grandes-ecoles 2024 QI

grandes-ecoles · France · geipi-polytech__maths Indices and Surds Simplifying Surd Expressions

Exercise I

I-A- $\quad \frac { ( 2 \sqrt { 3 } ) ^ { 2 } \times 12 ^ { 3 } \times 3 ^ { 2 } } { 3 ^ { - 4 } \times ( \sqrt { 2 } ) ^ { 4 } } = 3 ^ { 10 } \times 2 ^ { 8 }$. I-B- $\quad 2 \sqrt { 27 } - ( 2 \sqrt { 3 } - 1 ) ^ { 2 } = 10 \sqrt { 3 } - 13$. I-C- $\quad \ln \left( \frac { e } { 4 } \right) + \ln \left( \frac { 1 } { 9 e } \right) + \ln ( 36 e ) = 1$. I-D- $\quad e ^ { 2 \ln 3 + \ln 5 } + e ^ { - 2 \ln 5 } = 20$. I-E- For every real number $x$ different from $-2$ and from $2$, $\frac { 2 } { x + 2 } - \frac { 1 } { x - 2 } + \frac { 8 } { x ^ { 2 } - 4 } = \frac { 1 } { x - 2 }$. I-F- For every real number $x$, $\frac { e ^ { 2 x } + 2 e ^ { x } + 1 } { e ^ { x } + 1 } = e ^ { x } + 1$.
For each statement, indicate whether it is TRUE or FALSE.

\section*{Exercise I}
I-A- $\quad \frac { ( 2 \sqrt { 3 } ) ^ { 2 } \times 12 ^ { 3 } \times 3 ^ { 2 } } { 3 ^ { - 4 } \times ( \sqrt { 2 } ) ^ { 4 } } = 3 ^ { 10 } \times 2 ^ { 8 }$.\\
I-B- $\quad 2 \sqrt { 27 } - ( 2 \sqrt { 3 } - 1 ) ^ { 2 } = 10 \sqrt { 3 } - 13$.\\
I-C- $\quad \ln \left( \frac { e } { 4 } \right) + \ln \left( \frac { 1 } { 9 e } \right) + \ln ( 36 e ) = 1$.\\
I-D- $\quad e ^ { 2 \ln 3 + \ln 5 } + e ^ { - 2 \ln 5 } = 20$.\\
I-E- For every real number $x$ different from $-2$ and from $2$, $\frac { 2 } { x + 2 } - \frac { 1 } { x - 2 } + \frac { 8 } { x ^ { 2 } - 4 } = \frac { 1 } { x - 2 }$.\\
I-F- For every real number $x$, $\frac { e ^ { 2 x } + 2 e ^ { x } + 1 } { e ^ { x } + 1 } = e ^ { x } + 1$.

For each statement, indicate whether it is TRUE or FALSE.

Paper Questions

QI QII QIII QIV QSpec-I QSpec-II QV QVI QVII