Composite Numbers: A positive integer greater than 1 that has more than two distinct factors.
Represent 10,626 as a product of unique prime factors.
Solution: $10,626 = 2 \times 3 \times 7 \times 11 \times 23$
Prime Factorisation Method to find HCF and LCM:
- Find all prime factors of given numbers.
- HCF: Product of the smallest power of each common prime factor involved in the numbers.
- LCM: Product of the greatest power of each prime factor involved in the numbers.
Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Solution:
Prime factorisation of $6 = 2^1 \times 3^1$
Prime factorisation of $20 = 2^2 \times 5^1$
$\text{HCF}(6, 20) = 2^1 = 2$
$\text{LCM}(6, 20) = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60$
A number is called irrational if it cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples: $\sqrt{2}, \sqrt{3}, \pi$.
Theorem: Let $p$ be a prime number. If $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.
Solution:
Assume to the contrary that $\sqrt{2}$ is rational. Let $\sqrt{2} = \frac{p}{q}$, where $p$ and $q$ are co-prime integers and $q \neq 0$.
Squaring both sides: $2 = \frac{p^2}{q^2} \implies 2q^2 = p^2$.
Therefore, $2$ divides $p^2$, which implies $2$ divides $p$.
Let $p = 2r$ for some integer $r$.
Substituting $p$: $2q^2 = 4r^2 \implies q^2 = 2r^2$.
This implies $2$ divides $q^2$, which means $2$ divides $q$.
Thus, $2$ is a common factor of both $p$ and $q$, contradicting our assumption that $p$ and $q$ are co-prime.
Hence, $\sqrt{2}$ is irrational.
Zero of a Polynomial: A real number $k$ is said to be a zero of a polynomial $p(x)$ if $p(k) = 0$.
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = ax^2 + bx + c$, then:
For a quadratic expression $f(x) = ax^2 + bx + c$, the Discriminant is given by $D = b^2 - 4ac$.
General form of a pair of linear equations in two variables:
| Compare Ratios | Graphical Representation | Algebraic Interpretation | Consistency |
|---|---|---|---|
| $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ | Intersecting lines | Exactly one solution (Unique) | Consistent |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ | Coincident lines | Infinitely many solutions | Consistent (Dependent) |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ | Parallel lines | No solution | Inconsistent |
Standard Form: $ax^2 + bx + c = 0$, where $a \neq 0$ and $a, b, c$ are real numbers.
Nature of roots based on Discriminant $D = b^2 - 4ac$:
- If $D > 0$: Two distinct real roots.
- If $D = 0$: Two equal real roots (i.e., $x = -\frac{b}{2a}$).
- If $D < 0$: No real roots.
Standard AP sequence: $a, a+d, a+2d, a+3d, \dots$, where $a$ is the first term and $d$ is the common difference.
- Common Difference: $d = a_k - a_{k-1}$
- $n^{\text{th}}$ term from start:
$$a_n = a + (n - 1)d$$
- $n^{\text{th}}$ term from end:
$$a_n = l - (n - 1)d \quad \text{(where } l \text{ is the last term)}$$
- Sum of first $n$ terms:
$$S_n = \frac{n}{2}[2a + (n-1)d] \quad \text{or} \quad S_n = \frac{n}{2}[a + l]$$
- Distance Formula: Distance between points $P(x_1, y_1)$ and $Q(x_2, y_2)$:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
- Distance from Origin $(0,0)$:
$$d = \sqrt{x^2 + y^2}$$
- Section Formula (Internal Division): Coordinates of point dividing segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in ratio $m_1 : m_2$:
$$\left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right)$$
- Midpoint Formula:
$$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
Key Properties & Theorems:
- 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.
| $\angle A$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin A$ | $0$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | $1$ |
| $\cos A$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | $0$ |
| $\tan A$ | $0$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | Not defined |
| $\csc A$ | Not defined | $2$ | $\sqrt{2}$ | $\frac{2}{\sqrt{3}}$ | $1$ |
| $\sec A$ | $1$ | $\frac{2}{\sqrt{3}}$ | $\sqrt{2}$ | $2$ | Not defined |
| $\cot A$ | Not defined | $\sqrt{3}$ | $1$ | $\frac{1}{\sqrt{3}}$ | $0$ |
Angle of Elevation: The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.
Angle of Depression: The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.
- Circumference of a Circle: $C = 2\pi r$
- Area of a Circle: $A = \pi r^2$
- Length of an Arc of a Sector of Angle $\theta$:
$$l = \frac{\theta}{360^\circ} \times 2\pi r$$
- Area of a Sector of Angle $\theta$:
$$A = \frac{\theta}{360^\circ} \times \pi r^2$$
- Area of a Segment of a Circle:
$$\text{Area of Segment} = \text{Area of Sector} - \text{Area of Corresponding Triangle}$$$$\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 - \frac{1}{2} r^2 \sin \theta$$
| Solid Shape | Curved/Lateral Surface Area | Total Surface Area | Volume |
|---|---|---|---|
| Cuboid | $2h(l + b)$ | $2(lb + bh + hl)$ | $l \times b \times h$ |
| Cube | $4a^2$ | $6a^2$ | $a^3$ |
| Right Circular Cylinder | $2\pi r h$ | $2\pi r(h + r)$ | $\pi r^2 h$ |
| Right Circular Cone | $\pi r l \quad (l = \sqrt{h^2+r^2})$ | $\pi r(l + r)$ | $\frac{1}{3}\pi r^2 h$ |
| Sphere | $4\pi r^2$ | $4\pi r^2$ | $\frac{4}{3}\pi r^3$ |
| Hemisphere | $2\pi r^2$ | $3\pi r^2$ | $\frac{2}{3}\pi r^3$ |
- Direct Method for Mean:
$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$
- Assumed Mean Method for Mean:
$$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \quad \text{where } d_i = x_i - a$$
- Mode of Grouped Data:
$$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$($l = \text{lower limit of modal class}, f_1 = \text{frequency of modal class}, f_0 = \text{frequency of preceding class}, f_2 = \text{frequency of succeeding class}, h = \text{class size}$)
- Median of Grouped Data:
$$\text{Median} = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$$($l = \text{lower limit of median class}, N = \text{total frequency}, cf = \text{cumulative frequency of preceding class}, f = \text{frequency of median class}, h = \text{class size}$)
- The probability of a sure event is $1$.
- The probability of an impossible event is $0$.
- For any event $E$:
$$0 \le P(E) \le 1$$
- Elementary Events & Complementary Events:
$$P(E) + P(\bar{E}) = 1 \quad \text{or} \quad P(\text{not } E) = 1 - P(E)$$
