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.

Answer:

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 .

Answer:

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.

Answer:

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.

Answer:

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) .

Answer:

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.

Answer:

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.

Answer:

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 .

Answer:

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.

Answer:

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 .

Answer:

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 .

Answer:

(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}

Answer:

(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

Answer:

(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

Answer:

(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

Answer:

(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

Answer:

(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}

Answer:

(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}

Answer:

(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 .

Answer:

\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 .

Answer:

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}}

Answer:

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}

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