SAT/BAC math level vs this equivalent Watch

mimi112
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Report Thread starter 5 years ago
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this in the romanian SAT equivalent for this year (basically the math exam you take at the end of high school, don't know what's it called in the UK). sorry but can't translate, i hope most of it is somewhat obvious to math people. how does it compare to the UK exam in difficulty and level?

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BlueSam3
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Somewhat dodgy translation:

Section 1:
1) Determine the ratio of the GP (b_n) with b_1 = 1, b_4 = 27.
2) Determine the coordinates of the turning point (or maybe focus, I'm not sure) of the parabola associated to x^2-6x+8.
3) Find real solutions to 3^{x+2} = 9^{1-x}.
4) Calculate the probability of choosing a perfect square from the natural two digit numbers.
5) Let A, B, C \in \mathbb{R}^2 such that AB = 4i-3j, BC = 2i-5j (missing overarrows because I'm feeling lazy).
6) Calculate the angle at A in the triangle ABC in which AB = 4, BC = 5, \sin(C) =\frac{4}{5}.

Section 2:
1) For each real m, consider the matrix A(m) = \left( \begin{array}{ccc}1&1&1 \\ m&0&0 \\ m&0&m\end{array} \right).
a) Calculate \mathrm{det}(A(1)).
b) Determine m such that A(m) \cdot A(-m) = \left( \begin{array}{ccc} -1&1&0 \\ 1&1&1 \\ 0&1&0 \end{array} \right).
c) Show that \mathrm{det}(\sum_{m=1}^{101} A(m)) = -51^2 \cdot 101^3.
2) Define the binary operation \circ on the reals by x \circ y = xy - 4x -4y + 20.
a) Calculate 3 \circ 4.
b) Show that x \circ y = (x-4)(y-4)+4 \forall x,y \in \mathbb{R}.
c) Find x such that x \circ x \circ ... \circ x = 5, where there are 2013 x's.

Section 3:
1) Consider f: (0,+\infty) \to \mathbb{R}, f(x) = \frac{e^x}{x+e^x}.
a) Show that f'(x) = \frac{(x-1)e^x}{x+e^x}^2 \forall x \in (0,+\infty).
b) Determine the limit of f as x \rightarrow \infty.
c) Show that f(x) \geq \frac{e}{e+1} \forall x \in (0, +\infty).
2) For each natural n, consider I_n = \int^1_0xe^{-nx^2}dx.
a) Calculate I_0.
b) Show that I_{n+1} \leq I_n, \forall n \in \mathbb{N}.
c) Show that I_n = \frac{1}{2n}\left(1-\frac{1}{e^n}\right) \forall n \in \mathbb{N}.
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