3 Marks Question

Question 13 Marks

Use Identity 1A to find the values of $104^2, 37^2$. (Hint: Decompose 104 and 37 into sums or differences of numbers whose squares are easy to compute.) (Identity 1A $\quad(a+b)^2=a^2+2 a b+b^2$).

Answer

self

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Question 23 Marks

If $a$ and $b$ are any two integers, is $(a+b)^2$ always greater than $a^2+b^2$ ? If not, when is it greater?

Answer

self

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Question 33 Marks

Expand (i) $(a-b)(a+b)$, (ii) $(a-b)\left(a^2+a b+b^2\right)$, and (iii) $(a-b)\left(a^3+a^2 b+a b^2+b^3\right)$, Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

Answer

$\text { (i) }(a-b)(a+b)$
$\begin{aligned}= & (a-b) a+(a-b) b \\= & a^2-a b+a b-b^2 \\= & a^2-b^2 .\end{aligned}$
(ii) $(a-b)\left(a^2+a b+b^2\right)$
$\begin{array}{l}=(a-b) a^2+(a-b) a b+(a-b) b^2 \\=a^3-a^2 b+a^2 b-a b^2+a b^2-b^3 \\=a^3-b^3 .\end{array}$
(iii) $(a-b)\left(a^3+a^2 b+a b^2+b^3\right)$
$\begin{array}{l}=(a-b) a^3+(a-b) a^2 b+(a-b) a b^2+(a-b) b^3 \\=a^4-a^3 b+a^3 b-a^2 b^2+a^2 b^2-a b^3+a b^3-b^4 \\=a^4-b^4 .\end{array}$
The next identity would be : $(a-b)\left(a^4+a^3 b+a^2 b^2+a b^3+b^4\right)=a^5-b^5$.

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Question 43 Marks

Observe the multiplication grid below. Each number inside the grid is formed by multiplying two numbers. If the middle number of a 3 × 3 frame is given by the expression pq, as shown in the figure, write the expressions for the other numbers in the grid.

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MATHS 3 Marks Question

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