Algebraic Expressions and Identities Test

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Algebraic Expressions and Identities Test - 24

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Weekly Quiz Competition

Question 1
1 / -0

Simplify:
$$(2.5p -1.5q)^2- (1.5p-2.5q)^2$$.

A
$$4p^2 - 4q^2$$

B
$$6.25p^2 -4q^2$$

C
$$4p^2 - 2.25q^2$$

D
$$6.25p^2 - 2.25q^2$$

Solution

Given, $$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$$.

We know, $$(a-b)^2=a^2-2ab+b^2$$.

Then,
$$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$$
$$=[(2.5p)^2-2(2.5p)(1.5q)+(1.5q)^2]-[(1.5p)^2-2(1.5p)(2.5q)+(2.5q)^2]$$
$$=6.25p^2-7.5pq+2.25q^2-2.25p^2+7.5pq-6.25q^2$$
$$=6.25p^2+2.25q^2-2.25p^2-6.25q^2$$
$$=6.25p^2-2.25p^2+2.25q^2-6.25q^2$$
$$=4p^2-4q^2$$.

Therefore, option $$A$$ is correct.
Question 2
1 / -0

The value of the product $$(4a^{2}+3b)(9b^{2}+4a)$$ at $$a = 1,$$ b$$ = -2$$ is

A
$$60$$

B
$$-80$$

C
$$70$$

D
$$-50$$

Solution

At $$a = 1,$$ $$b = -2$$ we have $$4a^2 + 3b= 4 + 3(-2) = -2$$ and
$$9b^2 + 4a = 9(-2)^2 + 4(1) = 9 \times 4 + 4 = 40$$
Thus product =$$ -2 \times 40 = -80 $$
Question 3
1 / -0

Simplify : $$\displaystyle \left(\frac{3}{2}x - 0.45y\right)^2$$

A
$$2.25x^2-1.35xy+0.2015y^2$$

B
$$2.15x^2-1.35xy+0.2025y^2$$

C
$$2.25x^2-1.35xy+0.2025y^2$$

D
$$2.25x^2-1.25xy+0.2025y^2$$

Solution

$$\displaystyle \left(\frac{3}{2}x - 0.45y\right)^2$$
$$=\displaystyle \left(1.5x - 0.45y\right)^2$$
$$=\displaystyle (1.5x)^2 + (0.45y)^2- 2(1.5x)(0.45y)^2$$
$$= 2.25x^2 +0.2025 y^2 -1.35xy$$
Question 4
1 / -0

Evaluate (using formulae): $$\displaystyle \frac{2.43 \times 2.43 - 2 \times 2.43 \times 1.67 +1.67 \times 1.67}{2.43 - 1.67}$$

A
$$0.76$$

B
$$0.55$$

C
$$1$$

D
$$1.4$$

Solution

Given: $$\displaystyle \frac {2.43 \times 2.43 - 2 \times 2.43 \times 1.67 + 1.67 \times 1.67}{2.43-1.67}$$
$$=\displaystyle \frac {(2.43)^2 - 2 \times 2.43 \times 1.67 + (1.67)^2 }{0.76}$$
$$=\displaystyle \frac {(2.43-1.67)^2}{0.76}$$ [ $$ \because a^2-2ab+b^2=(a-b)^2 $$]

$$=\displaystyle \frac {(0.76)^2}{0.76}$$

$$=0.76$$
Question 5
1 / -0

Use the identity $$ (a+b)(a-b) = a^2-b^2$$ to evaluate:
$$9.8\times 10.2 $$.

A
$$98.96$$

B
$$99.86$$

C
$$98.86$$

D
$$99.96$$

Solution

We know, $$ 9.8 \times 10.2 = (10 -0.2) \times (10+0.2) $$ .

Applying the formula $$ (a+b)(a-b) = { a }^{ 2 }-{ b }^{ 2 } $$, where $$ a = 10 , b = 0.2 $$,
we get,
$$ 9.8 \times 10.2 = (10 -0.2) \times (10+0.2)=10^2-(0.2)^2 \\ =100-0.04=99.96 .$$
Therefore, option $$D$$ is correct.
Question 6
1 / -0

Use the identity $$ (a+b)(a-b) = a^2-b^2$$ to evaluate:
$$103\times 97 $$.

A
$$8881$$

B
$$9991$$

C
$$9981$$

D
$$9891$$

Solution

We know, $$ 103 \times 97 = (100 + 3) \times (100 - 3) $$ .

Applying the formula $$ (a+b)(a-b) = { a }^{ 2 }-{ b }^{ 2 } $$, where $$ a = 100 , b = 3 $$,
we get,
$$ 103 \times 97 = (100 + 3) \times (100 - 3) = { 100 }^{ 2 }-{ 3 }^{ 2 } = 10000 - 9
= 9991 $$.
Therefore, option $$B$$ is correct.
Question 7
1 / -0

Use the identity $$ (a+b)(a-b) = a^2-b^2$$ to evaluate:
$$8.3\times 7.7 $$.

A
$$63.91$$

B
$$63.81$$

C
$$64.91$$

D
$$64.21$$

Solution

We are given the formula $$(a+b)(a-b)=a^2-b^2$$.

We know, $$8.3\times7.7=(8+0.3)(8-0.3)$$,
where $$a=8$$ and $$b=0.3$$.

Now, solve as shown below:
$$8.3\times7.7$$
$$=(8+0.3)(8-0.3)=8^{ 2 }-(0.3)^{ 2 }\\ \Rightarrow 8.3\times 7.7=64-0.09\\ \Rightarrow 8.3\times 7.7=63.91.$$

Hence, $$8.3\times 7.7=63.91$$.

Therefore, option $$A$$ is correct.
Question 8
1 / -0

Use the identity $$ (a+b)(a-b)=a^2-b^2$$ to evaluate:
$$4.6\times 5.4 $$.

A
$$24.84$$

B
$$25.24$$

C
$$24.94$$

D
$$25.84$$

Solution

We are given the formula $$(a+b)(a-b)=a^2-b^2$$.

We know, $$4.6 \times 5.4 = (5-0.4)(5+0.4)$$,
where $$a=5$$ and $$b=0.4$$.

Now, solve as shown below:
$$4.6 \times 5.4 = (5-0.4)(5+0.4)$$
$$={ 5 }^{ 2 }-{ 0.4 }^{ 2 }=25-0.16=24.84$$.

Hence, $$4.6 \times 5.4 =24.84$$.

Therefore, option $$A$$ is correct.
Question 9
1 / -0

Evaluate using expansion of $$(a+b)^2$$ or $$(a-b)^2$$ :
$$(208)^2$$

A
45264

B
45294

C
43284

D
43264

Solution

$$ {208}^{2} = {(200 + 8)}^{2} $$

It is the form of $$ {(a+b)}^{2} $$, where $$ a = 200, b = 8 $$

Applying

the formula $$ { (a+b) }^{ 2 } = {a}^{2} + { b }^{ 2 } + 2ab $$
$$ {208}^{2} = {(200 + 8)}^{2} = {200}^{2} + { 8 }^{ 2 } + 2\times 200 \times 8 = 40000 + 64 + 3200 = 43264 $$
Question 10
1 / -0

Evaluate using the expansion of $$(a+b)^2$$ or $$(a-b)^2$$ :
$$(92)^2$$

A
8444

B
8464

C
8474

D
8414

Solution

Given: $$ {92}^{2} = {(90 + 2)}^{2} $$

It is in the form of $$ {(a+b)}^{2} $$, where $$ a = 90, b = 2 $$.

Applying the formula $$ { (a+b) }^{ 2 } = {a}^{2} + { b }^{ 2 } + 2ab $$ we get,

$$ {92}^{2} = {(90 + 2)}^{2} $$

$$= {90}^{2} + { 2 }^{ 2 } + 2\times 90 \times 2 $$

$$= 8100 + 4 + 360 = 8464 $$

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Algebraic Expressions and Identities Test - 24

Result

Algebraic Expressions and Identities Test - 24

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SHARING IS CARING

Question 1

$$4p^2 - 4q^2$$

$$6.25p^2 -4q^2$$

$$4p^2 - 2.25q^2$$

$$6.25p^2 - 2.25q^2$$

Solution

Question 2

$$60$$

$$-80$$

$$70$$

$$-50$$

Solution

Question 3

$$2.25x^2-1.35xy+0.2015y^2$$

$$2.15x^2-1.35xy+0.2025y^2$$

$$2.25x^2-1.35xy+0.2025y^2$$

$$2.25x^2-1.25xy+0.2025y^2$$

Solution

Question 4

$$0.76$$

$$0.55$$

$$1$$

$$1.4$$

Solution

Question 5

$$98.96$$

$$99.86$$

$$98.86$$

$$99.96$$

Solution

Question 6

$$8881$$

$$9991$$

$$9981$$

$$9891$$

Solution

Question 7

$$63.91$$

$$63.81$$

$$64.91$$

$$64.21$$

Solution

Question 8

$$24.84$$

$$25.24$$

$$24.94$$

$$25.84$$

Solution

Question 9

45264

45294

43284

43264

Solution

Question 10

8444

8464

8474

8414

Solution

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