Class 9 Maths — Important Questions with Answers

6 concept-first Class 9 Maths questions across 6 chapters — each with a clear model answer, the why behind it, and a memory trick to make it stick. NCERT-aligned and free. Build the base now, and your board exams, NEET and JEE feel a whole lot easier later.

6 questions•6 chapters•Concept-first•NCERT-aligned•For boards, NEET & JEE

Number Systems (1)Polynomials (1)Linear Equations in Two Variables (1)Heron's Formula (1)Surface Areas and Volumes (1)Statistics (1)

Number Systems1 question

MathematicsRationalising the denominatormedium

Rationalise the denominator of 1 / sqrt(2).

Reveal answer▾

What it is

Rationalising clears a surd (root) from the denominator by multiplying top and bottom by it.

Answer

Multiply the numerator and denominator by sqrt(2): (1/sqrt(2)) x (sqrt(2)/sqrt(2)) = sqrt(2) / 2. So 1/sqrt(2) = sqrt(2)/2, which is about 0.707. Rationalising removes the irrational number (surd) from the denominator.

1/sqrt(a) = sqrt(a)/a

•Multiply top and bottom by sqrt(2)
•= sqrt(2)/2
•Removes the surd from the denominator

Why learn this

It makes fractions easier to compute and compare - important in exams and in engineering maths.

💡 Memory trick

Multiply top and bottom by the same surd - the root moves upstairs.

Polynomials1 question

MathematicsRemainder theoremmedium

Using the remainder theorem, find the remainder when p(x) = x^3 + 3x^2 + 3x + 1 is divided by (x + 1).

Reveal answer▾

What it is

The remainder theorem says the remainder of p(x) divided by (x - a) is simply p(a).

Answer

By the remainder theorem, the remainder when p(x) is divided by (x + 1) is p(-1). p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0. Since the remainder is 0, (x + 1) is a factor of p(x).

Remainder theorem: remainder = p(a) when dividing by (x - a)

•Remainder = p(-1)
•p(-1) = -1 + 3 - 3 + 1 = 0
•Remainder 0 -> (x + 1) is a factor

Why learn this

It checks factors of big polynomials in one step - a shortcut used all through algebra.

💡 Memory trick

Dividing by (x + 1)? Just plug in x = -1. Set the bracket to zero, then substitute.

Linear Equations in Two Variables1 question

MathematicsIntercepts of a lineeasy

Where does the line x + y = 5 meet the x-axis and the y-axis?

Reveal answer▾

What it is

A straight line meets the axes at its intercepts - put y = 0 for the x-axis, x = 0 for the y-axis.

Answer

On the x-axis, y = 0, so x + 0 = 5 gives x = 5; the line meets the x-axis at (5, 0). On the y-axis, x = 0, so 0 + y = 5 gives y = 5; it meets the y-axis at (0, 5). These two intercept points are enough to draw the straight line.

Intercepts: set y = 0 for x-axis, x = 0 for y-axis

•x-axis: put y = 0 -> (5, 0)
•y-axis: put x = 0 -> (0, 5)
•Two points define the straight line

Why learn this

Two intercepts are the fastest way to draw any line - used in graphs, economics and coding.

💡 Memory trick

Zero the other one: y = 0 gives the x-cut, x = 0 gives the y-cut.

Heron's Formula1 question

MathematicsArea of a trianglemedium

Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm using Heron's formula.

Reveal answer▾

What it is

Heron's formula finds a triangle's area from just its three sides, using the semi-perimeter.

Answer

First find the semi-perimeter s = (3 + 4 + 5)/2 = 6 cm. Heron's formula gives Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[6(6 - 3)(6 - 4)(6 - 5)] = sqrt[6 x 3 x 2 x 1] = sqrt[36] = 6 square cm.

Area = sqrt[s(s-a)(s-b)(s-c)], s = (a+b+c)/2

•s = (a + b + c)/2 = 6 cm
•Area = sqrt[s(s-a)(s-b)(s-c)]
•= sqrt[36] = 6 cm^2

Why learn this

It measures land, plots and any triangle where you cannot easily find the height.

💡 Memory trick

Half the perimeter is 's'; Area = sqrt of s times the three (s - side) gaps.

Surface Areas and Volumes1 question

MathematicsVolume of a spheremedium

Find the volume of a sphere of radius 7 cm. (Take pi = 22/7.)

Reveal answer▾

What it is

A sphere's volume grows with the cube of its radius: V = (4/3) pi r^3.

radius r = 7 cm

V = 4/3 π r³ ≈ 1437.3 cm³

Sphere — slide the radius; volume grows with r³ (π = 22/7)

Answer

Volume of a sphere = (4/3) x pi x r^3 = (4/3) x (22/7) x 7^3 = (4/3) x (22/7) x 343 = (4 x 22 x 49)/3 = 4312/3, which is about 1437.3 cubic cm.

V(sphere) = (4/3) pi r^3

•V = (4/3) pi r^3
•= (4/3)(22/7)(343)
•= 4312/3 = about 1437.3 cm^3

Why learn this

It measures balls, planets, bubbles and tanks - anywhere something is round.

💡 Memory trick

Four-thirds pi r-cubed: V = (4/3) pi r^3 - the roundest formula in maths.

Statistics1 question

MathematicsMean of dataeasy

Find the mean of the data: 10, 15, 20, 25, 30.

Reveal answer▾

What it is

The mean (average) is the total of all values divided by how many values there are.

Answer

Mean = (sum of all observations) / (number of observations) = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20.

Mean = (sum of observations) / (number of observations)

•Mean = sum of values / number of values
•Sum = 100, count = 5
•Mean = 20

Why learn this

It summarises data in one number - used in marks, cricket averages, economics and science.

💡 Memory trick

Add them all up, divide by how many. Mean = sum / count.

Start early. Stay ahead.

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