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Given ratio of variables, evaluate an expression in them

This type of question, asked 6 times: CDS 1 2017, CDS 2 2019, CDS 1 2023, CDS 2 2023, CDS 1 2024, CDS 1 2025. All 7 questions below, oldest first.

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  1. Question 1

    CDS 1 2017

    12
    If A:B=3:4A : B = 3 : 4, then what is the value of the expression (3A2+4B3A−4B2)\left( \dfrac{3A^2 + 4B}{3A - 4B^2} \right) ?

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  2. Question 2

    CDS 2 2019

    27
    If a:b=c:d=1:6a : b = c : d = 1 : 6, then what is the value of a2+c2b2+d2\frac{a^2 + c^2}{b^2 + d^2} ?

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  3. Question 3

    CDS 1 2023

    11
    If a, b, c, d, e and f satisfy 2a=3b=6c=9d=12e=18f2a = 3b = 6c = 9d = 12e = 18f, then what is the value of (a+b)/(c+d+e+f)(a+b)/(c+d+e+f)?

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  4. Question 4

    CDS 2 2023

    8
    If 2a3=4b5=3c4\dfrac{2a}{3} = \dfrac{4b}{5} = \dfrac{3c}{4}, then what is the value of 18aa2+c2−b2\dfrac{18}{a}\sqrt{a^2 + c^2 - b^2} ?

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  5. Question 5

    CDS 1 2024

    3
    If m:n=1:2m : n = 1 : 2 and p:q=3:4p : q = 3 : 4, then what is (2m+4p):(n+3q)(2m + 4p) : (n + 3q) equal to ?

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  6. Question 6

    CDS 1 2024

    1
    If a:b:c:d=4:3:2:1a : b : c : d = \sqrt{4} : \sqrt{3} : \sqrt{2} : \sqrt{1}, then what is the value of (−a2+b2+c2+d2)(a2−b2+c2−d2)\frac{(-a^2 + b^2 + c^2 + d^2)}{(a^2 - b^2 + c^2 - d^2)} ?

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  7. Question 7

    CDS 1 2025

    45
    The ratio of sum of two numbers to their difference is 5:15 : 1. What is the ratio of the sum of their squares to the difference of their squares ?

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