Class 8 CBSE Maths: Cubes, Coordinate Geometry, Factorisation & Identities: 35 flashcards
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1. What is a perfect cube?
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A number that can be written as $n \times n \times n$ for some integer $n$.
2. How do you test whether a number is a perfect cube from its prime factorisation?
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Every prime's exponent must be a multiple of 3.
3. Is 9261 a perfect cube? Justify.
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Yes — $9261 = 3^3 \times 7^3 = 21^3$, so all exponents are multiples of 3.
4. What is the smallest number by which 8788 must be divided to get a perfect cube?
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$4$ (since $8788 = 2^2 \times 13^3$, dividing by $2^2$ leaves $13^3$).
5. What is the smallest number by which 2560 must be multiplied to get a perfect cube?
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$25$ (since $2560 = 2^9 \times 5^1$, multiplying by $5^2$ completes the triple).
6. Rule for making a cube by multiplication vs division?
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Multiply to add missing primes; divide to remove extra primes.
7. What is the abscissa of a point?
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The $x$-coordinate — the horizontal distance from the origin.
8. What is the ordinate of a point?
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The $y$-coordinate — the vertical distance from the origin.
9. For the point $(-10, -7)$, what are the abscissa and ordinate?
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Abscissa $= -10$, ordinate $= -7$; the point lies in the third quadrant.
10. When are three or more points collinear?
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When the slope between every consecutive pair is equal.
11. What is the slope formula?
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$\text{slope} = \dfrac{y_2 - y_1}{x_2 - x_1}$
12. Are $A(0,2)$, $B(1,4)$, $C(2,6)$, $D(4,6)$ collinear?
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No — $A$, $B$, $C$ lie on $y = 2x + 2$, but $D$ does not.
13. Which property of a rhombus helps find a missing vertex?
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Diagonals of a rhombus bisect each other — their midpoints coincide.
14. For rhombus $A(1,2)$, $B(3,4)$, $C(5,2)$, what is the fourth vertex $D$?
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$D(3, 0)$ — found using the shared midpoint of the diagonals.
15. If the diagonals of a rhombus are equal, what is it actually?
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A square.
16. What is the first step in factorising any algebraic expression?
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Pull out the HCF of coefficients and the lowest powers of common variables.
17. Factorise: $18a^3b^3 - 27a^2b^3 + 36a^3b^2$
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$9a^2b^2(2ab - 3b + 4a)$
18. Factorise: $24x^3 - 36xy$
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$12x(2x^2 - 3y)$
19. How do you factorise $x(a-3) + y(3-a)$?
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$(a-3)(x - y)$, using $3 - a = -(a-3)$.
20. State the identity $(a+b)^2$.
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$(a+b)^2 = a^2 + 2ab + b^2$
21. State the identity $(a-b)^2$.
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$(a-b)^2 = a^2 - 2ab + b^2$
22. State the difference of squares identity.
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$a^2 - b^2 = (a+b)(a-b)$
23. Evaluate $(197)^2$ using an identity.
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$(200-3)^2 = 40000 - 1200 + 9 = 38809$
24. Evaluate $(82)^2 - (18)^2$ using an identity.
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$(82+18)(82-18) = 100 \times 64 = 6400$
25. Simplify $(3x-2y)(3x+2y)(9x^2+4y^2)$.
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$81x^4 - 16y^4$
26. Given $x - y$ and $xy$, how do you find $x^2 + y^2$?
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$x^2 + y^2 = (x-y)^2 + 2xy$
27. If $x - y = 7$ and $xy = 9$, find $x^2 + y^2$.
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$49 + 18 = 67$
28. If $x + \frac{1}{x} = 4$, find $x^2 + \frac{1}{x^2}$.
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$16 - 2 = 14$
29. How do you recognise a hidden perfect square?
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Check whether the middle term equals $2ab$ where the outer terms are $a^2$ and $b^2$.
30. Evaluate $64x^2 + 81y^2 + 144xy$ at $x = 11$, $y = \frac{4}{3}$.
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$(8x + 9y)^2 = 100^2 = 10000$
31. Is the cube of an odd number odd or even?
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Odd — the cube of an odd number is always odd.
32. If the side of a cube is doubled, by what factor does the volume change?
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By a factor of 8, not 2.
33. A cubical tank has volume $9261$ m³. What is its side?
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$21$ m, since $9261 = 21^3$.
34. Rectangle $P(1,1)$, $Q(5,1)$, $R(5,4)$, $S(1,4)$ — where do the diagonals meet?
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$(2.5, 2.5)$ — the centre of the rectangle.
35. What is the length and breadth of rectangle $P(1,1)$, $Q(5,1)$, $R(5,4)$, $S(1,4)$?
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Length $= 4$ units, breadth $= 3$ units.
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