Kit Library / Mathematics / Algebra
⚡ Topic Learning KitShared by a Veda teacher · Generated with Veda AI
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
💡 Key Idea
Cubes hide in prime factor triples.
A number is a perfect cube only when every prime factor appears a multiple of 3 times in its prime factorisation.
↳ To test or build a cube, count prime exponents in groups of three.
📖 Definition
Two numbers, two names — don't mix them.
↳ In (x, y), x is the abscissa and y is the ordinate.
➗ Formula
Four identities, endless questions.
↳ Recognising the pattern is faster than expanding every term.
⚙️ Process
One missing prime ruins the cube.
↳ The multiplier is the product of primes whose exponents fall short of a multiple of 3.
⚙️ Process
Divide away the leftovers.
↳ Dividing removes surplus primes; multiplying fills missing ones.
💡 Key Idea
Always hunt the common factor first.
↳ HCF-outside, quotient-inside — the universal first move in factorisation.
⚠️ Common Mistake
Flip the sign, save the factor.
↳ Whenever you see (a − b) and (b − a), flip one sign and factor.
197² in one line — no calculator.
↳ Round the number, apply the identity, adjust back.
Big squares cancel in one step.
↳ Spot a² − b², split into sum × difference.
💡 Key Idea
Multiply in pairs, not all at once.
When you see $(3x-2y)(3x+2y)(9x^2+4y^2)$, pair the conjugates first — they collapse into a difference of squares.
↳ Conjugate pairs always multiply into a difference of squares.
➗ Formula
Two facts, one missing square.
↳ Whenever a question gives (x ± y) and xy, use the squared identity.
💡 Key Idea
That ugly expression is a square.
Expressions like $64x^2 + 144xy + 81y^2$ look messy but are actually $(8x + 9y)^2$ in disguise.
↳ Spot the pattern, substitute once, done.
⚙️ Process
Three points, one line — check the slope.
↳ Equal slopes between consecutive points = collinear.
💡 Key Idea
Find the fourth vertex with one midpoint.
In a rhombus $ABCD$, the diagonals $AC$ and $BD$ share the same midpoint — use it to find a missing vertex.
↳ Midpoint of one diagonal = midpoint of the other in any parallelogram.
🎯 Exam Tip
Skip the working, lose the marks.
↳ Examiners award method marks — never jump to the answer.
🧠 Memory Trick
Parity survives cubing.
↳ Cubing never changes whether a number is odd or even.
🌍 Real World
Cubes aren't just textbook shapes.
↳ Cube roots turn volume back into side length — a real engineering move.
⚠️ Common Mistake
Expanding wastes time and invites errors.
↳ Identity-spotting is faster and safer than brute-force expansion.
A Class 8 CBSE half-yearly scope covering cubes and cube roots, coordinate geometry (plotting and reading points), factorisation by common factors, and algebraic identities. The focus is on multi-step problem solving — p…
Full explanation, worked examples & memory tricks in the app
All 18 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. A number is a perfect cube only when every prime in its prime factorisation has an exponent that is a multiple of which number?
BeginnerB · 3
A perfect cube is $n \times n \times n$, so every prime appears three times over — its exponent must be a multiple of 3.
Q2. Which of the following numbers is a perfect cube?
BeginnerC · 9261
$9261 = 3^3 \times 7^3 = 21^3$, so all exponents are multiples of 3. The others have primes with exponents not divisible by 3.
Q3. For the point $(-10, -7)$, what is the abscissa?
BeginnerC · $-10$
The abscissa is the $x$-coordinate, which is $-10$ in the ordered pair $(-10, -7)$.
Q4. What is the first step when factorising any algebraic expression?
BeginnerA · Pull out the HCF of coefficients and lowest powers of common variables
Factorisation always begins by removing the highest common factor — this is the step that makes the remaining expression manageable.
Q5. State the identity for $(a+b)^2$.
BeginnerB · $a^2 + 2ab + b^2$
The square of a sum expands to $a^2 + 2ab + b^2$ — the middle term is positive.
Q6. A number is a perfect cube if and only if every prime in its prime factorisation has an exponent that is a multiple of 3 — this single test decides every cube question in the chapter.
BeginnerA · True
A number is a perfect cube if and only if every prime in its prime factorisation has an exponent that is a multiple of 3 — this single test decides every cube question in the chapter.
Q7. A student factors $24x^3 - 36xy$ as $6x(4x^2 - 6y)$. What is the error?
IntermediateC · The HCF $12x$ was not fully removed
The HCF of $24$ and $36$ is $12$, not $6$, so $6x$ is not the full HCF. The correct factorisation is $12x(2x^2 - 3y)$.
Q8. Given $x - y = 7$ and $xy = 9$, find $x^2 + y^2$.
IntermediateB · $67$
Use $x^2 + y^2 = (x-y)^2 + 2xy = 49 + 18 = 67$.
Q9. Evaluate $(82)^2 - (18)^2$ using a suitable identity.
IntermediateA · $6400$
Using $a^2 - b^2 = (a+b)(a-b)$: $(82+18)(82-18) = 100 \times 64 = 6400$.
Q10. Factorise: $x(a-3) + y(3-a)$
IntermediateB · $(a-3)(x-y)$
Since $3 - a = -(a-3)$, the expression becomes $x(a-3) - y(a-3) = (a-3)(x-y)$.
Q11. For rhombus $A(1,2)$, $B(3,4)$, $C(5,2)$, what is the fourth vertex $D$?
IntermediateD · $(3, 0)$
Diagonals of a rhombus bisect each other. Midpoint of $AC$ is $(3, 2)$, so $D = (3, 0)$.
Q12. A cubical water tank has volume $9261$ m³. What is the length of its side?
IntermediateA · $21$ m
$9261 = 3^3 \times 7^3 = 21^3$, so the side is $21$ m.
Q13. If the side of a cubical tank is doubled, by what factor does its volume increase?
IntermediateB · 8
Volume scales as the cube of the side, so doubling the side multiplies volume by $2^3 = 8$.
Q14. Evaluate $(197)^2$ using a binomial identity.
AdvancedA · $38809$
Write $197 = 200 - 3$: $(200-3)^2 = 40000 - 1200 + 9 = 38809$.
Q15. Simplify: $(3x-2y)(3x+2y)(9x^2+4y^2)$
AdvancedA · $81x^4 - 16y^4$
First pair gives $9x^2 - 4y^2$. Multiplying by $9x^2 + 4y^2$ gives $81x^4 - 16y^4$.
Q16. Evaluate $64x^2 + 81y^2 + 144xy$ when $x = 11$ and $y = \frac{4}{3}$.
AdvancedA · $10000$
Recognise the perfect square: $64x^2 + 144xy + 81y^2 = (8x + 9y)^2$. Substituting gives $(88 + 12)^2 = 100^2 = 10000$.
Q17. Given the expression $64x^2 + 81y^2 + 144xy$, which of the following statements is true when substituting $x = 11$ and $y = \frac{4}{3}$?
AdvancedC · The expression evaluates to $10000$
The expression can be recognized as a perfect square: $(8x + 9y)^2$. Substituting $x = 11$ and $y = \frac{4}{3}$ gives $8(11) + 9(\frac{4}{3}) = 100$, so the expression evaluates to $100^2 = 10000$.
Q18. For the points $A(0,2)$, $B(1,4)$, $C(2,6)$, and $D(4,6)$, determine if these points are collinear and justify your answer.
AdvancedB · Only points A, B, and C are collinear
To check for collinearity, calculate the slopes: $AB = 2$, $BC = 2$, and $CD = 0$. Points A, B, and C share the same slope, indicating they are collinear, while point D has a different slope, confirming it is not collinear with the others.
35 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
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Word Match
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True False
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Speed Quiz
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Sequence Builder
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Revision Battle
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