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Cím:	Practice problems for the entrance exam
Szerző(k):	Imre Rábai
Füzet:	2003/októberi melléklet, 24. oldal	PDF \| MathML
Témakör(ök):	Felvételi előkészítő feladatsor

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In memory of György Geréb, psychologist and college professor

1. A trapezium has both an inscribed circle and a circumscribed circle. One of the parallel sides is 10 units long and the radius of the incircle is

ϱ = 5 \sqrt[]{3}

units. What percentage of the area of the trapezium is the area of the incircle?

2. Two cities are

560 km

apart. A car takes 1 hour less to cover this distance than another car because its speed is

10 \frac{km}{h}

greater than that of the other one. Find the speeds of the two cars.

3. The points

P (b, - 2)

and

Q (16 b,6)

lie on the graph of the logarithm function

f : x \mapsto log_{a} x

x \in R^{+}

(

a > 0

a \neq 1

). Determine the values of

a

and

b

and the equation of the line

P Q

4. Find the domain and range of the expression

\begin{matrix} \frac{sin 2 x + 2 sin x}{sin 2 x - 2 sin x} \cdot tan^{2} \frac{x}{2} . \end{matrix}

5. Find the equation of the circle of radius

\sqrt[]{8}

passing through the point

P (0,2)

and touched from the outside by the circle of equation

\begin{matrix} x^{2} + y^{2} + 2 x + 6 y + 8 = 0. \end{matrix}

6. Solve the following inequality on the set of real numbers:

\begin{matrix} 2 \sqrt[]{log_{3} x} + \sqrt[]{log_{x} 3} \geq 3. \end{matrix}

7. A convex hexagon with three sides of length

2 a

and three sides of length

5 a

is inscribed in a circle of radius

\sqrt[]{13}

. Find the area of the hexagon.

8. The first term of a sequence is

a_{1} = 1

, and

\begin{matrix} a_{n + 1} = (1 - \frac{1}{{(n + 2)}^{2}}) \cdot a_{n} \end{matrix}

for

n \geq 1

. Express (in closed form) the

n

th term of the sequence and the product of the first

n

terms.