[5 marks sum]

Question 515 Marks

Solve the following equations for the unknown: $\frac{6 x+7}{3 x+2}=\frac{4 x+5}{2 x+3}$

Answer

$\frac{6 x+7}{3 x+2}=\frac{4 x+5}{2 x+3} $
$ \Rightarrow(6 x+7)(2 x+3)=(4 x+5)(3 x+2) $
$ \Rightarrow 6 x(2 x+3)+7(2 x+3)=4 x(3 x+2)+5(3 x+2) $
$ \Rightarrow 12 x^2+18 x+14 x+21=12 x^2+8 x+15 x+10 $
$\Rightarrow 18 x+14 x+21=8 x+15 x+10$
$ \Rightarrow 18 x-8 x+14 x-15 x=10-21$
$\Rightarrow 9 x=-11 $
$\Rightarrow x=-\frac{11}{9}$

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Question 525 Marks

Solve the following equations for the unknown: $\frac{7}{x-2}-\frac{5}{3}=3, x \neq 2$

Answer

$\frac{7}{x-2}-\frac{5}{3}=3, x \neq 2 $
$ \Rightarrow \frac{7}{x-2}=\frac{5}{3}+3 $
$ \Rightarrow \frac{7}{x-2}=\frac{5+9}{3}$
$\Rightarrow \frac{7}{x-2}=\frac{14}{3} $
$\Rightarrow 21=14(x-2)$
$ \Rightarrow 21=14 x-28 $
$ \Rightarrow 49=14 x $
$ \Rightarrow x=\frac{49}{14} $
$ \Rightarrow x=\frac{7}{2} .$

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Question 535 Marks

Solve the following equations for the unknown: $\frac{1}{x-1}+\frac{4}{5}=\frac{2}{3}, x \neq 1$

Answer

$ \frac{1}{x-1}+\frac{4}{5}=\frac{2}{3}, x \neq 1$
$\Rightarrow \frac{1}{x-1}=\frac{2}{3}-\frac{4}{5}$
$\Rightarrow \frac{1}{x-1}=\frac{10}{15}-\frac{12}{15}$
$\Rightarrow \frac{1}{x-1}=\frac{-2}{15}$
$\Rightarrow 15=-2(x-1)$
$\Rightarrow 15=-2 x+2$
$\Rightarrow 2 x=-13$
$\Rightarrow x=\frac{-13}{2}$
$\Rightarrow x=-6 \frac{1}{2} .$

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Question 545 Marks

If $m(x-1)=40$, find the value of $m$ when $\frac{x-1}{2}=1+\frac{x+1}{3}$.

Answer

Given $\ m(x-1)=40$
$\Rightarrow x-1=\frac{40}{\ m} $
$\frac{x-1}{2}=1+\frac{x+1}{3} $
$\frac{x-1}{2}=1+\frac{x-1+1+1}{3} \ldots($Adding and subtracting $1$ in the $\text{R.H.S.})$
$ \Rightarrow \frac{\left(\frac{40}{ \ m }\right)}{2}=1+\frac{\left(\frac{40}{ \ m }\right)^3+2}{3}$
$\Rightarrow \frac{40}{2\ m }=1+\frac{40+2\ m }{\frac{\ m }{3}} $
$\Rightarrow \frac{40}{2\ m }=1+\frac{40+2\ m }{3\ m } $
$ \Rightarrow \frac{40}{2\ m }=\frac{3\ m +40+2\ m }{3} $
$ \Rightarrow \frac{40}{2\ m }=\frac{5\ m +40}{3\ m }$
$ \Rightarrow 40(3\ m )=2\ m (5\ m +40) $
$ \Rightarrow 120 \ m =10 \ m ^2+80\ m $
$\Rightarrow 10\ m ^2+80\ m -120\ m =0 $
$ \Rightarrow 10\ m ^2=40\ m $
$\Rightarrow \ m =4 \text {. } $

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Question 555 Marks

If $x+\frac{6}{a}=11$, find the value of a when $4 \frac{1}{3}-\frac{3 x-4}{5}=\frac{x-7}{3}$.

Answer

$x+\frac{6}{a}=11$
$\Rightarrow \frac{a x+6}{a}=11$
On Cross$-$Multiplying, we get:
$ax +6=11 a $
$\Rightarrow x =\frac{11 a -6}{ a } \cdots \cdots(1) $
$4 \frac{1}{3}-\frac{3 x-4}{5}=\frac{x-7}{3} $
$\Rightarrow \frac{13}{3}-\frac{3 x-4}{5}=\frac{x-7}{3} $
$\Rightarrow \frac{13 \times 5-3(3 x-4)}{15}=\frac{x-7}{3} $
$\Rightarrow \frac{65-9 x+12}{5}= x -7$
Cross multiplying:
$5(x-7)=77-9 x $
$\Rightarrow 5 x+9 x=77+35 $
$\Rightarrow 14 x=112 $
$\Rightarrow x=\frac{112}{14}=8 \dots .$
From $(1)$ and $(2),$
$\frac{11 a-6}{a}=8$
$\Rightarrow 8 a =11 a -6$
$\Rightarrow 6=3 a$ or $a =2$.

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Question 565 Marks

If $\frac{1}{x}-\frac{2}{3 b}+1=0$, find the value of $b$ when $\frac{2 x+4}{8}-\frac{3-2 x}{12}=\frac{x-3}{6}$

Answer

$\frac{1}{x}-\frac{2}{3 b}+1=0$
Taking $\text{LCM},$
$\Rightarrow \frac{3 b -2 x+3 b x}{3 b x}=0 $
$\Rightarrow 3 b = x (-3 b +2) $
$\Rightarrow x =\frac{3 b }{2-3 b } \cdots(1)$
Solving $\frac{2 x+4}{8}-\frac{3-2 x}{12}=\frac{x-3}{6}$ for $x$ :
$\Rightarrow \frac{x+2}{4}-\frac{3-2 x}{12}-\left(\frac{x-3}{6}\right)^6=0$
Taking $\text{LCM},$
$\frac{3(x+2)-(3-2 x)-2(x-3)}{12}=0 $
$\Rightarrow 3 x+6-3+2 x-2 x+6=0 $
$\Rightarrow 3 x=-9 $
$\Rightarrow x=-3 \dots(2)$
From $(1)$ and $(2),$
$\Rightarrow \frac{3 b }{2-3 b }=-3$
Cross multiplying,
$\Rightarrow 3 b=-6+9 b $
$\Rightarrow-6 b=-6 $
$\Rightarrow b=1 .$

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Question 575 Marks

In the following equations, verify if the given value is a solution of the equation: $2 \frac{1}{2} x+3 \frac{1}{2} x=56,2 x ; x=7$

Answer

$2 \frac{1}{2} x+3 \frac{1}{2} x=56-2 x _{ i } x =7$
Simplifying, we get
$\Rightarrow \frac{5}{2} x+\frac{7}{2} x=56-2 x $
$\Rightarrow \frac{12}{2} x+2 x=56 $
$\Rightarrow 8 x =56 $
$\Rightarrow x =7$
Put $x=7$ in above gives, $\text{L.H.S. = R.H.S.}$
Thus, $x =7$ is a solution of the equation $2 \frac{1}{2} x+3 \frac{1}{2} x=56-2 x _{ i } x =7$.

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Question 585 Marks

If $m=x-3$ and $\frac{4 m-3}{2}-\frac{3 m-1}{5}=\frac{3}{2}$, find $x$.

Answer

We first solve $\frac{4\ m-3}{2}-\frac{3\ m-1}{5}=\frac{3}{2}$ for $m$ :
Taking $\text{LCM},$
$\frac{5(4\ m-3)-2(3\ m-1)}{10}=\frac{3}{2} $
$\Rightarrow \frac{14 m-13}{10}=\frac{3}{2}$
$\Rightarrow$ On Cross $-$ multiplying, we get:
$2(14\ m-13)=30$
$\Rightarrow 14\ m-13=15$
$\Rightarrow m =2$
Now, given $m = x -3$
$\Rightarrow 2=x-3 $
$\Rightarrow x=5 .$

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Question 595 Marks

If $x=p+1$ and $2.5+\frac{2 p+1}{3}=1.5(2 x-1)$, find the value of $' p\ '.$

Answer

$x=p+1$ and $2.5+\frac{2 p+1}{3}=1.5(2 x-1) $
$ \Rightarrow 2.5+\frac{2 p+1}{3}=1.5[2(p+1)-1]$
$ \Rightarrow 2.5+\frac{2 p+1}{3}=3(p+1)-1.5 $
$ \Rightarrow 2.5+\frac{2 p+1}{3}=3 p+3-1.5$
$ \Rightarrow 2.5+\frac{2 p+1}{3}=3 p+1.5 $
$ \Rightarrow \frac{2 p+1}{3}-3 p=1.5-2.5$
$ \Rightarrow \frac{2 p+1-9 p}{3}=-1$
$\Rightarrow 2 p+1-9 p=-3 $
$\Rightarrow-7 p=-3-1 $
$ \Rightarrow-7 p=-4 $
$ \Rightarrow p=\frac{4}{7}$

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MATHEMATICS [5 marks sum]