SS 1 Revision and Examination for Mathematics Second Term

Mathematics in SS 1 covers a wide range of topics that require a thorough understanding of various concepts and techniques. This revision guide provides a comprehensive analysis of the key topics from the second term, offering a detailed breakdown that makes the content easy to grasp for students, even those who may not have prior knowledge in the subject.

1. Construction of Angles and Geometrical Concepts

This section covers various geometrical constructions and the application of the properties of triangles, angles, and points.

• Construction of Angles: The construction of angles such as 30°, 45°, 60°, and 90° is fundamental in geometry. These angles can be constructed using a compass and ruler. For instance, to construct a 60° angle, you can use an equilateral triangle where each angle is 60°.
• Triangles: Understanding the different types of triangles is important. A triangle’s angles add up to 180°. Various methods exist for constructing triangles, such as using the lengths of sides (SSS), the angle and side (SAS), or two angles and a side (ASA).
• Equivalent Angles: These are angles that have the same measure but may be located in different positions. For example, alternate interior angles when two parallel lines are cut by a transversal.
• Equidistance from Points: This refers to the concept where a point is equidistant from two other points, meaning the distances from the point to the other two points are the same. This concept is useful in geometrical constructions like bisectors.
• Locus of Points: A locus is the set of all points satisfying a particular condition. For example, the locus of points that are equidistant from two given points is the perpendicular bisector of the line joining the points.

2. Proofs of Basic Theorems in Euclidean Geometry

Understanding and proving theorems in geometry helps strengthen logical reasoning skills.

• Angle Sum of a Triangle: This theorem states that the sum of the three interior angles of any triangle is always 180°. This is fundamental to understanding the properties of triangles.
• The Angle of a Triangle Equals the Sum of Two Opposite Interior Angles: This theorem shows the relationship between the angles of a triangle and provides insight into parallel lines and transversals.

3. Proofs of Properties of Angles

This section covers various important geometric properties.

• Angles on Parallel Lines: When two parallel lines are cut by a transversal, several pairs of angles are formed, such as corresponding angles, alternate interior angles, and alternate exterior angles, which are all congruent.
• Angles in a Polygon: The sum of the interior angles of a polygon with $n$ sides is given by $180(n-2)$. For a quadrilateral, for example, the sum of angles is 360°.
• Congruent Angles: Congruent angles are angles that have the same measure, and they play a crucial role in proving other geometric properties.
• Parallelograms and the Intercept Theorem: The intercept theorem relates to the proportionality of segments created by intersecting lines in a triangle or parallelogram.

4. Trigonometric Ratios in Right-Angled Triangles

Trigonometry deals with the relationships between the angles and sides of right-angled triangles.

• Sine, Cosine, and Tangent: The basic trigonometric ratios for right-angled triangles are:
  • Sine (sin) = Opposite / Hypotenuse
  • Cosine (cos) = Adjacent / Hypotenuse
  • Tangent (tan) = Opposite / Adjacent

These ratios are vital for solving various geometric problems, such as finding the length of sides or the angle measures in right-angled triangles.

5. Derivation of Trigonometric Ratios for Specific Angles

Trigonometric ratios for specific angles like 30°, 45°, and 60° can be derived using the properties of special triangles, such as the 30-60-90 triangle or the 45-45-90 triangle.

6. Angles of Elevation and Depression

These angles arise when objects are observed from a height, like when looking up to a tall building (angle of elevation) or looking down at the ground (angle of depression). These angles are critical in solving real-world problems involving heights and distances.

7. Trigonometric Ratios in Relation to the Unit Circle

The unit circle provides a way to understand the values of sine and cosine for various angles. The sine of an angle is the y-coordinate, and the cosine is the x-coordinate of the point on the unit circle that corresponds to the angle.

8. Graphs of Sine and Cosine

The graphs of sine and cosine functions are periodic, repeating every 360° (or 2π radians). These graphs can be drawn by plotting the values of sine or cosine for different angles, such as 15°, 30°, 45°, and 60°.

9. Length of Arc of a Circle

The length of an arc of a circle can be calculated using the formula:

Length of Arc=θ360°×2πr\text{Length of Arc} = \frac{\theta}{360°} \times 2\pi rLength of Arc=360°θ​×2πr

where $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.

10. Revision of First Term Work

A review of first-term work will help reinforce the foundational concepts learned, such as algebraic expressions, linear equations, and basic geometry.

11. Quadratic Equations

Quadratic equations are a key part of mathematics. The standard form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

The solutions to a quadratic equation can be found using several methods:

• Factorization: Finding factors of the quadratic expression.
• Completing the Square: Rewriting the equation in the form (x+p)2=q(x + p)^2 = q(x+p)2=q.
• Quadratic Formula: Using the formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​.
• Graphical Method: Plotting the equation on a graph and finding the points where it intersects the x-axis.

12. Sets and Venn Diagrams

The concept of sets is crucial in mathematics. A set is a collection of distinct objects. Topics include:

• Universal Sets: The set that contains all elements under consideration.
• Finite and Infinite Sets: A finite set has a definite number of elements, while an infinite set has an unlimited number of elements.
• Union and Intersection: Union refers to the combination of two sets, while intersection refers to the elements common to both sets.
• Venn Diagrams: Diagrams used to represent sets and their relationships visually.

13. Logic and Statements

Understanding logical statements is key in forming valid mathematical arguments. This topic covers:

• Simple and Compound Statements: A simple statement makes a single assertion, while a compound statement combines multiple assertions.
• Negation, Conjunction, and Disjunction: These are logical operations used to form new statements from existing ones.
• Conditional and Biconditional Statements: Conditional statements express an “if-then” relationship, while biconditional statements express “if and only if.”

Likely 10 Questions for Each Topic

1. Construction of Angles:
   • How do you construct a 45° angle using a compass and ruler?
   • What is the significance of the perpendicular bisector in geometry?
2. Proofs of Basic Theorems:
   • Prove that the sum of angles in a triangle is 180°.
   • Explain why the angle of a triangle is equal to the sum of two opposite interior angles.
3. Trigonometric Ratios:
   • Calculate the sine, cosine, and tangent of a 30° angle.
   • How do the trigonometric ratios help solve problems in right-angled triangles?
4. Quadratic Equations:
   • Solve x2−5x+6=0x^2 – 5x + 6 = 0x2−5x+6=0 using factorization.
   • Derive the quadratic formula from the general form of a quadratic equation.
5. Sets and Venn Diagrams:
   • If set A = {2, 4, 6} and set B = {4, 6, 8}, find $A\cup B$ and $A\cap B$.
   • Draw a Venn diagram to show the relationship between two sets.