Question 1: Frame a formula for each of the following statements:

(i) The area $(A)$  of a rectangle is equal to the product of its length $(l)$ and breadth $(b)$

$A = l \times b$

(ii) The area $(A)$  of a triangle is half the product of its base $(b)$ and height $(h)$

$A =$ $\frac{1}{2}$ $\times b \times h$

(iii) The volume $(V)$ of the cone is one third the product of $\pi$, square of the radius $(r)$ and height $(h)$

$V =$ $\frac{1}{3}$ $\pi r^2 h$

(iv) The perimeter $(P)$ of the rectangle is twice the sum of its length $(l)$ and breadth $(b)$

$P = 2(l+b)$

(v)  The perimeter $(P)$ of a square is four times it size $(s)$

$P = 4s$

(vi) The distance $(s)$ through which the body falls freely under gravity is $4.81$ times the square of the time $(t)$

$S = 4.81 t^2$

(vii) The reciprocal of focal length $(f)$ is equal to the sum of reciprocals of the object distance $(u)$ and the image distance $(v)$

$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

(viii) $2$ years ago, a man whose present age is $x$ years was three times as old as his son, whose present age is $y$ years

$(x-2) = 3(y -2)$

(ix) Two digit number having $x$ as ten’s digit and $y$ as units’s digit is $5$ times the sum of the digits

$10x + y = 5(x+y)$

(x) The number of diagonals $(d)$ that can be drawn from one vertex of an $n$-side polygon to all other vertices is $3$ less than $n$

$d = (n-3)$

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Question 2: A workman is paid $Rs. \ x$ for each day he works and fined $Rs. \ y$ for each day he is absent. If he works for $N$ days in a month of $30$ days, find the expression of his total earnings $(E)$ in rupees.

$E = N \times x -(30-N) \times y$

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Question 3: A purse contains $x$ notes of $Rs. 10$ each, $y$ notes of $Rs. 5$ each, $z$ coins of $50$ paisa each and $t$ coins of $5$ paisa each. Find the total money $M$ in $Rs$.

$M = 10x + 5y +$$\frac{1}{2}$ $z +$ $\frac{1}{20}$ $t$

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Question 4: A shopkeeper buys $m$ kg. of rice at $Rs. \ x$ per kg and another $n$ kg of rice at $Rs. \ y$ per kg. He mixes the two quantities and sells the mixture at $Rs. \ z$ per kg. Find the expression of his (i) total profit and (ii) profit per cent.

Total cost $= (mx + ny)$

Total Sale $= (m+n)z$

(i) Total profit $= (m+n)z - (mx + ny)$

(ii) Profit percentage $=$ $\{ \frac{(m+n)z - (mx + ny)}{mx + ny}$ $\times 100$ $\}$

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Question 5: A man cycles for $p$ hours at $x$ km per hour and for another $q$ hours at $y$ km per hour. Find his average speed $(A)$ for the whole journey.

$A =$ $\frac{px+qy}{p+q}$

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Question 6: The average age of $x$ boys in a class is $y$ years. A new boy of age $z$ years joins the class. Find the present average age $(A)$.

$A =$ $\frac{xy+z}{x+1}$

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Question 7: A cricketer has an average score of $85$ runs per innings in $x$ innings and an average of $63$ runs per innings in $y$ innings. Find the average score $(A)$ per innings.

$A =$ $\frac{85x+63y}{x+y}$

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Question 8: In a class of $x$ children, each one of y children pays $Rs. \ 10$ and each of the remaining pays $Rs.\ 6$ for a charity show. Find the total amount $(C)$ in rupees.

$C = 10y +( x-y)\times 6 = 4y+6x$

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Question 9: A shopkeeper marks each article at $Rs.\ m$ and gives $20\%$ discount on the marked price. If the cost price of an article $Rs. \ C$, find the formula for profit $(P)$. Find P when $m = 300$ and $C = 200$.

$P = 0.8m - C$

$P = 0.8 \times 300 - 200 = 40$

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Question 10: The hiring charges $(h)$ of a taxi were $Rs. \ 100$ plus $Rs.\ 8$ per km for distances traveled beyond $25$ km. If the distance traveled is $x$ km, write a formula for the hiring charges.

$h = 100 + (x - 25) \times 8$

If $h = 140 \Rightarrow x = 25 +$ $\frac{140 - 100}{8}$ $= 30$

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Question 11: Make $b$ as a subject in the formula, $P = 2 (l + b)$. Find $b$, when $P = 66$ and $l = 18$.

$P = 2 (l + b)$

$\Rightarrow b = ($ $\frac{P}{2}$ $- l)$

$b = ($ $\frac{66}{2}$ $- 18) = 15$

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Question 12: Given: $s = ut -$ $\frac{1}{2}$ $gt^2$

(i) Make $g$, the subject of the formula

(ii) Find $g$, when $t = 20, s = 10$ and $u = 50$.

(i) $s = ut -$ $\frac{1}{2}$ $gt^2$

$\Rightarrow$ $\frac{1}{2}$ $gt^2 = ut - S$

$\Rightarrow g =$ $\frac{2}{t^2}$ $(ut-S)$

(ii) when $t = 20, s = 10$ and $u = 50$

$g =$ $\frac{2}{10^2}$ $(50 \times 10 -10) = 9.8$

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Question 13: Given: $T = 2 \pi$ $\sqrt{\frac{l}{g}}$

(i) Make $l$ as the subject of the formula

(ii) Find $l$, when $T = 2, g = 9.8$ and $\pi = \sqrt{10}$

(i) $T = 2 \pi$ $\sqrt{\frac{l}{g}}$

$\Rightarrow T^2 = 4 \pi^2$ $\frac{l}{g}$

$l =$ $\frac{gT^2}{4 \pi^2}$

(ii) When $T = 2, g = 9.8$ and $\pi = \sqrt{10}$

$l =$ $\frac{9.8 \times 2^2}{4 (\sqrt{10})^2}$ $= 0.98$

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Question 14: Let $S =$ $\frac{n}{2}$ $\{2a + (n-1) d \}$

(i) Find $a$ when $S = 185, n = 10$ and $d = 3$

(ii) Find $d$ when $a = 11, n = 10$ and $S = 380$

(i)   $S =$ $\frac{n}{2}$ $\{2a + (n-1) d \}$

$\Rightarrow$ $\frac{2S}{n}$ $= 2a + (n-1)d$

$\Rightarrow a =$ $\frac{1}{2} \{ \frac{2S}{n}$ $- (n-1)d$ $\}$

$\Rightarrow a =$ $\frac{1}{2} \{ \frac{2 \times 185}{10}$ $- (10-1) \times 3$ $\}$ $= 5$

(ii)  $S =$ $\frac{n}{2}$ $\{2a + (n-1) d \}$

$\Rightarrow$ $\frac{2S}{n}$ $= 2a + (n-1)d$

$d =$ $\frac{1}{n-1} \{ \frac{2S}{n}$ $- 2a$ $\}$

$d =$ $\frac{1}{10-1} \{ \frac{2 \times 380}{10}$ $- 2 \times 11$ $\}$ $= 6$

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Question 15: Let $v = \sqrt{u^2+2ax}$

(i) Make $x$, the subject of the formula

(ii) Find $x$, when $v = 35, u = 25$ and $a = 50$

(i)  $v = \sqrt{u^2+2ax}$

$v^2 = u^2 + 2ax$

$\Rightarrow x =$ $\frac{1}{2a}$ $(v^2 - u^2)$

(ii)  $x =$ $\frac{1}{2 \times 50}$ $(35^2 - 25^2) = 6$

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Question 16: The volume $(V)$ of a hollow cylindrical pipe with outer radius $(R)$, inner radius $(r)$ and length $(h)$ is given by the formula, $V = \pi (R^2 - r^2)h$.

(i) Make $h$ the subject of the formula

(ii) Find $h$, when $R = 2.6, r = 2.3, \pi =$ $\frac{22}{7}$ and $V = 115.5$

(i)   $V = \pi (R^2 - r^2)h$

$h =$ $\frac{V}{\pi (R^2 - r^2)}$

(ii)  $h =$ $\frac{115.5 \times 7}{22 \times (2.6^2 - 2.3^2)}$ $= 25$

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Question 17:  Let $x =$ $\frac{m+n}{m-n}$,

(i) Make $n$ the subject of the formula

(ii) Find $n$, when $m = 36$ and $x = 2$

(i)   $x =$ $\frac{m+n}{m-n}$

$mx-nx = m+ n$

$\Rightarrow m(x-1) = (x+1) n$

$\Rightarrow n =$ $(\frac{x-1}{x+1})$ $m$

(ii) $\Rightarrow n =$ $(\frac{2-1}{2+1})$ $\times 36 = 12$

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Question 18: Let $x =$ $\frac{3-4p}{p+2q}$,

(i) Make $p$ the subject of the formula

(ii) Find $p$, when $q =$ $\frac{1}{2}$ and $x =$ $\frac{1}{5}$

(i)   $x =$ $\frac{3-4p}{p+2q}$

$px + 2qx = 3 - 4p$

$px + 4p = 3 - 2qx$

$p =$ $\frac{3 -2qx}{x+4}$

(ii)  $p =$ $\frac{3 -2 \times \frac{1}{2} \times \frac{1}{5}}{\frac{1}{5}+4}$ $=$ $\frac{2}{3}$

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Question 19: Let $R =$ $\sqrt{\frac{3V}{\pi h}}$,

(i) Make $h$ the subject formula

(ii) Find $h$, when $V = 13.5, R = 2.5$ and $\pi=\frac{22}{7}$

(i)   $R =$ $\sqrt{\frac{3V}{\pi h}}$

$\pi R^2 h = 3V$

$\Rightarrow h =$ $\frac{3V}{\pi R^2}$

(ii)  $\Rightarrow h =$ $\frac{3 \times 13.5}{\pi 2.5^2}$ $= 2.06$

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Question 20: If $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$, find $u$ in terms of $v$ and $f$. Find $u$ when $v = 32$ and $f = 24$.

$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

$\Rightarrow \frac{1}{u} = \frac{1}{f} - \frac{1}{v}$

$\Rightarrow \frac{1}{u} = \frac{v-f}{fv}$

$\Rightarrow u =$ $\frac{fv}{v-f}$

$\Rightarrow u =$ $\frac{24 \times 32}{32-24}$ $=96$

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Question 21: If $r = \sqrt{x^2 +y^2}$, express $y$ in terms of $r$ and $x$. Find $y$, when $r = 17$ and $x = 8$.

$r = \sqrt{x^2 +y^2}$

$y^2 = r^2 - x^2$

$y = \sqrt{r^2 - x^2}$

$y = \sqrt{17^2 - 8^2} = 15$

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Question 22: Make $b$ the subject of $x =$ $\sqrt{\frac{a-b}{a+b}}$

$x =$ $\sqrt{\frac{a-b}{a+b}}$
$x^2 =$ $\frac{a-b}{a+b}$
$ax^2+bx^2 = a- b$
$b(x^2 + 1) = a (1 - x^2)$
$\Rightarrow b =$ $\frac{a(1-x^2)}{1+ x^2}$