Class 10 Mathematics Case Study Revision Notes

Real Numbers

• Fundamental Theorem of Arithmetic states every composite number can be uniquely expressed as a product of prime factors.

• For any two positive integers $a$ and $b$:     $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

Polynomials

• Standard form of a quadratic polynomial:     $f(x) = a x^2 + b x + c$     where $a \neq 0$ and $a, b, c$ are real numbers.

• A quadratic polynomial has equal roots when the discriminant $D = 0$, where $D = b^2 - 4 a c$.

• The graph of a quadratic polynomial forms a parabola:

  • Opens upwards if $a > 0$

  • Opens downwards if $a < 0$

• Quadratic polynomial expression given sum ($S$) and product ($P$) of zeroes:     $k(x^2 - S x + P)$

Linear Equations in Two Variables

• General form of a pair of linear equations in two variables:     $a_1 x + b_1 y + c_1 = 0$     $a_2 x + b_2 y + c_2 = 0$

• System of linear equations represents linear relationships used to solve problems involving rates, scoring systems, and physical dimensions.

Quadratic Equations

• Relation between speed, distance, and time:     $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Upstream and downstream motion equations:

  • $\text{Upstream speed} = \text{Speed of boat} - \text{Speed of stream}$

  • $\text{Downstream speed} = \text{Speed of boat} + \text{Speed of stream}$

Arithmetic Progression

• An Arithmetic Progression (AP) is a sequence in which the difference between consecutive terms remains constant.

• General $n^{\text{th}}$ term formula:     $a_n = a + (n - 1)d$

• Sum of first $n$ terms formula:     $S_n = \frac{n}{2}[2a + (n - 1)d]$

Similar Triangles and Scale Factor

• Scale factor formula:     $\text{Scale factor} = \frac{\text{Length in image}}{\text{Corresponding length in object}}$

• Two shapes are similar when one can become another through resizing, flipping, sliding, or turning.

• Pythagoras Theorem for right-angled triangles:     $a^2 + b^2 = c^2$

Coordinate Geometry

• Distance between two points $(x_1, y_1)$ and $(x_2, y_2)$:     $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint formula for a line segment connecting two points:     $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Circles

• A tangent to a circle is perpendicular to the radius at the point of contact.

• Lengths of tangents drawn from an external point to a circle are equal.

Applications of Trigonometry

• Angle of Elevation: The angle formed by the line of sight with the horizontal line when the object is above horizontal level.

• Angle of Depression: The angle formed by the line of sight with the horizontal line when the object is below horizontal level.

• Basic trigonometric ratio relation:     $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Areas Related to Circles

• Area of a circle:     $A = \pi r^2$

• Circumference of a circle:     $C = 2 \pi r$

• Area of a sector with central angle $\theta$:     $A = \frac{\theta}{360^\circ} \times \pi r^2$

Surface Areas and Volumes

• Volume of a cylinder:     $V = \pi r^2 h$

• Volume of a hemisphere:     $V = \frac{2}{3} \pi r^3$

• Volume of a sphere:     $V = \frac{4}{3} \pi r^3$

• Slant height $l$ of a cone:     $l = \sqrt{r^2 + h^2}$

Statistics

• Modal class is the class interval containing the highest frequency in a dataset.

• Median represents the middle observation of grouped or ungrouped numerical data.

Probability

• Theoretical probability of an event $E$:     $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

• The probability of an impossible event is $0$, and the probability of a sure event is $1$.