GATE 2018 EE GA Q.4: solution and answer

GATE 2018 · EEGA Q.4 · 1 mark · MCQ

For what values of $k$ given below is $\frac{( k + 2 ) ^{2}}{k - 3}$ an integer?

(A)
4, 8, 18
(B)
4, 10, 16
(C)
4, 8, 28
(D)
8, 26, 28

See the question in the paper

Answer and solution

ANSWER

(C) 4, 8, 28. Since $\frac{( k + 2 ) ^{2}}{k - 3} = (k - 3) + 10 + \frac{25}{k - 3}$ , the expression is an integer only when $k - 3$ divides 25, which gives $k - 3 = 1, 5, 25$ for these three values.

Official answer: CSource: Official answer key, GATE 2018 EE (IIT Guwahati)

Write the numerator in terms of the denominator: $k + 2 = (k - 3) + 5$ .
Expand: $(k + 2)^{2} = ((k - 3) + 5)^{2} = (k - 3)^{2} + 10 (k - 3) + 25$ .
Divide term by term: $\frac{( k + 2 ) ^{2}}{k - 3} = (k - 3) + 10 + \frac{25}{k - 3}$
Every option lists integer values of $k$ , so $(k - 3) + 10$ is an integer. The whole expression is therefore an integer exactly when $\frac{25}{k - 3}$ is an integer, that is, when $(k - 3)$ divides $25$ .
The divisors of 25 are $\pm 1, \pm 5, \pm 25$ . So $k - 3 \in {1, - 1, 5, - 5, 25, - 25}$ , which gives $k \in {4, 2, 8, - 2, 28, - 22}$ .
Check the options. The values 18, 10, 16 and 26 give $k - 3 = 15, 7, 13, 23$ , and none of these divides 25, so options (A), (B) and (D) each contain a value that fails. Only option (C), $4, 8, 28$ , has all three values in the set.
Verify directly: $k = 4$ gives $\frac{36}{1} = 36$ ; $k = 8$ gives $\frac{100}{5} = 20$ ; $k = 28$ gives $\frac{900}{25} = 36$ . All three are integers, so the answer is (C).

KEY IDEA

Write the numerator in powers of the denominator (put $k + 2 = (k - 3) + 5$ ). This leaves a polynomial part plus a constant remainder over $(k - 3)$ , so the fraction is an integer exactly when $(k - 3)$ divides that remainder, here 25.

WATCH OUT FOR

Checking only one or two values in an option. Option (A) passes for 4 and 8, but $k = 18$ gives $\frac{400}{15}$ , which is not an integer. Every value in the option has to work. A second slip is to expand $(k + 2)^{2}$ and divide by $k$ instead of by $(k - 3)$ , which gives a wrong remainder.

From the GATE 2018 EE paper. Every question of this paper · Download the full paper · All 58 Numerical questions, with notes

GATE 2018 EE GA Q.4: solution and answer

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