (AB/PQ)^2 = (BC/QR)^2 = (CA/RP)^2$. Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($h^2 = p^2 + b^2$). 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. 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: Point $P(x,y)$ dividing line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in ratio $m:n$ is: $x = (mx_2 + nx_1) / (m+n)$, $y = (my_2 + ny_1) / (m+n)$. Mid-point Formula: $( (x_1+x_2)/2, (y_1+y_2)/2 )$. Area of a Triangle: Vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $1/2 |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. If area is 0, points are collinear. Introduction to Trigonometry Trigonometric Ratios (Right Triangle): $\sin A = \text{Opposite} / \text{Hypotenuse}$ $\cos A = \text{Adjacent} / \text{Hypotenuse}$ $\tan A = \text{Opposite} / \text{Adjacent}$ $\csc A = 1/\sin A$ $\sec A = 1/\cos A$ $\cot A = 1/\tan A$ Identities: $\sin^2 A + \cos^2 A = 1$ $1 + \tan^2 A = \sec^2 A$ $1 + \cot^2 A = \csc^2 A$ 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 Table (Common 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 Some Applications of Trigonometry (Heights and Distances) Line of Sight: The line drawn from the eye of an observer to the object. Angle of Elevation: The angle between the horizontal level and the line of sight when the object is above the horizontal level. Angle of Depression: The angle between the horizontal level and the line of sight when the object is below the horizontal level. Circles Tangent: A line that intersects the circle at exactly one point. Radius & Tangent: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Length of Tangents: The lengths of tangents drawn from an external point to a circle are equal. Constructions Division of a line segment in a given ratio (internally). Construction of a triangle similar to a given triangle (scale factor). Construction of tangents to a circle from an external point. Areas Related to Circles Circumference: $2\pi r$ Area of Circle: $\pi r^2$ Area of Sector: $(\theta/360^\circ) \times \pi r^2$ (where $\theta$ is angle in degrees) Length of Arc: $(\theta/360^\circ) \times 2\pi r$ Area of Segment: Area of sector - Area of corresponding triangle ($1/2 r^2 \sin\theta$ for triangle with angle $\theta$) 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: $\pi r^2 h$ Cone: CSA: $\pi rl$ (where $l = \sqrt{r^2+h^2}$ is slant height) TSA: $\pi r(r+l)$ Volume: $1/3 \pi r^2 h$ Sphere: Surface Area: $4\pi r^2$ Volume: $4/3 \pi r^3$ Hemisphere: CSA: $2\pi r^2$ TSA: $3\pi r^2$ Volume: $2/3 \pi r^3$ Frustum of a Cone: (radii $r_1, r_2$, height $h$, slant height $l$) Volume: $1/3 \pi h (r_1^2 + r_2^2 + r_1r_2)$ CSA: $\pi l (r_1+r_2)$ where $l = \sqrt{h^2 + (r_1-r_2)^2}$ TSA: $\pi l (r_1+r_2) + \pi r_1^2 + \pi r_2^2$ Statistics Mean (Grouped Data): 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 $u_i = (x_i - a) / h$ Mode (Grouped Data): $Mode = l + ((f_1 - f_0) / (2f_1 - f_0 - f_2)) \times h$ (where $l=$ lower limit of modal class, $h=$ class size, $f_1=$ freq. of modal class, $f_0=$ freq. of class preceding modal class, $f_2=$ freq. of class succeeding modal class) Median (Grouped Data): $Median = l + ((n/2 - cf) / f) \times h$ (where $l=$ lower limit of median class, $n=$ total freq., $cf=$ cumulative freq. of class preceding median class, $f=$ freq. of median class, $h=$ class size) Empirical Relationship: $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$ Ogive: Cumulative frequency curve. Intersection of 'less than' and 'more than' ogives gives the median. Probability Definition: $P(E) = \text{Number of favorable outcomes} / \text{Total number of possible outcomes}$ $0 \le P(E) \le 1$ $P(E) + P(\text{not } E) = 1$ Impossible Event: $P(\text{impossible event}) = 0$ Sure Event: $P(\text{sure event}) = 1$