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Class 10 Math Variations 1. Prove that a: b=c: d if(iv) \frac{a^{2} c+b^{2} d}{a^{2} c-b^{2} d}=\frac{a c^{2}+b d^{2}}{a c^{2}-b d^{2}}


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1. Prove that a: b=c: d if(iv) \frac{a^{2} c+b^{2} d}{a^{2} c-b^{2} d}=\frac{a c^{2}+b d^{2}}{a c^{2}-b d^{2}}

(iv) In a proportion a: b:: c: d b and c are called(a) means(b) extremes(c) fourth proportional(d) None of these
(iv) In a proportion  a: b:: c: d b  and  c  are called(a) means(b) extremes(c) fourth proportional(d) None of these

(iv) In a proportion a: b:: c: d b and c are called(a) means(b) extremes(c) fourth proportional(d) None of these

Example 1: If y varies jointly as x^{2} and z and y=6 when x=4 z=9 . Write y as a function of x and z and determine the value of y when x=-8 and z=12 .
Example 1: If  y  varies jointly as  x^{2}  and  z  and  y=6  when  x=4 z=9 . Write  y  as a function of  x  and  z  and determine the value of  y  when  x=-8  and  z=12 .

Example 1: If y varies jointly as x^{2} and z and y=6 when x=4 z=9 . Write y as a function of x and z and determine the value of y when x=-8 and z=12 .

10. Complete the following:(iii) If \frac{9 p q}{2 l m}=\frac{18 p}{5 m} then 5 q=
10. Complete the following:(iii) If  \frac{9 p q}{2 l m}=\frac{18 p}{5 m}  then  5 q=

10. Complete the following:(iii) If \frac{9 p q}{2 l m}=\frac{18 p}{5 m} then 5 q=

3. If y varies directly as x^{3} and inversely as z^{2} and t and y \doteq 16 when x=4 z=2 t=3 . Find the value of y when x=2 z=3 and t=4 .
3. If  y  varies directly as  x^{3}  and inversely as  z^{2}  and  t  and  y \doteq 16  when  x=4 z=2   t=3 . Find the value of  y  when  x=2 z=3  and  t=4 .

3. If y varies directly as x^{3} and inversely as z^{2} and t and y \doteq 16 when x=4 z=2 t=3 . Find the value of y when x=2 z=3 and t=4 .

4. Find the value of p if the ratios 2 p+5: 3 p+4 and 3: 4 are equal.
4. Find the value of  p  if the ratios  2 p+5: 3 p+4  and  3: 4  are equal.

4. Find the value of p if the ratios 2 p+5: 3 p+4 and 3: 4 are equal.

8. If y \propto \frac{1}{x} and y=4 when x=3 find x when y=24 .
8. If  y \propto \frac{1}{x}  and  y=4  when  x=3  find  x  when  y=24 .

8. If y \propto \frac{1}{x} and y=4 when x=3 find x when y=24 .

4. If u varies directly as x^{2} and inversely as the product y z^{3} and u=2 when x=8 y=7 z=2 . Find the value of u when x=6 y=3 z=2 .
4. If  u  varies directly as  x^{2}  and inversely as the product  y z^{3}  and  u=2  when  x=8 y=7   z=2 . Find the value of  u  when  x=6 y=3 z=2 .

4. If u varies directly as x^{2} and inversely as the product y z^{3} and u=2 when x=8 y=7 z=2 . Find the value of u when x=6 y=3 z=2 .

9. The kinetic energy (K.E.) of a body varies jointly as the mass " m " of the body and the square of its velocity " v ". If the kinetic energy is 4320 \mathrm{ft} / \mathrm{lb} when the mass is 45 \mathrm{lb} and the velocity is 24 \mathrm{ft} / \mathrm{sec} determine the kinetic energy of a 3000 \mathrm{lb} automobile travelling 44 \mathrm{ft} / \mathrm{sec} .
9. The kinetic energy (K.E.) of a body varies jointly as the mass "  m  " of the body and the square of its velocity "  v  ". If the kinetic energy is  4320 \mathrm{ft} / \mathrm{lb}  when the mass is  45 \mathrm{lb}  and the velocity is  24 \mathrm{ft} / \mathrm{sec}  determine the kinetic energy of a  3000 \mathrm{lb}  automobile travelling  44 \mathrm{ft} / \mathrm{sec} .

9. The kinetic energy (K.E.) of a body varies jointly as the mass " m " of the body and the square of its velocity " v ". If the kinetic energy is 4320 \mathrm{ft} / \mathrm{lb} when the mass is 45 \mathrm{lb} and the velocity is 24 \mathrm{ft} / \mathrm{sec} determine the kinetic energy of a 3000 \mathrm{lb} automobile travelling 44 \mathrm{ft} / \mathrm{sec} .

1. Prove that a: b=c: d if(iv) \frac{a^{2} c+b^{2} d}{a^{2} c-b^{2} d}=\frac{a c^{2}+b d^{2}}{a c^{2}-b d^{2}}
1. Prove that  a: b=c: d  if(iv)  \frac{a^{2} c+b^{2} d}{a^{2} c-b^{2} d}=\frac{a c^{2}+b d^{2}}{a c^{2}-b d^{2}}
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1. Prove that a: b=c: d if(iv) \frac{a^{2} c+b^{2} d}{a^{2} c-b^{2} d}=\frac{a c^{2}+b d^{2}}{a c^{2}-b d^{2}}

11. Find x in the following proportions.(v) 8-x: 11-x:: 16-x: 25-x
11. Find  x  in the following proportions.(v)  8-x: 11-x:: 16-x: 25-x

11. Find x in the following proportions.(v) 8-x: 11-x:: 16-x: 25-x

(ii) In a ratio x: y y is called(a) relation(b) antecedent(c) consequent(d) None of these
(ii) In a ratio  x: y y  is called(a) relation(b) antecedent(c) consequent(d) None of these

(ii) In a ratio x: y y is called(a) relation(b) antecedent(c) consequent(d) None of these

3. Fill in the blanks(i) The simplest form of the ratio \frac{(x+y)\left(x^{2}+x y+y^{2}\right)}{x^{3}-y^{3}} is(ii) In a ratio x: y ; x is called(iii) In a ratio a: b ; b is called(iv) In a proportion a: b:: x: y ; a and y are called(v) In a proportion p: q:: m: n ; q and m are called(vi) In proportion 7: 4:: p: 8 p= (vii) If 6: m: 0: 9: 12 then m= (viii) If x and y varies directly then x= (ix) If v varies directly as u^{3} then u^{3}= (x) If w varies inversely as p^{2} then k= (xi) A third proportional of 12 and 4 is(xii) The fourth proportional of 1565 is(xiii) The mean proportional of 4 m^{2} n^{4} and p^{6} is(xiv) The continued proportion of 4 m and 9 is
3. Fill in the blanks(i) The simplest form of the ratio  \frac{(x+y)\left(x^{2}+x y+y^{2}\right)}{x^{3}-y^{3}}  is(ii) In a ratio  x: y ; x  is called(iii) In a ratio  a: b ; b  is called(iv) In a proportion  a: b:: x: y ; a  and  y  are called(v) In a proportion  p: q:: m: n ; q  and  m  are called(vi) In proportion 7:  4:: p: 8 p= (vii) If  6: m: 0: 9: 12  then  m= (viii) If  x  and  y  varies directly then  x= (ix) If  v  varies directly as  u^{3}  then  u^{3}= (x) If  w  varies inversely as  p^{2}  then  k= (xi) A third proportional of 12 and 4  is(xii) The fourth proportional of  1565  is(xiii) The mean proportional of  4 m^{2} n^{4}  and  p^{6}  is(xiv) The continued proportion of  4 m  and 9 is

3. Fill in the blanks(i) The simplest form of the ratio \frac{(x+y)\left(x^{2}+x y+y^{2}\right)}{x^{3}-y^{3}} is(ii) In a ratio x: y ; x is called(iii) In a ratio a: b ; b is called(iv) In a proportion a: b:: x: y ; a and y are called(v) In a proportion p: q:: m: n ; q and m are called(vi) In proportion 7: 4:: p: 8 p= (vii) If 6: m: 0: 9: 12 then m= (viii) If x and y varies directly then x= (ix) If v varies directly as u^{3} then u^{3}= (x) If w varies inversely as p^{2} then k= (xi) A third proportional of 12 and 4 is(xii) The fourth proportional of 1565 is(xiii) The mean proportional of 4 m^{2} n^{4} and p^{6} is(xiv) The continued proportion of 4 m and 9 is

4. Find the values of the letter involved in the following continued proportions.(ii) 8 x 18
4. Find the values of the letter involved in the following continued proportions.(ii)   8 x 18

4. Find the values of the letter involved in the following continued proportions.(ii) 8 x 18

5. If V \propto R^{3} and V=5 when R=3 find R when V=625 .
5. If  V \propto R^{3}  and  V=5  when  R=3  find  R  when  V=625 .

5. If V \propto R^{3} and V=5 when R=3 find R when V=625 .

1. Find a third proportional to(v) (x+y)^{2} x^{2}-x y-2 y^{2}
1. Find a third proportional to(v)  (x+y)^{2} x^{2}-x y-2 y^{2}

1. Find a third proportional to(v) (x+y)^{2} x^{2}-x y-2 y^{2}

2. Find a fourth proportional to(iv) x^{2}-11 x+24(x-3) 5 x^{4}-40 x^{3}
2. Find a fourth proportional to(iv)  x^{2}-11 x+24(x-3) 5 x^{4}-40 x^{3}

2. Find a fourth proportional to(iv) x^{2}-11 x+24(x-3) 5 x^{4}-40 x^{3}

Example 2: Find fourth proportional of a^{3}-b^{3} a+b and a^{2}+a b+b^{2}
Example 2: Find fourth proportional of  a^{3}-b^{3} a+b  and  a^{2}+a b+b^{2}

Example 2: Find fourth proportional of a^{3}-b^{3} a+b and a^{2}+a b+b^{2}

1. Find a third proportional to(i) 612
1. Find a third proportional to(i) 612

1. Find a third proportional to(i) 612

Example 4: Find the ratio 3 a+4 b: 5 a+7 b if a: b=5: 8 .
Example 4: Find the ratio  3 a+4 b: 5 a+7 b  if  a: b=5: 8 .
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Example 4: Find the ratio 3 a+4 b: 5 a+7 b if a: b=5: 8 .

3. Find a mean proportional between(iv) x^{2}-y^{2} \frac{x-y}{x+y}
3. Find a mean proportional between(iv)  x^{2}-y^{2} \frac{x-y}{x+y}
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3. Find a mean proportional between(iv) x^{2}-y^{2} \frac{x-y}{x+y}

1. Express the following as a ratio a: b and as a fraction in its simplest (lowest) form.(ii) 450 \mathrm{~cm} 3 \mathrm{~m}
1. Express the following as a ratio  a: b  and as a fraction in its simplest (lowest) form.(ii)  450 \mathrm{~cm} 3 \mathrm{~m}
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1. Express the following as a ratio a: b and as a fraction in its simplest (lowest) form.(ii) 450 \mathrm{~cm} 3 \mathrm{~m}

2. Find a fourth proportional to(i) 5815
2. Find a fourth proportional to(i)  5815
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2. Find a fourth proportional to(i) 5815

1. If y varies directly as x and y=8 when x=2 find(iii) x when y=28
1. If  y  varies directly as  x  and  y=8  when  x=2  find(iii)  x  when  y=28
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1. If y varies directly as x and y=8 when x=2 find(iii) x when y=28

2. Using theorem of componendo-dividendo(i) Find the value of \frac{x+2 y}{x-2 y}+\frac{x+2 z}{x-2 z} if x=\frac{4 y z}{y+z}
2. Using theorem of componendo-dividendo(i) Find the value of  \frac{x+2 y}{x-2 y}+\frac{x+2 z}{x-2 z}  if  x=\frac{4 y z}{y+z}
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2. Using theorem of componendo-dividendo(i) Find the value of \frac{x+2 y}{x-2 y}+\frac{x+2 z}{x-2 z} if x=\frac{4 y z}{y+z}

Example 3: Find the mean proportional of 9 p^{6} q^{4} and r^{8} .
Example 3: Find the mean proportional of  9 p^{6} q^{4}  and  r^{8} .
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Example 3: Find the mean proportional of 9 p^{6} q^{4} and r^{8} .

2. If \frac{a}{b}=\frac{c}{d}=\frac{e}{f}(a b c d e f \neq 0) then show that(iii) \frac{a c}{b d}+\frac{c e}{d f}+\frac{e a}{f b}=\frac{a^{2}}{b^{2}}+\frac{c^{2}}{d^{2}}+\frac{e^{2}}{f^{2}}
2. If  \frac{a}{b}=\frac{c}{d}=\frac{e}{f}(a b c d e f \neq 0)  then show that(iii)  \frac{a c}{b d}+\frac{c e}{d f}+\frac{e a}{f b}=\frac{a^{2}}{b^{2}}+\frac{c^{2}}{d^{2}}+\frac{e^{2}}{f^{2}}
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2. If \frac{a}{b}=\frac{c}{d}=\frac{e}{f}(a b c d e f \neq 0) then show that(iii) \frac{a c}{b d}+\frac{c e}{d f}+\frac{e a}{f b}=\frac{a^{2}}{b^{2}}+\frac{c^{2}}{d^{2}}+\frac{e^{2}}{f^{2}}

1. Express the following as a ratio a: b and as a fraction in its simplest (lowest) form.(iii) 4 \mathrm{~kg} 2 \mathrm{~kg} 750 \mathrm{gm}
1. Express the following as a ratio  a: b  and as a fraction in its simplest (lowest) form.(iii)  4 \mathrm{~kg} 2 \mathrm{~kg} 750 \mathrm{gm}
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1. Express the following as a ratio a: b and as a fraction in its simplest (lowest) form.(iii) 4 \mathrm{~kg} 2 \mathrm{~kg} 750 \mathrm{gm}

2. Write short answers of the following questions.(vi) Find x if 6: x:: 3: 5 .
2. Write short answers of the following questions.(vi) Find  x  if  6: x:: 3: 5 .
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2. Write short answers of the following questions.(vi) Find x if 6: x:: 3: 5 .

2. Write short answers of the following questions.(xii) If y \propto \frac{x^{2}}{z} and y=28 when x=7 z=2 then find y .
2. Write short answers of the following questions.(xii) If  y \propto \frac{x^{2}}{z}  and  y=28  when  x=7 z=2  then find  y .
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2. Write short answers of the following questions.(xii) If y \propto \frac{x^{2}}{z} and y=28 when x=7 z=2 then find y .

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