## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 7 Ratio and Proportion Chapter Test

Question 1.

Find the compound ratio of:

(a + b)^{2} : (a – b )^{2}

(a^{2} – b^{2}) : (a^{2} + b^{2})

(a^{4} – b^{4}) : (a + b)^{4}

Solution:

Question 2.

If (7p + 3q) : (3p – 2q) = 43 : 2 find p : q

Solution:

Question 3.

If a : b = 3 : 5, find (3a + 5b) : (7a – 2b).

Solution:

Question 4.

The ratio of the shorter sides of a right-angled triangle is 5 : 12. If the perimeter of the triangle is 360 cm, find the length of the longest side.

Solution:

Question 5.

The ratio of the pocket money saved by Lokesh and his sister is 5 : 6. If the sister saves Rs 30 more, how much more the brother should save in order to keep the ratio of their savings unchanged?

Solution:

Question 6.

In an examination, the number of those who passed and the number of those who failed were in the ratio of 3 : 1. Had 8 more appeared, and 6 less passed, the ratio of passed to failures would have been 2 : 1. Find the number of candidates who appeared.

Solution:

Question 7.

What number must be added to each of the numbers 15, 17, 34 and 38 to make them proportional?

Solution:

Question 8.

If (a + 2b + c), (a – c) and (a – 2b + c) are in continued proportion, prove that b is the mean proportional between a and c.

Solution:

Question 9.

If 2, 6, p, 54 and q are in continued proportion, find the values of p and q.

Solution:

Question 10.

If a, b, c, d, e are in continued proportion, prove that: a : e = a^{4} : b^{4}.

Solution:

Question 11.

Find two numbers whose mean proportional is 16 and the third proportional is 128.

Solution:

Question 12.

If q is the mean proportional between p and r, prove that:

\({ p }^{ 2 }-{ 3q }^{ 2 }+{ r }^{ 2 }={ q }^{ 4 }\left( \frac { 1 }{ { p }^{ 2 } } -\frac { 3 }{ { q }^{ 2 } } +\frac { 1 }{ { r }^{ 2 } } \right) \)

Solution:

Question 13.

If \(\frac { a }{ b } = \frac { c }{ d } = \frac { e }{ f } \), prove that each ratio is

(i) \(\sqrt { \frac { { 3a }^{ 2 }-{ 5c }^{ 2 }+{ 7e }^{ 2 } }{ { 3b }^{ 2 }-{ 5d }^{ 2 }+{ 7f }^{ 2 } } } \)

(ii) \({ \left[ \frac { { 2a }^{ 3 }+{ 5c }^{ 3 }+{ 7e }^{ 3 } }{ { 2b }^{ 3 }+{ 5d }^{ 3 }+{ 7f }^{ 3 } } \right] }^{ \frac { 1 }{ 3 } } \)

Solution:

Question 14.

If \(\frac { x }{ a } = \frac { y }{ b } = \frac { z }{ c } \), prove that

\(\frac { { 3x }^{ 3 }-{ 5y }^{ 3 }+{ 4z }^{ 3 } }{ { 3a }^{ 3 }-{ 5b }^{ 3 }+{ 4c }^{ 3 } } ={ \left( \frac { 3x-5y+4z }{ 3a-5b+4c } \right) }^{ 3 }\)

Solution:

Question 15.

If x : a = y : b, prove that

\(\frac { { x }^{ 4 }+{ a }^{ 4 } }{ { x }^{ 3 }+{ a }^{ 3 } } +\frac { { y }^{ 4 }+{ b }^{ 4 } }{ { y }^{ 3 }+{ b }^{ 3 } } =\frac { { \left( x+y \right) }^{ 4 }+{ \left( a+b \right) }^{ 4 } }{ { \left( x+y \right) }^{ 3 }+{ \left( a+b \right) }^{ 3 } } \)

Solution:

Question 16.

If \(\frac { x }{ b+c-a } =\frac { y }{ c+a-b } =\frac { z }{ a+b-c } \) prove that each ratio’s equal to :

\(\frac { x+y+z }{ a+b+c } \)

Solution:

Question 17.

If a : b = 9 : 10, find the value of

(i) \(\frac { 5a+3b }{ 5a-3b } \)

(ii) \(\frac { { 2a }^{ 2 }-{ 3b }^{ 2 } }{ { 2a }^{ 2 }+{ 3b }^{ 2 } } \)

Solution:

Question 18.

If (3x² + 2y²) : (3x² – 2y²) = 11 : 9, find the value of \(\frac { { 3x }^{ 4 }+{ 25y }^{ 4 } }{ { 3x }^{ 4 }-{ 25y }^{ 4 } } \) ;

Solution:

Question 19.

If \(x=\frac { 2mab }{ a+b } \) , find the value of

\(\frac { x+ma }{ x-ma } +\frac { x+mb }{ x-mb } \)

Solution:

Question 20.

If \(x=\frac { pab }{ a+b } \) ,prove that \(\frac { x+pa }{ x-pa } -\frac { x+pb }{ x-pb } =\frac { 2\left( { a }^{ 2 }-{ b }^{ 2 } \right) }{ ab } \)

Solution:

Question 21.

Find x from the equation \(\frac { a+x+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } }{ a+x-\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } =\frac { b }{ x } \)

Solution:

Question 22.

If \(x=\frac { \sqrt [ 3 ]{ a+1 } +\sqrt [ 3 ]{ a-1 } }{ \sqrt [ 3 ]{ a+1 } -\sqrt [ 3 ]{ a-1 } } \), prove that :

x³ – 3ax² + 3x – a = 0

Solution:

Question 23.

If \(\frac { by+cz }{ b^{ 2 }+{ c }^{ 2 } } =\frac { cz+ax }{ { c }^{ 2 }+{ a }^{ 2 } } =\frac { ax+by }{ { a }^{ 2 }+{ b }^{ 2 } } \), prove that each of these ratio is equal to \(\frac { x }{ a } =\frac { y }{ b } =\frac { z }{ c } \)

Solution: