Question 12 Marks
Simplify$:\left(x-\frac{1}{x}\right)\left(x^2+1+\frac{1}{x^2}\right)$
Answer$\left(x-\frac{1}{x}\right)\left(x^2+1+\frac{1}{x^2}\right) $
$ =x\left(x^2+1+\frac{1}{x^2}\right)-\frac{1}{x}\left(x^2+1+\frac{1}{x^2}\right) $
$ =x^3+x+\frac{1}{x}-x-\frac{1}{x}-\frac{1}{x^3}$
$=x^3-\frac{1}{x^3} .$
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Simplify$:(4x + 5y)^2 + (4x - 5y)^2$
Answer$(4x + 5y)^2 + (4x - 5y)^2$
$= (4x)^2 + (5y)^2 + 2(4x)(5y) + (4x)^2+ (5y)^2 - 2(4x)(5y)$
$= 16x^2 + 25y^2 + 40xy + 16x^2 + 25y^2 - 40xy$
$= 32x^2 + 50y^2.$
View full question & answer→Question 32 Marks
Find the cube of$: \frac{2 m}{3 n}+\frac{3 n}{2 m}$
Answer$\left(\frac{2 m }{3 n }+\frac{3 n }{2 m }\right)^3 $
$ =\left(\frac{2 m }{3 n }\right)^3+\left(\frac{3 n }{2 m }\right)^3+3\left(\frac{2 m }{3 n }\right)\left(\frac{3 n }{2 m }\right)\left(\frac{2 m }{3 n }+\frac{3 n }{2 m }\right)$
$ =\frac{8 m ^3}{27 n ^3}+\frac{27 n ^3}{8 m ^3}+\frac{2 m }{ n }+\frac{9 n }{2 m } .$
View full question & answer→Question 42 Marks
Find the cube of$: 4 p-\frac{1}{p}$
Answer$\left(4 p-\frac{1}{p}\right)^3 $
$=(4 p)^3-\left(\frac{1}{p}\right)^3-3(4 p)\left(\frac{1}{p}\right)\left(4 p-\frac{1}{p}\right) $
$ =64 p^3-\frac{1}{p^3}-48 p+\frac{12}{p}$
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Find the cube of$: 3 a+\frac{1}{3 a}$
Answer$\left(3 a+\frac{1}{3 a}\right)^3 $
$=(3 a)^3+\left(\frac{1}{3 a}\right)^3+3(3 a)\left(\frac{1}{3 a}\right)\left(3 a+\frac{1}{3 a}\right) $
$=27 a^3+\frac{1}{27 a^3}+9 a+\frac{1}{a} .$
View full question & answer→Question 62 Marks
If $a+\frac{1}{a}=6;$find $a^2-\frac{1}{a^2}$
Answer$a^2-\frac{1}{a^2}$
$=\left(a+\frac{1}{a}\right)\left(a-\frac{1}{a}\right) $
$ =(6)( \pm 4 \sqrt{2})$
$ = \pm 24 \sqrt{2} .$
View full question & answer→Question 72 Marks
Simplify by using formula $:\left(a+\frac{2}{a}-1\right)\left(a-\frac{2}{a}-1\right)$
Answer$\left(a+\frac{2}{a}-1\right)\left(a-\frac{2}{a}-1\right) $
$=(a-1)^2-\left(\frac{2}{a}\right)^2 $
$=a^2+1-2 a-\frac{4}{a^2}$
$($Using identity$: \left.(a+b)(a-b)=a^2-b^2\right)$.
View full question & answer→Question 82 Marks
Simplify by using formula $:(x + y - 3) (x + y + 3)$
Answer$(x + y - 3) (x + y + 3)$
$= (x + y)^2 - (3)^2$
$= x^2 + y^2 + 2xy - 9$
$($Using identity : $(a + b)(a - b) = a^2 - b^2).$
View full question & answer→Question 92 Marks
Simplify by using formula $:(2x + 3y) (2x - 3y)$
Answer$(2x + 3y) (2x - 3y)$
$= (2x)^2 - (3y)^2$
$= 4x^2 - 9y^2$
$($Using identity : $(a + b) (a - b) = a^2 - b^2).$
View full question & answer→Question 102 Marks
Simplify by using formula $:(5x - 9) (5x + 9)$
Answer$(5x - 9) (5x + 9)$
$= (5x)^2- (9)^2$
$= 25x^2 - 81$
$($Using identity : $(a + b) (a - b) = a^2 - b^2).$
View full question & answer→Question 112 Marks
Find the squares of the following$:\frac{7 x}{9 y}-\frac{9 y}{7 x}$
Answer$\left(\frac{7 x}{9 y}-\frac{9 y}{7 x}\right)^2 $
$ =\left(\frac{7 x}{9 y}\right)^2+2\left(\frac{7 x}{9 y}\right)\left(\frac{9 y}{7 x}\right)+\left(\frac{9 y}{7 x}\right)^2$
$ =\frac{49 x^2}{81 y^2}+2+\frac{81 y^2}{49 x^2} .$
View full question & answer→Question 122 Marks
Find the squares of the following$:3p - 4q^2$
Answer$(3p - 4q)^2$
$= (3p)^2 - 2(3p)(4q) + (4q)^2$
$= 9p^2 - 24pq + 16q^2.$
View full question & answer→Question 132 Marks
Find the squares of the following$:9m - 2n$
AnswerUsing $(x + y)^2$
$= x^2 + 2xy + y^2$,
we get $(9m - 2n)^2$
$= (9m)^2 + 2(9m)(-2n) + (-2n)^2$
$= 81m^2- 36mn + 4n^2.$
View full question & answer→Question 142 Marks
Expand the following$:(x - 3y - 2z)^2$
AnswerUsing $(a + b + c)^2$
$= a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$
$(x - 3y - 2z)^2$
$= x^2 + (3y)^2 + (2z)^2 + 2(x)(-3y) + 2(-3y)(-2z) + 2(x)(-2z)$
$= x^2 + 9y^2 + 4z^2- 6xy + 12yz - 4xz.$
View full question & answer→Question 152 Marks
Expand the following$:\left(2 a+\frac{1}{2 a}\right)^2$
Answer$\left(2 a+\frac{1}{2 a}\right)^2 $
$=(2 a)^2+2(2 a)\left(\frac{1}{2 a}\right)+2 a^2$
$ =4 a^2+2+\frac{1}{4 a^2}$
View full question & answer→Question 162 Marks
Expand the following$:(2p - 3q)^2$
Answer$(2p - 3q)^2$
$= (2p)^2 - 2(2p)(3q) + (3q)^2$
$= 4p^2 - 12pq + 9q^2.$
View full question & answer→Question 172 Marks
Expand the following$:(a + 3b)^2$
AnswerUsing $(x + y)^2$
$= x^2 + 2xy + y^2, $
we get $(a + 3b)^2$
$= a^2+ 2(a)(3b) + (3b)^2$
$= a^2 + 6ab + 9ab^2.$
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If $a^2-3 a-1=0$ and $a \neq 0,$ find $: a^2-\frac{1}{a^2}$
Answer$ a ^2-\frac{1}{ a ^2}$
$ =\left( a +\frac{1}{ a }\right)\left( a -\frac{1}{ a }\right) $
$ =( \pm \sqrt{13})(3) $
$ = \pm 3 \sqrt{13} .$
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If $a^2-3 a-1=0$ and $a \neq 0,$ find $: a-\frac{1}{a}$
AnswerDividing the given equation by a we get
$a-3-\frac{1}{a}=0$
$\Rightarrow a-\frac{1}{a}=3$
View full question & answer→Question 202 Marks
If $a ^2-7 a +1=0$ and $a =\neq 0$, find$:a +\frac{1}{ a }$
AnswerDividing the given equation by $a,$ we get$:$
$\frac{ a ^2}{ a }-\frac{7 a }{ a }+\frac{1}{ a }=0,$
$a -7+\frac{1}{ a }=0 $
$\Rightarrow a +\frac{1}{ a }=7$
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Expand the following$:(2x - 5) (2x + 5) (2x- 3)$
Answer$(2x - 5) (2x + 5) (2x- 3)$
$= (4x^2 - 25) (2x - 3)$
$= 8x^3- 12x^2 - 50x + 75$
$($Using identity $: (x - a) (x + b))$
$= x^2 - (a - b) x - ab).$
View full question & answer→Question 222 Marks
If $x + y = 1$ and $xy = -12;$ find$:x^2 - y^2.$
Answer$x^2 - y^2$
$= (x + y) (x - y)$
$= (1) (±7)$
$= ±7.$
View full question & answer→Question 232 Marks
Expand the following$:(3x + 4) (2x - 1)$
Answer$(3x + 4) (2x - 1)$
$= 6x^2- 3x + 8x - 4$
$= 6x^2 + 5x - 4$
Using identity : $(x+ a) (x -b)$
$= x^2 + (a - b) x - ab).$
View full question & answer→Question 242 Marks
If $m - n = 0.9$ and $mn = 0.36,$ find$:m^2 - n^2.$
Answer$m^2 - n^2$
$= (m + n)(m - n)$
$= (±1.5)(0.9)$
$= ±1.35.$
View full question & answer→Question 252 Marks
Expand the following$:(x - 5) (x - 4)$
Answer$(x - 5) (x - 4)$
$= x^2 -5x - 4x + 20$
$= x^2 - 9x + 20$
$($Using identify : $(x - a) (x - b))$
$= x^2 - (a + b) x + ab).$
View full question & answer→Question 262 Marks
If $p + q = 8$ and $p - q = 4,$ find$:p^2+ q^2$
AnswerPutting $pq = 12$ in
$(i)$ we get :$p^2 + q^2$
$= 64 - 2(12)$
$= 64 - 24$
$= 40.$
View full question & answer→Question 272 Marks
Expand the following$:(m + 8) (m - 7)$
Answer$(m + 8) (m - 7)$
$= m^2+ 8m - 7m - 56$
$= m^2 + m - 56$
$($Using identify : $(x+ a) (x - b))$
$= x^2 + (a - b) x - ab).$
View full question & answer→Question 282 Marks
If $x+\frac{1}{x}=3 ;$ find $x^4+\frac{1}{x^4}$
Answersquaring both sides of the equation $\left(x^2+\frac{1}{x^2}\right)=7$, we get:
$x^4+\frac{1}{x^4}+2 =49 $
$x^4+\frac{1}{x^4} $
$=47 .$
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Expand the following$:(a + 4) (a + 7)$
Answer$(a + 4) (a + 7)$
$= a^2+ 4a + 7a + 28$
$= a^2 + 11a + 28$
$($Using identity : $(x + a) (x + b))$
$= x^2 + (a + b) x + ab).$
View full question & answer→Question 302 Marks
If $a -\frac{1}{ a }=10 ;$ find $a ^2-\frac{1}{ a ^2}$
Answer$a ^2-\frac{1}{ a ^2} $
$=\left( a +\frac{1}{ a }\right)\left( a -\frac{1}{ a }\right) $
$ =( \pm 2 \sqrt{26})(10) $
$= \pm 20 \sqrt{26} .$
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