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CAT 2024 Slot 1 — QA Question 1

Square root of SurdsEasy

If (a + b√n) is the positive square root of (29 - 12√5), where a and b are integers, and n is a natural number, then the maximum possible value of (a + b + n) is

Answer & solution

  • A

    4

  • 18

  • C

    6

  • D

    22

Solution

Medium

Turn the nested surd 29−12529-12\sqrt5 into a perfect square (x−y5)2(x-y\sqrt5)^2 to get its square root in surd form. Then notice a+bna+b\sqrt n is not unique — bn=b2nb\sqrt n=\sqrt{b^2n} — so nn can be pushed as large as possible to maximise a+b+na+b+n.

1

Write it as a perfect square and match the rational and surd parts:

29−125=(x−y5)2=(x2+5y2)−2xy5⇒ x2+5y2=29(rational parts)⇒ 2xy=12(surd parts)⇒ xy=6\begin{aligned} &29-12\sqrt5=(x-y\sqrt5)^2=(x^2+5y^2)-2xy\sqrt5\\ &\Rightarrow\ x^2+5y^2=29 \quad\text{(rational parts)}\\ &\Rightarrow\ 2xy=12 \quad\text{(surd parts)}\\ &\Rightarrow\ xy=6 \end{aligned}
2

Solve the two equations from step 1 by inspection:

try x=3, y=2xy=3⋅2=6 ✓(matches xy=6)x2+5y2=9+20=29 ✓⇒ 29−125=(3−25)2\begin{aligned} &\text{try } x=3,\ y=2\\ &xy=3\cdot2=6\ \checkmark \quad\text{(matches } xy=6\text{)}\\ &x^2+5y^2=9+20=29\ \checkmark\\ &\Rightarrow\ 29-12\sqrt5=(3-2\sqrt5)^2 \end{aligned}
3

Take the positive root of the square from step 2:

3−25≈−1.47<0⇒ 29−125=25−3(flip sign for positive root)\begin{aligned} &3-2\sqrt5\approx-1.47\lt 0\\ &\Rightarrow\ \sqrt{29-12\sqrt5}=2\sqrt5-3 \quad\text{(flip sign for positive root)} \end{aligned}
4

Match step 3 to a+bna+b\sqrt n and maximise:

25−3=a+bn⇒ a=−3(rational part)⇒ bn=25, so b2n=20⇒ b=1, n=20(largest n)⇒ a+b+n=−3+1+20=18\begin{aligned} &2\sqrt5-3=a+b\sqrt n\\ &\Rightarrow\ a=-3 \quad\text{(rational part)}\\ &\Rightarrow\ b\sqrt n=2\sqrt5,\ \text{so } b^2n=20\\ &\Rightarrow\ b=1,\ n=20 \quad\text{(largest } n\text{)}\\ &\Rightarrow\ a+b+n=-3+1+20=18 \end{aligned}
a+b+n=18a+b+n=\mathbf{18}

The whole game is choosing the larger nn: write 252\sqrt5 as 1⋅201\cdot\sqrt{20} instead of 252\sqrt5 — that single move turns the answer 4 into 18.

Need a hint?

For the square root of p−qkp-q\sqrt{k}, try to write it as (x−yk)2(x-y\sqrt{k})^2 and match the rational part and the k\sqrt{k} part.

CAT 2024 Slot 1 QA Q1: If (a + b&radic;n) is the positive square root of (29 - 12&radic;5), where a and b are integers, and n is a na — Solution | TheCATExam