[2 Mark Question Answer]

Question 512 Marks

Use identities to evaluate : $(502)^2$

Answer

$(502)^2$
$(502)^2= (500 + 2)^2$
We know that,
$( a + b )^2 = a^2 + b^2 + 2ab$
$\therefore ( 500 + 2 )^2$
$= 500^2 + 2^2 + 2 \times 500 \times 2$
$= 250000 + 4 + 2000$
$= 2,52,004$

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Question 522 Marks

Use identities to evaluate : $(101)^2$

Answer

$(101)^2$
$(101)^2 = (100 + 1)^2$
We know that,
$(a + b)^2 = a^2 + b^2 + 2ab$
$\therefore (100 + 1)^2$
$= 100^2 + 1^2 + 2 \times 100\times 1$
$= 10,000 + 1 + 200$
$= 10,201$

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Question 532 Marks

The difference between two positive numbers is $5$ and the sum of their squares is $73.$ Find the product of these numbers.

Answer

Let the two positive numbers be $a$ and $b.$
Given difference between them is $5$ and sum of squares is $73.$
So $a - b = 5, a^2+ b^2= 73$
Squaring on both sides gives
$(a - b)^2= 5^2$
$a^2+ b^2- 2ab = 25$
$but a^2+ b^2= 73$
so $2ab = 73 - 25 = 48$
$ab= 24$
So, the product of numbers is $24.$

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Question 542 Marks

The number $x$ is $2$ more than the number $y.$ If the sum of the squares of $x $ and $y$ is $34,$ then find the product of $x$ and $y.$

Answer

Given $x$ is $2$ more than $y,$ so $x = y + 2$
Sum of squares of $x$ and $y$ is $34$, so $x^2+ y^2= 34.$
Replace $x = y + 2$ in the above equation and solve for $y.$
We get $(y + 2)^2+ y^2= 34$
$2y^2+ 4y - 30 = 0$
$y^2+ 2y - 15 = 0$
$(y + 5)(y - 3) = 0$
So $y = -5$ or $3$
For $y = -5, x =-3$
For $y = 3, x = 5$
Product of $x$ and $y$ is $15$ in both cases.

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Question 552 Marks

Find the square of $: \frac{3 a}{2 b}-\frac{2 b}{3 a}$

Answer

We know that,
$(\mathrm{a}-\mathrm{b}) 2=\mathrm{a} 2+\mathrm{b} 2-2 \mathrm{ab}$
$ \frac{3 a}{2 b}-\frac{2 b}{3 a}=\left[\frac{3 a}{2 b}\right]^2+\left[\frac{2 b}{3 a}\right]^2-2 \times \frac{3 a}{2 b} \times \frac{2 b}{3 a}$
$ =\frac{9 a^2}{4 b^2}+\frac{4 b^2}{9 a^2}-2$

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Question 562 Marks

Find the square of : $3a - 4b$

Answer

We know that
$( a - b )^2 = a^2 + b^2 - 2ab$
$( 3a - 4b )^2 = 9a^2 + 16b^2 - 2 \times 3a \times 4b$
$= 9a^2 + 16b^2 - 24ab$

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Question 572 Marks

Find the square of :$3a + 7b$

Answer

We know that
$( a + b )^2 = a^2 + b^2 + 2ab$
$( 3a + 7b )^2 = 9a^2 + 49b^2 + 2 \times 3a \times 7b$
$= 9a^2 + 49b^2 + 42ab$

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Question 582 Marks