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

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=53 units. What percentage of the area of the trapezium is the area of the incircle?
 

2. Two cities are 560km apart. A car takes 1 hour less to cover this distance than another car because its speed is 10kmh greater than that of the other one. Find the speeds of the two cars.
 

3. The points P(b,-2) and Q(16b,6) lie on the graph of the logarithm function f:xlogax, xR+ (a>0, a1). Determine the values of a and b and the equation of the line PQ.
 

4. Find the domain and range of the expression
sin2x+2sinxsin2x-2sinxtan2x2.
 

5. Find the equation of the circle of radius 8 passing through the point P(0,2) and touched from the outside by the circle of equation
x2+y2+2x+6y+8=0.
 

6. Solve the following inequality on the set of real numbers:
2log3x+logx33.
 

7. A convex hexagon with three sides of length 2a and three sides of length 5a is inscribed in a circle of radius 13. Find the area of the hexagon.
 

8. The first term of a sequence is a1=1, and
an+1=(1-1(n+2)2)an
for n1. Express (in closed form) the nth term of the sequence and the product of the first n terms.