Polynomials Test

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Polynomials Test - 28

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Weekly Quiz Competition

Question 1
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 2
1 / -0

If $$f(x) = 2x^3 - 13x^2 +17 x+12$$, then find out the value of $$f(-2)$$

A
$$-90$$

B
$$-85$$

C
$$-70$$

D
$$-55$$

Solution

$$f(x) = 2x^3 - 13x^2 + 17x+12$$
$$f(-2)=2(-2)^3-13(-2)^2+17(-2)+12$$
$$=-16 - 52 - 34 +12 =-90$$
Question 3
1 / -0

Zero of a polynomial is always zero.

A
True

B
False

C
Cannot be determined

D
None of the above

Solution

We know that the zero of the polynomial is defined as any real value of $$x$$, for which the value of the polynomial becomes zero.

For example:
Let $$f(x)=− 4x + 7$$ be the polynomial.
To find the zero of the polynomial, we substitute $$f(x)=0$$ as shown below:
$$f(x)=−4x+7\\ \Rightarrow 0=−4x+7\\ \Rightarrow 4x=7$$
$$\Rightarrow x=\dfrac { 7 }{ 4 }$$.

Therefore, $$\dfrac {7}{4}$$ is the zero of the polynomial $$f(x)=− 4x + 7$$, which contradicts the statement that zero of a polynomial is always zero.

Hence, the zero of a polynomial is not always zero.
Therefore, the given statement is false.

That is, option $$B$$ is correct.
Question 4
1 / -0

If the two zeroes of a quadratic polynomial $$ax^{2} +

bx + c$$ are negative, which of the following statements holds true?

A
a and c are of opposite signs.

B
a, b and c have the same sign.

C
a and b are of opposite signs.

D
b and c are of opposite signs.
Question 5
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 6
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 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:
$$9.8\times 10.2 $$.

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 9
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 10
1 / -0

Simplify: $$\left (\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2} \right)$$.

A
$$3$$

B
$$5$$

C
$$6$$

D
$$8$$

Solution

Given, $$(\sqrt { 5 } +\sqrt { 2 } )(\sqrt { 5 } -\sqrt { 2 } )$$.

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

Applying the identity,

$$(\sqrt { 5 } +\sqrt { 2 } )(\sqrt { 5 } -\sqrt { 2 } )$$

$$=(\sqrt { 5 })^2- (\sqrt { 2 })^2=5-2=3$$.

Hence, $$(\sqrt { 5 } +\sqrt { 2 } )(\sqrt { 5 } -\sqrt { 2 } )=3$$.

Therefore, option $$A$$ is correct.

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Polynomials Test - 28

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Polynomials Test - 28

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Question 1

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

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

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

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

Solution

Question 2

$$-90$$

$$-85$$

$$-70$$

$$-55$$

Solution

Question 3

True

False

Cannot be determined

None of the above

Solution

Question 4

a and c are of opposite signs.

a, b and c have the same sign.

a and b are of opposite signs.

b and c are of opposite signs.

Question 5

$$0.76$$

$$0.55$$

$$1$$

$$1.4$$

Solution

Question 6

$$24.84$$

$$25.24$$

$$24.94$$

$$25.84$$

Solution

Question 7

$$63.91$$

$$63.81$$

$$64.91$$

$$64.21$$

Solution

Question 8

$$98.96$$

$$99.86$$

$$98.86$$

$$99.96$$

Solution

Question 9

$$8881$$

$$9991$$

$$9981$$

$$9891$$

Solution

Question 10

$$3$$

$$5$$

$$6$$

$$8$$

Solution

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