If x,y,z are in continued proportion; then prove that (x+y)²/(y+z)² = x/z. [icse 2010]

 

If (7m + 2n)/(7m - 2n) = 5/3 ; use the properties of proportion to find:(i) m:n. (ⅱ) (m ^ 2 + n ^ 2)/(m ^ 2 - n ^ 2). [icse 2017]

(i) m:n 

Given that ,

(7m + 2n)/(7m - 2n) = 5/3

Applying componendo and dividendo, we get, ((7m + 2n) + (7m - 2n))/((7m + 2n) - (7m - 2n)) = (5 + 3)/(5 - 3)

[By using (Num. + Deno.)/(Num. - Deno.) on both sides]

=) (7m + 2n + 7m - 2n)/(7m + 2n - 7m + 2n) = 8/2 

=> 14m/4n = 8/2 => (7m)/(2n) = 4/1

Multiplying both sides by 2/7 , we get,

 2/7 * 7/2 * m/n = 2/7 * 4/1 

=> m/n = 8/7 => m:n = 8:7

(ii) Given that (7m + 2n)/(7m - 2n) = 5/3

Applying componendo and dividendo, we get, 

 ((7m + 2n) + (7m - 2n))/((7m + 2n) - (7m - 2n)) = (5 + 3)/(5 - 3)

=> (7m + 2n + 7m - 2n)/(7m + 2n - 7m + 2n) = 8/2

=> 14m/4n = 4 => 7/2(m/n) = 4/1 Multiplyning both sides by 2/7 , 

we get. m/n = 8/7

Squaring sides, we get (m²)/(n²) = 64/49

Applying componendo and devedando, we get,

m²+n²/m²-n² = 64+49/ 64-49

=> (m²+n²)/(m²-n²) = 113/15 Ans.

If (x-2) is a -factor of the expression 2x ^ 3 + a * x ^ 2 + bx - 14 and when the expression in divided by (x - 3) , it leaves 52, find a the values of a and b. [icse 2013]

What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional ? [ICSE 2013]

 

Find the equation of a line passing through the points (0,-3) and (5,0) by the method of determinant.

 

Solve the following equation for x correct to 2 decimal places, 4x²-5x-3=0 [ICSE 2017].

 

Find the values of p and q for which the following system of equations has infinite number of solutions: 2x + 3y = 7 , (p + q) * x + (2p - q) * y = 21 [cbse 10th 2001]

 

Solve x/a + y/b = 2 ax - by = a ^ 2 - b ^ 2 [CBSE.10th 2005 ]

 

Find the value of x if 0.75:x :: 5:8.