Exercise 1:
Either
a. Show that
first of all
Let be the function , this function is strictly increasing on
.
so
thus
and therefore
b. Calculate .
c. Deduce the value of There are two possible values X=2 or X=-2
or
from 1 X<0 , we deduce that X = – 2 .
Exercise 2:
Calculate:
Exercise 3:
Exercise 4:
1. Develop and reduce:
a= (5x+1)(2x+3)
b= (4x-5)(7x-1)
c= (2x+5)(7x-3)
d= (-4x-6)(2x-1)
2. Expand and reduce:
e= (5x+1)(2x+3)+(5x+1)(x+2)
f= (4x-5)(7x-1)-(4x-5)(3x+2)
g= (-4x-6)(2x-1)+(2x-3)(8x-11)
h= (x-8)(5+3x)-(x-8)(7-x)
Exercise 5:
Frame when
.
then
Exercise 6:
Factor the following expressions:
Exercise 7:
Exercise 8:
1. In what form is an even number written ?
2. In what form is an odd number written?
3. Show that the square of an even number is an even number.
with
Conclusion: the square of an even number is an even number.
Exercise 9:
1. Calculate the sum of 5 consecutive integers.
It’s up to you to do some tests.
2. Show that the sum of five consecutive integers is a multiple of 5.
Let x be the smallest integer .
Either
So the sum of 5 consecutive integers is a multiple of 5.
Exercise 10:
1. Calculate the product of four consecutive integers and add 1.
What do we notice? (Make several attempts)
2x3x4x5+1=121=11²
5x6x7x8+1=1681=41²
6x7x8x9+1=3025=552
2. Show that, for any real x, we have :
Explain the result observed in question 1.
The product of 4 consecutive numbers plus 1 can be written as the square of a number.
Exercise 11:
a. Decompose into the product of prime numbers 220 and 284.
b. Check that 220 and 284 are friendly.
The divisors of 220 are 1;2;4;5;10;11;20;22;44;55;110;220.
The divisors of 284 are 1;2;4;71;142;284
or let’s calculate the sum of the divisors except m :
and
Conclusion: 220 and 284 are two friendly integers.
Exercise 12:
Write in interval form :
Exercise 13:
1. Hervé must factor A.
Here’s his answer:
Test the equality obtained by Hervé for
What can we conclude from this?
and if we replace 0 in the original expression, we obtain :
Conclusion : Hervé made a mistake when factoring.
2. To factor A, we can think of writing :
Then factor A correctly.
Exercise 14:
Factor each expression by highlighting a common factor.
Exercise 15:
Exercise 16:
Exercise 17:
Exercise 18:
Exercise 19:
Exercise 20:
The golden number is the number .
Check the following equalities:
1. .
2. .
Let’s multiply by :
(this is the equality of question 1.)
3.
We know that :
Let’s multiply this equality by :
(because
)
so
Exercise 21:
The speed of light is estimated at m/s
and the average distance Earth-Sun at 149 million kilometers.
Calculate the time needed for a light signal from the Earth to reach the
Sun.
Exercise 22:
For , calculate :
Exercise 23:
Prove that the diagonal of a square of side is
.
In the right-angled triangle ABC at B, according to the direct part of the Pythagorean theorem :
(a length is positive or zero)
Exercise 24:
Exercise 25:
Exercise 26:
Exercise 28:
1. For each line, reconstruct the sentence using Si……then …. or …if and only if ….:
a.
IF IT RAINS THEN I take my umbrella.
IF I in the middle of [AB] THEN AI=BI
IF AND ONLY IF
.
IF THEN
.
IF THEN ABC is isosceles at A.
Exercise 29:
For n natural numbers, compare the following numbers:
Exercise 30:
Let ,compare the numbers :
Suppose that
So
So
So
So
this being always true since
so
Exercise 31:
1. Complete using the symbols
a.
b.
2. Specify the interval corresponding to :
a.
b.
c.
Exercise 32:
a.
b.
2. Simplify and then write the following fraction in scientific form:
3. Simplify the following entries:
a.
b.
c.
4. Give the prime factor product decomposition of the number .
Exercise 33:
a. Indicate the nature of the following numbers:
: rational
: rational
: integer D=1+
: irrational
b. Simplify the writing of the following number:
After having consulted the answers to these exercises on calculations and problems in grade 2, you can return to the exercises ingrade 2.
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