Class IX Mathematics Sample Paper 2024-25

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Given the points (2, -3) and (-5, 1), describe the steps to find the equation of the line that passes through them.

First, calculate the slope using the formula: $m = (y_2 - y_1) / (x_2 - x_1)$. Then, use the point-slope form of a linear equation: $y - y_1 = m(x - x_1)$, substituting the slope and one of the points. Finally, convert to slope-intercept or standard form.

Explain how to determine if two lines, given in the form $ax + by = c$, are parallel or perpendicular.

Convert both equations to slope-intercept form ($y = mx + b$) to compare their slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular.

Describe the relationship between the mean, median, and mode in a symmetrical distribution.

In a perfectly symmetrical distribution, the mean, median, and mode are all equal. They represent the central point around which the data is evenly distributed.

Explain the difference between the probability of an event occurring and the odds of the same event.

Probability is the ratio of favorable outcomes to total possible outcomes, while odds are the ratio of favorable outcomes to unfavorable outcomes. Probability ranges from 0 to 1, while odds can range from 0 to infinity. Signup and view all the answers

Describe how to find the area of a triangle when you only know the coordinates of its vertices.

Use the determinant formula: Area = $1/2 |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$, where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the vertices. Signup and view all the answers

Summarize the key differences between rectangular prisms and cubes.

Both are three-dimensional shapes with six faces, but a rectangular prism has rectangular faces (which may or may not be square), while a cube has all faces as congruent squares. Signup and view all the answers

Explain how the properties of angle bisectors can be used to solve geometric problems.

An angle bisector divides an angle into two equal angles. A point on the angle bisector is equidistant from the sides of the angle. These properties help in finding unknown angles and proving congruency. Signup and view all the answers

Outline the steps to prove that two triangles are congruent using the ASA (Angle-Side-Angle) postulate.

Show that two angles and the included side of one triangle are congruent to the corresponding two angles and included side of the other triangle. State the congruence based on the ASA postulate. Signup and view all the answers

Briefly explain how to determine if a number is rational or irrational.

A rational number can be expressed as a fraction $p/q$, where $p$ and $q$ are integers and $q \ne 0$. Irrational numbers cannot be expressed in this form; their decimal representation is non-terminating and non-repeating. Signup and view all the answers

How does knowing that two sides of a triangle are congruent help in determining the measure of angles in the triangle?

If two sides of a triangle are congruent (isosceles triangle), then the angles opposite those sides are also congruent. This allows for calculating unknown angles if one of these angles or the vertex angle is known. Signup and view all the answers

Describe the impact of multiplying a linear inequality by a negative number.

Multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign. For example, if $a < b$, then $-2a > -2b$. Signup and view all the answers

Explain the relationship between the radius, diameter, and circumference of a circle.

The radius is the distance from the center to any point on the circle. The diameter is twice the radius ($d = 2r$), and the circumference is $\pi$ times the diameter ($C = \pi d = 2 \pi r$). Signup and view all the answers

Describe how the surface area of a sphere changes when its radius is doubled.

The surface area of a sphere is given by $4 \pi r^2$. If the radius is doubled, the new surface area becomes $4 \pi (2r)^2 = 4 \pi (4r^2) = 16 \pi r^2$. Thus, the surface area becomes four times the original. Signup and view all the answers

How do you find the midpoint of a line segment given the coordinates of its endpoints?

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is found by averaging the x-coordinates and the y-coordinates: Midpoint $= ((x_1 + x_2)/2, (y_1 + y_2)/2)$. Signup and view all the answers

What is the relationship between the area of a parallelogram and the area of a rectangle with the same base and height?

A parallelogram and a rectangle with the same base and height have equal areas. The area of both is given by the formula, Area = base × height. Signup and view all the answers

Describe how to use the Pythagorean Theorem to determine if a triangle is a right triangle.

If the square of the length of the longest side (c) of a triangle is equal to the sum of the squares of the lengths of the other two sides (a and b), i.e., $a^2 + b^2 = c^2$, then the triangle is a right triangle. Signup and view all the answers

Explain the difference between permutations and combinations and when each is used.

Permutations are used when the order of selection matters, while combinations are used when the order does not matter. The formula for permutations is $nPr = n! / (n-r)!$, and for combinations, it is $nCr = n! / (r!(n-r)!)$. Signup and view all the answers

Describe how to factor a quadratic expression of the form $ax^2 + bx + c$.

Find two numbers that multiply to $ac$ and add up to $b$. Rewrite the middle term using these two numbers and then factor by grouping. Signup and view all the answers

What is the effect of adding a constant to every data point in a set on the mean and standard deviation?

Adding a constant to every data point increases the mean by the same constant, but it does not change the standard deviation. Signup and view all the answers

Explain the connection between the exterior angle of a triangle and its two remote interior angles.

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. This is known as the Exterior Angle Theorem. Signup and view all the answers

Signup and view all the answers

Flashcards

What are the coordinates of a point on the x-axis?

A point on the x-axis, 3 units to the right of the origin.

Area of a triangle with sides 5cm, 7cm, and 8cm

10√3 cm²