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

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

$\displaystyle A = l \times b$

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

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

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

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

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

$\displaystyle P = 2(l+b)$

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

$\displaystyle P = 4s$

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

$\displaystyle S = 4.81 t^2$

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

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

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

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

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

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

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

$\displaystyle d = (n-3)$

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

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

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

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

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

Total cost $\displaystyle = (mx + ny)$

Total Sale $\displaystyle = (m+n)z$

$\displaystyle \text{(i) Total profit } = (m+n)z - (mx + ny)$

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

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

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

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

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

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

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

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

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

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

$\displaystyle P = 0.8m - C$

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

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

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

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

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

$\displaystyle P = 2 (l + b)$

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

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

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

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

(ii) Find $\displaystyle g$, when $\displaystyle t = 20, s = 10 \text{ and } u = 50$.

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

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

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

$\displaystyle \text{(ii) } t = 20, s = 10 \text{ and } u = 50$

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

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

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

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

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

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

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

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

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

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

(i) Find $\displaystyle a$ when $\displaystyle S = 185, n = 10 \text{ and } d = 3$

(ii) Find $\displaystyle d$ when $\displaystyle a = 11, n = 10 \text{ and } S = 380$

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

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

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

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

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

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

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

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

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

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

(ii) Find $\displaystyle x$, when $\displaystyle v = 35, u = 25 \text{ and } a = 50$

$\displaystyle \text{(i) } v = \sqrt{u^2+2ax}$

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

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

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

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

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

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

$\displaystyle \text{(i) } V = \pi (R^2 - r^2)h$

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

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

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

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

(ii) Find $\displaystyle n$, when $\displaystyle m = 36 \text{ and } x = 2$

$\displaystyle \text{(i) } x = \frac{m+n}{m-n}$

$\displaystyle mx-nx = m+ n$

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

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

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

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

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

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

$\displaystyle \text{(i) } x = \frac{3-4p}{p+2q}$

$\displaystyle px + 2qx = 3 - 4p$

$\displaystyle px + 4p = 3 - 2qx$

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

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

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

(i) Make $\displaystyle h$ the subject formula

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

$\displaystyle \text{(i) } R = \sqrt{\frac{3V}{\pi h}}$

$\displaystyle \pi R^2 h = 3V$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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