MSc Biophotonics
Mathematical Tools In Photonics and Biology
Self Test
1.- Express the following fractions in their simplest form:
4
a)
52
b)
15
90
15 8
+
90 52
c)
5
+1
4 2
−
3
6
2
d)
2.- Re express the following expressions:
a)
a b
+
b a
a 1/ b
−
b 1/ a
b)
c)
x+
1
x
1 250 x
2x − +
−
2 500 y
d)
3.- Simplify each of the following and write the answers with only positive exponents:
−4
5 3
a) ( 4 x y )
 24a 3b −8 


−5
 6a b 
b)
−2
c) e x e −2 x
x
d) ln  + 2 ln( y )
 y
4.- Evaluate each of the following expressions:
a) 8
2
3
b)
625
3
4
c)
 4
− 
 5
−3
 32 
 
 4 
d)
−1 / 2
5.- Solve the following quadratic equations by factorizing them or using the quadratic formula:
a)
x 2 − 10 x − 24 = 0
b) x 2 − 10 x + 24 = 0
c) x 2 + 4 x − 4 = 0
d) ax 2 + ax + a = 0
6.- Expand the following factorized polynomials:
a) ( a + b)( a − b) b) ( a + b) 2
c) ( a − b)( a 2 + ab + b 2 )
d) ( a + b + c )( p − q )
7.- Sketch the graph of the following functions (without the help of computer tools):
a)
y = 1/ x
b) y = − ln(x )
c) y = e x
f)
y= x
g) y = sin( x)
h) y = cos( x − π 4)
e) y = 1 + x 2
d) y = 1 − e − x
(NOTE: for incises h) and i) the argument is expressed in radians where
π
i) y = e − x
2
o
rad = 180 )
8.- Using an equilateral triangle, with side lengths a = b = c = 2 , and the Pythagorean Theorem find the value of
the following trigonometric functions (express the result in both fractions and decimal):
a) sin(π / 6)
b) cos(π / 6)
c) tan(π / 6)
d) sin(π / 3)
e) cos(π / 3)
f) tan( π / 3 )
9.-Using a right angle triangle, with hypotenuse c = 2 , and the Pythagorean Theorem find the value of the
following trigonometric functions (express the result in both fractions and decimal):
a) sin(π / 4)
b) cos(π / 4)
c) tan(π / 4)
10. Find the set of values of x for which each of the following inequalities holds:
b) x 2 − 3 x − 2 < 10 − 2 x
a) 3( 2 − x ) ≥ 4 − x
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