Oxford University MAT 2021 Solved Papers

© examsiri.com

Question : 2 of 10

Marks: +1, -0

The positive number

a

satisfies

∫↙0^a(√{x}+x^2) {d} x=5

if

$a=(√{21}-1)^{1 \/ 3}, \;\;$
$a=√{3}, \;\;$
$a=3^{2 \/ 3}$ ,
$a=(√{6}-1)^{2 \/ 3}$ ,
$a=5^{2 \/ 3}$

Solution:

The integral is

∫↙0^a √{x}+x^2 {d} x=∫↙0^a x^{1 \/ 2}+x^2 {d} x=[\;{2}/{3} x^{3 \/ 2}+\;{1}/{3} x^3]_0^a=\;{2}/{3} a^{3 \/ 2}+\;{1}/{3} a^3 .

So we have

2 a^{3 \/ 2}+a^3=15

. This factorises as

(a^{3 \/ 2}-3)(a^{3 \/ 2}+5)=0

. Since

a>0

we want

a^{3 \/ 2}>0

, so it's 3 , so

a=3^{2 \/ 3}

.
The answer is (c)

© examsiri.com

Go to Question:

Prev Question

Next Question