NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

$a, b, \text{ and } c$ are positive integers such that $a^2 + b^2 - 2bc = 100$ and $2ab - c^2 = 100$ . Then, the value of $\frac{a+b}{c}$ is:

Choose the correct answer:

Explanation

1. Dono equations ko equate karo:

Chunki dono $100$ ke barabar hain:

a^2 + b^2 - 2bc = 2ab - c^2

2. Saari terms ek side le aao:

a^2 + b^2 - 2ab + c^2 - 2bc = 0

3. Formula mein set karo:

$(a^2 - 2ab + b^2)$ ko $(a-b)^2$ likh sakte hain:

(a - b)^2 + c^2 - 2bc = 0

Isko balance karne ke liye dono taraf $b^2$ add karo:

(a - b)^2 + (c^2 - 2bc + b^2) = b^2

(a - b)^2 + (c - b)^2 = b^2

4. Pythagorean Triplet check karo:

Hume pata hai $3^2 + 4^2 = 5^2$ .

Maano $b = 5$ :

$(a - b) = 3 \implies a - 5 = 3 \implies \mathbf{a = 8}$
$(c - b) = 4 \implies c - 5 = 4 \implies \mathbf{c = 9}$

Ab ye values original equation mein check karo:

$2ab - c^2 = 2(8)(5) - 9^2 = 80 - 81 = -1$ (Hume $100$ chahiye, so ye triplet fit nahi hai).

5. Shortcut Method (Adding Equations):

Dono equations ko add kar do:

(a^2 + 2ab + b^2) - (c^2 + 2bc) = 100 + 100

(a+b)^2 - (c^2 + 2bc) = 200

Ab options se value try karo. Agar $\frac{a+b}{c} = 2$ (Option c) rakhein:

Toh $a+b = 2c$ . Isse equation mein put karo:

(2c)^2 - (c^2 + 2bc) = 200

4c^2 - c^2 - 2bc = 200

3c^2 - 2bc = 200

Agar $c = 10$ aur $b = 5$ lein:

3(100) - 2(5)(10) = 300 - 100 = 200

Ye perfectly match kar raha hai!

Conclusion:

Jab $c=10, b=5$ aur $a=15$ , toh $\frac{a+b}{c} = \frac{15+5}{10} = \mathbf{2}$ .

Correct Option: (c) 2