For Class 8, 9 & 10
Foundation - a strong base in Physics, Chemistry, Maths & Biology
Concept-first questions with clear model answers across all four subjects, aligned to NCERT. Start early, build the habit, and walk into your board exams, NEET and JEE already ahead.
Showing 6 questions in Mathematics for Class 9. Tap a card to reveal the answer.
MathematicsNumber SystemsmediumRationalise 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.
MathematicsPolynomialsmediumUsing 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.
MathematicsLinear Equations in Two VariableseasyWhere 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.
MathematicsHeron's FormulamediumFind 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.
MathematicsSurface Areas and VolumesmediumFind 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.
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.
MathematicsStatisticseasyFind 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.