All questions of Mathematics for Class 9 Exam

Explanation:

Given:
x2 = 2

Find:
x3

Solution:

Step 1:
Given x2 = 2, we need to find x3.

Step 2:
To find x3, we can simply multiply x2 by x:
x3 = x * x2

Step 3:
Substitute the value of x2 from the given information:
x3 = x * 2

Step 4:
Since x2 = 2, we can rewrite the above equation as:
x3 = x * √2

Step 5:
Therefore, x3 = 2√2

Conclusion:
Therefore, if x is a positive real number and x2 = 2, then x3 = 2√2. Hence, the correct answer is option 'C'.

Explanation:

Initial Area of Scalene Triangle:
- Let the sides of the scalene triangle be a, b, and c.
- The area of a scalene triangle can be found using Heron's formula:
Area = √(s(s-a)(s-b)(s-c)), where s = (a + b + c) / 2.
- Let the initial area of the triangle be A.

Area of Doubled Triangle:
- If the side of the scalene triangle is doubled, the new sides would be 2a, 2b, and 2c.
- The new area of the triangle can be found using Heron's formula with the new sides.
- Let the new area of the triangle be A'.

Calculating the Increase in Area:
- The percentage increase in area can be calculated using the formula:
Percentage Increase = ((A' - A) / A) * 100%.

Solving the Problem:
- Since the sides of the triangle are doubled, the new sides are 2a, 2b, and 2c.
- The new area, A', can be calculated using Heron's formula with the new sides.
- The percentage increase in area can then be calculated using the formula mentioned above.
- Upon calculation, it is found that the area is increased by 300%.
Therefore, the correct answer is option 'D' - 300%.

Deepak drove in which quadrants

Introduction:
When a transversal intersects a pair of parallel lines, it forms various angles. These angles have specific properties and relationships. In this case, we need to determine the number of angles formed by a transversal with a pair of parallel lines.

Explanation:
When a transversal intersects a pair of parallel lines, it forms eight angles. Let's label these angles for better understanding:

1. Alternate Interior Angles: These angles are formed on opposite sides of the transversal and inside the parallel lines.
- Angle 1 and Angle 5 are alternate interior angles.
- Angle 3 and Angle 7 are alternate interior angles.

2. Alternate Exterior Angles: These angles are formed on opposite sides of the transversal and outside the parallel lines.
- Angle 2 and Angle 6 are alternate exterior angles.
- Angle 4 and Angle 8 are alternate exterior angles.

3. Corresponding Angles: These angles are formed on the same side of the transversal and in the same position relative to the parallel lines.
- Angle 1 and Angle 3 are corresponding angles.
- Angle 2 and Angle 4 are corresponding angles.
- Angle 5 and Angle 7 are corresponding angles.
- Angle 6 and Angle 8 are corresponding angles.

4. Consecutive Interior Angles: These angles are formed on the same side of the transversal and inside the parallel lines.
- Angle 3 and Angle 5 are consecutive interior angles.
- Angle 7 and Angle 1 are consecutive interior angles.

Summary:
In total, there are eight angles formed by a transversal with a pair of parallel lines. These include four pairs of alternate interior/exterior angles and four pairs of corresponding angles. Therefore, the correct answer is option 'B' - 8.

Given:
The linear equation is 2x + 3y = k
The solution is (2, 0)

To find:
The value of k

Solution:
Step 1: Substitute the given solution (2, 0) into the equation
2(2) + 3(0) = k
4 + 0 = k
4 = k

Step 2: Compare the value of k obtained in step 1 with the options provided
The value of k is 4

Step 3: Determine the correct option
The correct option is (B) 4

30

Explanation:

Given:
x = 2, y = 3

Linear Equations:
Linear equations are equations of the form ax + by = c, where a, b, and c are constants.

Number of Equations:
When x = 2 and y = 3, these values can satisfy many linear equations. This is because there are infinite number of linear equations that can have x = 2 and y = 3 as solutions.

Example Equations:
- 2x + 3y = 13
- x + 2y = 8
- 3x - y = 4

Explanation of Answer:
The values x = 2 and y = 3 can satisfy an infinite number of linear equations because any equation that holds true for x = 2 and y = 3 is a valid solution. Therefore, the correct answer is option 'C' - many.

Understanding Congruence in Triangles
In the given problem, we need to identify which congruence condition applies between triangles ABC and PQR.
Given Information:
- AB = QP
- ∠B = ∠P
- BC = PR
Analyzing the Conditions:
- Side AB = Side QP: This indicates that one pair of corresponding sides are equal.
- Angle ∠B = Angle ∠P: This shows that one pair of corresponding angles are equal.
- Side BC = Side PR: This indicates that another pair of corresponding sides are also equal.
Applying the SAS Congruence Condition:
The SAS (Side-Angle-Side) congruence condition states that if two sides and the angle between them in one triangle are equal to two sides and the included angle in another triangle, then the triangles are congruent.
- Here, we have:
- One side (AB) equal to another side (QP)
- The angle (∠B) between those sides equal to (∠P)
- And the second side (BC) equal to the second side (PR)
Thus, we can conclude that triangles ABC and PQR are congruent by the SAS condition.
Conclusion:
Therefore, the correct answer is option 'A' – SAS. This condition effectively demonstrates the congruence between the two triangles based on the information provided.

Explanation:

Given:
- Sides of the triangle: 11 m, 60 m, and 61 m

Altitude to the smallest side:
To find the altitude to the smallest side of the triangle, we can use the formula for the area of a triangle: Area = 1/2 * base * height.

Calculations:
Since the smallest side of the triangle is 11 m, we can consider the altitude to this side as the height of the triangle.
Using the formula for the area of a triangle and the given sides, we can calculate the area of the triangle as follows:
Area = 1/2 * 11 * height (where height is the altitude to the smallest side)
Since the area of the triangle can also be calculated using Heron's formula, which is given by:
Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter of the triangle and a, b, c are the sides of the triangle.
By substituting the given values, we get:
s = (11 + 60 + 61)/2 = 132/2 = 66
Area = √[66(66-11)(66-60)(66-61)]
Area = √[66*55*6*5]
Area = √(108900) = 330
Now, equating the two expressions for the area of the triangle, we get:
1/2 * 11 * height = 330
height = 330 / (1/2 * 11) = 60 m
Therefore, the altitude to the smallest side of the triangle is 60 m. Hence, the correct answer is option 'A'.

Explanation:
To determine whether the point (a, -a) lies on the graph of a given equation, we substitute the values of x and y from the point into the equation and check if the equation holds true.

a) y = x:
When we substitute x = a and y = -a into the equation y = x, we get:
- a = a

This equation is true, which means the point (a, -a) lies on the graph of y = x.

b) xy = 0:
When we substitute x = a and y = -a into the equation xy = 0, we get:
- a * (-a) = 0
a^2 = 0

This equation is not true unless a = 0. Therefore, the point (a, -a) does not lie on the graph of xy = 0.

c) x = a:
When we substitute x = a and y = -a into the equation x = a, we get:
a = a

This equation is true, which means the point (a, -a) lies on the graph of x = a.

d) y = -a:
When we substitute x = a and y = -a into the equation y = -a, we get:
- a = -a

This equation is true, which means the point (a, -a) lies on the graph of y = -a.

Conclusion:
From the analysis above, we can see that the point (a, -a) does not lie on the graph of xy = 0, so the correct answer is option A.

In a histogram, the area of each rectangle is proportional to the frequency of the corresponding class interval.

Understanding a Histogram:
A histogram is a graphical representation of data that is grouped into intervals or classes. It consists of a series of rectangles, where the height of each rectangle represents the frequency (or number of occurrences) of data falling within a particular class interval.

Explanation:
To understand why the area of each rectangle in a histogram is proportional to the frequency, let's break it down step by step:

1. Class Intervals:
When data is collected, it is often grouped into intervals or classes to make it more manageable. These intervals represent a range of values. For example, if we are collecting data on the heights of students, we may have class intervals like 150-160 cm, 160-170 cm, etc.

2. Frequency:
The frequency of a class interval refers to the number of data points that fall within that interval. It represents how many times a particular value or range occurs in the data set.

3. Rectangle Height:
In a histogram, the height of each rectangle corresponds to the frequency of the corresponding class interval. The taller the rectangle, the higher the frequency of data falling within that interval.

4. Rectangle Width:
The width of each rectangle is determined by the class interval. The width represents the range of values covered by the interval. The wider the interval, the wider the rectangle.

5. Area Calculation:
To calculate the area of a rectangle, we multiply its height by its width. In the case of a histogram, the area of each rectangle is proportional to both the frequency (height) and the class interval (width). However, since the width is the same for all rectangles in a histogram, it cancels out when comparing the areas of different rectangles.

6. Proportional Area:
Therefore, the only remaining factor that determines the area of each rectangle is the frequency. The larger the frequency, the larger the area of the rectangle. This proportionality allows us to visually compare the frequencies of different class intervals in the histogram.

Conclusion:
In conclusion, the area of each rectangle in a histogram is proportional to the frequency of the corresponding class interval. This property enables us to visually represent and compare the distribution of data across different intervals in a graphical and intuitive manner.

Understanding the Problem
The question states that the mean weight of six boys is 48 kg. This means that the total weight of all six boys can be calculated using the formula for mean:
- Mean = Total Weight / Number of Boys
- Total Weight = Mean * Number of Boys
- Total Weight = 48 kg * 6 = 288 kg
Calculating the Total Weight of Five Boys
We have the individual weights of five boys:
- Boy 1: 51 kg
- Boy 2: 45 kg
- Boy 3: 49 kg
- Boy 4: 46 kg
- Boy 5: 44 kg
Now, we calculate the total weight of these five boys:
- Total Weight of Five Boys = 51 + 45 + 49 + 46 + 44
- Total Weight of Five Boys = 235 kg
Finding the Weight of the 6th Boy
To find the weight of the 6th boy, we subtract the total weight of the first five boys from the total weight of all six boys:
- Weight of 6th Boy = Total Weight of Six Boys - Total Weight of Five Boys
- Weight of 6th Boy = 288 kg - 235 kg
- Weight of 6th Boy = 53 kg
Conclusion
Thus, the weight of the 6th boy is 53 kg, which corresponds to option 'D'.