Class 9 Mathematics – Interactive Questions

Step-by-step explanation:
- Divide 1 by 7: 1 ÷ 7 = 0.142857142857...
- The digits 142857 repeat infinitely.
- This is a non-terminating recurring decimal.

Conclusion:
Since the decimal repeats, 1/7 is a rational number.
Fun Fact: The repeating block 142857 has cyclic properties. Try multiplying it by 2, 3, ..., 6!

Construction Steps:
1. Draw a number line and mark point O (0) and A (2 units).
2. At point A, draw a perpendicular and mark point B such that AB = 1 unit.
3. Connect OB. By Pythagoras theorem:
    OB² = OA² + AB² = 2² + 1² = 4 + 1 = 5
    ⇒ OB = √5
4. Use a compass to draw an arc with radius OB and center O.
5. The arc intersects the number line at point P, which represents √5.

Visual Insight: This method uses a right triangle to geometrically construct irrational lengths.

Step-by-step:
Multiply numerator and denominator by the conjugate:
(1 / (√7 + √6)) × ((√7 - √6) / (√7 - √6))
= (√7 - √6) / ((√7)² - (√6)²)
= (√7 - √6) / (7 - 6) = √7 - √6

Why this works: Multiplying by the conjugate removes the surds from the denominator.

√2 ≈ 1.414, √3 ≈ 1.732
Rational numbers between them: 1.5 and 1.6

Note: Rational numbers have terminating or repeating decimals, unlike irrational numbers.

Let x = 0.333...
Multiply both sides by 10: 10x = 3.333...
Subtract: 10x - x = 3.333... - 0.333...
⇒ 9x = 3 ⇒ x = 1/3

Conclusion: Every recurring decimal is a rational number.

π = 3.1415926535...
- The decimal expansion is non-terminating and non-repeating.
- It cannot be expressed as a ratio of two integers.

Conclusion: π is an irrational number.

Explanation:
- A polynomial’s degree is the highest power of the variable.
- In 5x³ - 2x² + x - 7, the highest exponent of x is 3.

Conclusion: Degree = 3

Step-by-step evaluation:
- p(1) = 1² - 3×1 + 2 = 1 - 3 + 2 = 0
- p(2) = 2² - 3×2 + 2 = 4 - 6 + 2 = 0
- p(0) = 0² - 3×0 + 2 = 0 + 0 + 2 = 2

Conclusion: p(1) = 0, p(2) = 0, p(0) = 2

Method:
- Find two numbers whose product is 6 and sum is -5.
- Numbers: -2 and -3
- Rewrite: x² - 2x - 3x + 6
- Group and factor: x(x - 2) - 3(x - 2)
- Final factorization: (x - 2)(x - 3)

Conclusion: x² - 5x + 6 = (x - 2)(x - 3)

Verification:
- Substitute x = 2 into the polynomial:
- x² - 4x + 4 = 2² - 4×2 + 4 = 4 - 8 + 4 = 0

Conclusion: Since the value is 0, x = 2 is a zero of the polynomial.

Observation:
- The expression x³ + 3x² + 3x + 1 matches the identity:
- (x + 1)³ = x³ + 3x² + 3x + 1

Conclusion: Quotient = (x + 1)², Remainder = 0
Alternate Method: You can also use long division to verify.

The Cartesian plane is a two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin (0, 0). The plane is divided into four quadrants:

• Quadrant I: (+x, +y)
• Quadrant II: (−x, +y)
• Quadrant III: (−x, −y)
• Quadrant IV: (+x, −y)

The quadrants are numbered in an anticlockwise direction starting from the top-right.

Coordinates are written as an ordered pair (x, y), where:

• x is the horizontal distance from the origin (called the abscissa)
• y is the vertical distance from the origin (called the ordinate)

For example, the point (3, −2) lies 3 units to the right of the origin and 2 units down.

The point (−4, 5) lies in Quadrant II because:

• x is negative (−4), indicating a position to the left of the origin
• y is positive (5), indicating a position above the origin

Quadrant II contains all points where x is negative and y is positive.

The origin is the point where the x-axis and y-axis intersect. It serves as the reference point for all other coordinates on the plane. Its coordinates are always (0, 0).

This means there is no horizontal or vertical displacement from the center.

Standard Form:
Ax + By + C = 0
- A, B, and C are real numbers
- A and B ≠ 0 (to ensure both variables are present)

Example: 2x + 3y - 6 = 0

Solution 1: Let x = 0 ⇒ 2(0) + 3y = 6 ⇒ y = 2
Solution 2: Let x = 3 ⇒ 2(3) + 3y = 6 ⇒ y = 0

Solutions: (0, 2) and (3, 0)

Choose values:
- x = 0 ⇒ y = 4
- x = 2 ⇒ y = 2
- x = 4 ⇒ y = 0

Plot: Points (0,4), (2,2), (4,0)
Graph: Draw a straight line through these points.

Substitute x = 1:
3(1) + 2y = 7 ⇒ 3 + 2y = 7 ⇒ 2y = 4 ⇒ y = 2

Answer: y = 2

Find slope:
m = (0 - 2)/(2 - 0) = -1
Use point-slope form: y - y₁ = m(x - x₁)
⇒ y - 2 = -1(x - 0) ⇒ y = -x + 2
Standard form: x + y = 2

To solve the inequality 3x − 5 < 7, follow these steps:

1. Add 5 to both sides to isolate the term with x:
   3x − 5 + 5 < 7 + 5 → 3x < 12
2. Divide both sides by 3 to solve for x:
   3x ÷ 3 < 12 ÷ 3 → x < 4

Final Answer: x is any number less than 4.
✅ This means x can be 3.9, 0, −5, etc., but not 4.

To graph x ≥ 2:

• Place a solid dot at x = 2 to show that 2 is included in the solution.
• Shade the number line to the right of 2 to represent all values greater than or equal to 2.

Visual cues:

• Solid dot → the number is included (≥ or ≤)
• Arrow or shading → shows the direction of the inequality

🧠 Example values that satisfy x ≥ 2: 2, 3, 10, 100...

To check if x = −1 is a solution to 2x + 3 > 0, substitute −1 into the inequality:

• 2(−1) + 3 = −2 + 3 = 1
• Since 1 > 0, the inequality is true.

✅ Conclusion: x = −1 is a solution to the inequality.

The word 'geometry' comes from the ancient Greek words:

• geo meaning “earth”
• metrein meaning “to measure”

🔍 Interpretation: Geometry originally referred to measuring land and physical space.

“A straight line may be drawn from any one point to any other point.”

✏️ This means that for any two distinct points, there exists exactly one straight line connecting them.

“That which has no part.”

📌 A point has no length, width, or thickness. It represents a precise location.

“If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two lines, if extended indefinitely, meet on that side.”

🔍 This describes how lines behave when intersected by a transversal.

When two lines intersect, the angles opposite each other are equal.
Example: If ∠A and ∠B are opposite, then ∠A = ∠B

Supplementary angles sum to 180°:
180° - 65° = 115°

Two adjacent angles whose non-common arms form a straight line.
Sum: 180°

In a right triangle:
(Hypotenuse)² = (Base)² + (Height)²
Example: If base = 3, height = 4 ⇒ hypotenuse = 5

If AB = AC, triangle ABC is isosceles.
By congruence (SAS or SSS), ∠B = ∠C
Conclusion: Angles opposite equal sides are equal

Always: 180°

A quadrilateral with both pairs of opposite sides parallel and equal.
Examples: Rectangle, Rhombus

Sum of interior angles = 360°

Adjacent angles in a parallelogram are supplementary:
180° - 70° = 110°

A chord is a line segment that joins any two points on the circumference of a circle.

Note: The longest chord in a circle is the diameter, which passes through the center.

A diameter subtends a right angle (90°) at the circumference of the circle.

Why? This is a special case of the angle in a semicircle theorem.
If A and B are endpoints of the diameter and C is any point on the circle forming triangle ABC, then ∠ACB = 90°.

The diameter is twice the radius:
Diameter = 2 × Radius

Example: If radius = 5 cm, then diameter = 10 cm.

The mean is the average value of a data set.
Mean = (Sum of all observations) ÷ (Number of observations)

Example: For data 2, 4, 6 ⇒ Mean = (2 + 4 + 6)/3 = 12/3 = 4

The mode is the value that occurs most frequently.
In the given data: 4 appears three times.

Answer: Mode = 4

The median is the middle value when data is arranged in order.
Ordered data: 5, 7, 9, 11, 13
Middle value = 9

Answer: Median = 9

Probability measures the likelihood of an event occurring.
Probability = (Number of favorable outcomes) ÷ (Total number of outcomes)

Note: Probability values range from 0 to 1.

A coin has two outcomes: Head and Tail.
Favorable outcome = Head
Total outcomes = 2

Probability: 1/2

A die has 6 faces: 1, 2, 3, 4, 5, 6
Even numbers = 2, 4, 6 ⇒ 3 favorable outcomes
Total outcomes = 6

Probability: 3/6 = 1/2

When two lines intersect, the angles opposite each other are equal.
Example: If ∠A and ∠B are opposite, then ∠A = ∠B

Supplementary angles sum to 180°:
180° - 65° = 115°

Two adjacent angles whose non-common arms form a straight line.
Sum: 180°

1. Vertically Opposite Angles:
When two lines intersect, they form two pairs of vertically opposite angles.
These angles are equal in measure.

2. Supplementary Angles:
Two angles whose measures add up to 180°.
Example: If one angle is 65°, the other is 115°.

3. Linear Pair:
A pair of adjacent angles formed when two lines intersect.
Their non-common arms form a straight line, and their sum is 180°.

Diagram:


This diagram illustrates vertically opposite angles, supplementary angles, and linear pairs.