Tag: using brackets in algebraic expressions

Questions Related to using brackets in algebraic expressions

Subtract  $ - \frac{2}{3}{y^3}-\frac{2}{7}{y^2} - 5$ from $\frac{1}{3}{y^3} + \frac{5}{7}{y^2} - 2$, then the resultant value is . 

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. ${y^3} + {y^2} - 7$

  2. ${y^3} + {y^2} + 3$

  3. ${y^3} + {y^2} - 3$

  4. ${y^3} + {y^2} + 7$

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Correct Option: A

If r and s are zeroes of the polynomial $t^2-4t+3$, then $\dfrac{1}{r}+\dfrac{1}{s}-2rs+\dfrac{14}{3}$ is equal to

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. 0

  2. 1

  3. 2

  4. -1

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Correct Option: A
Explanation:
Quadratic Equation
If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$
then $\alpha + \beta=\dfrac{-b}{a}$ and $\alpha\beta=\dfrac{c}{a}$

$f(t)=t^2-4t+3$
$r+s=4$
$rs=3$
Now,
$\dfrac{1}{r}+\dfrac{1}{s}-2rs+\dfrac{14}{3}=\dfrac{r+s}{rs}-2rs+\dfrac{14}{3}$

$=\dfrac{4}{3}-6+\dfrac{14}{3}=\dfrac{4-18+14}{3}=0$

State whether True or False.

Simplify: $(3y+4z)(3y-4z)+(2y+7z)(y+z) $.
The answer is $11y^2+9yz-9z^2$.

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. True

  2. False

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Correct Option: A
Explanation:

$Multiplying\quad (3y+4z)(3y-4z),\quad (2y+7z)(y+z),\ =-16z^{ 2 }+12yz-12yz+9y^{ 2 }\quad +\quad 7z^{ 2 }+7yz+2yz+2y^{ 2 }\ =-9z^{ 2 }+9yz+11y^{ 2 }$

Simplify $(a + b) (c -d) + (a-  b) (c + d) + 2 (ac + bd)$

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. $4ac$

  2. $4ac - 4bd$

  3. $4bd$

  4. $4ac+4bd$

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Correct Option: A
Explanation:

$(a+b)(c-d)+(a-b)(c+d)+2(ac+bd)$
$=a(c-d)+b(c-d)+a(c+d)-b(c+d)+2(ac+bd)$
$=ac-ad+bc-bd+ac+ad-bc-db+2ac+2bd$
$=4ac$

Simplify: $(x + y)(x^2 -xy + y^2)$

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. $x^3 - y^3$

  2. $x^3 + y^3$

  3. $x^3 + y^3 + 3ab$

  4. $x^3 + y^3-3ab$

Show answer
Correct Option: B
Explanation:

$(x + y)(x^2 - xy + y^2)$
$=x(x^2 - xy + y^2)+y(x^2 - xy + y^2)$
$=x^3-x^2y+xy^2+x^2y^-xy^2+y^3$
$=x^3+y^3$

The simplified form of the expression given below is :$\dfrac{\dfrac{y^4-x^4}{x(x+y)}-\dfrac{y^3}{x}}{y^2-xy+x^2}$

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. $1$

  2. $0$

  3. $-1$

  4. $2$

Show answer
Correct Option: C
Explanation:

The given expression $\dfrac { \dfrac { { y }^{ 4 }-{ x }^{ 4 } }{ x(x+y) } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } }$ can be simplified as follows:

 
$\dfrac { \dfrac { { y }^{ 4 }-{ x }^{ 4 } }{ x(x+y) } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \ =\dfrac { \dfrac { ({ y }^{ 2 })^{ 2 }-({ x }^{ 2 })^{ 2 } }{ x(x+y) } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \ =\dfrac { \dfrac { ({ y }^{ 2 }-{ x }^{ 2 })({ y }^{ 2 }+{ x }^{ 2 }) }{ x(x+y) } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \quad \quad \quad \quad \quad \quad \quad \left( \because \quad a^{ 2 }-b^{ 2 }=(a+b)(a-b) \right)$
$=\dfrac { \dfrac { ({ y }+x)(y-x)({ y }^{ 2 }+{ x }^{ 2 }) }{ x(x+y) } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \ =\dfrac { \dfrac { (y-x)({ y }^{ 2 }+{ x }^{ 2 }) }{ x } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \ =\dfrac { \dfrac { { y }^{ 3 }-xy^{ 2 }+yx^{ 2 }-{ x }^{ 3 }-{ y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } }$
$=\dfrac { \dfrac { -x(y^{ 2 }-xy+{ x }^{ 2 }) }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \ =-\dfrac { y^{ 2 }-xy+{ x }^{ 2 } }{ { y }^{ 2 }-xy+{ x }^{ 2 } } \ =-1$

Hence, $\dfrac { \dfrac { { y }^{ 4 }-{ x }^{ 4 } }{ x(x+y) } -\dfrac { { y }^{ 3 } }{ x }  }{ { y }^{ 2 }-xy+{ x }^{ 2 } }=-1$  

In the equation $4x+y=10$, if the value of $x$ ins increased by $3$, then what would be the effect on the corresponding value of $y$

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. The value of $y$ is decreased by $12$

  2. The value of $y$ is decreased by $2$

  3. The value of $y$ is increased by $3$

  4. The value of $y$ will be $3$ times as large

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Correct Option: A

Evaluate:
$( b - c + d + a ) ( d + a - b + c ) + ( c - d + a + b ) ( b + c + d - a )$

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. $4 ( a d + b c )$

  2. $2 ( a d + b c )$

  3. $3 ( a d + b c )$

  4. $ ( a d + b c )$

Show answer
Correct Option: A
Explanation:
$(b-c+d+a)(d+a-b+c)+(c-d+a+b)(b+c+d-a)$

$=[(d+a)+(b-c)][(d+a)-(b-c)]+[(b+c)+(a-d)][(b+c)-(a-d)]$

$[\because (a-d)^2=(d-a)^2]$

$=(d+a)^2-(b-c)^2+(b+c)^2-(a-d)^2$

$=(d+a)^2-(d-a)^2+(b+c)^2-(b-c)^2$

$=4ad+4bc$

$=4(ad+bc)$.

Evaluate $\sqrt {13+\sqrt {44+10^2}}$.

maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. $12$

  2. $5$

  3. $25$

  4. None

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Correct Option: B
Explanation:

$\sqrt {13+\sqrt {44+10^2}}\$

$=\sqrt{13+\sqrt {44+100}}\$
$=\sqrt {13+144}\$
$=\sqrt {13+12}\$
$=\sqrt {25}=5$

If $f(x)=x^2+x+5$ then find $f(1)$
maths brackets order operations and algebra using brackets in algebraic expressions order of operations

  1. $7$

  2. $5$

  3. $4$

  4. None of the above

Show answer
Correct Option: A
Explanation:
$f(x)=x^2+x+5$

Put $x=1$

$\Rightarrow$$f(1)=1+1+5$

$\Rightarrow$$f(1)=7$