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1. Prove that if ab > 0, then

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2 LO Batory September 5, 2017

YEAR 1 REVISION

1. Prove that if ab > 0, then

ab

+

ab

­ 2.

2. Find all natural numbers n, for which

2n+5n+1

is also a natural number.

3. Prove that 2

1

+ 2

2

+ 2

3

+ ... + 2

88

is divisible by 3 and 5.

4. Prove that the fraction

15k+26k+1

is irreducible for all integer values k.

5. Find all values of x for which the sum |x − 1| + |x + 2| has the least value.

6. Solve the inequality

q

(2 − x)

2

> 3.

7. Find the value of k, for which the equation:

|x − 6| + |x + 1| = k

has two solutions. Find these solutions for k = 10.

8. Calculate log

7

16, given that log

14

2 = a.

9. Calculate log

ab

(

ab

), given that log

a

b = 2.

10. Show that log

3

12 × (2 − log

12

48) is an integer.

11. Prove that 3

log 2

− 2

log 3

= 0.

12. Sketch the following functions:

a) f (x) = max(4, |x + 1|);

b) g(x) = min(1, x

2

);

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