Algebra problem solutions (60..69)

60.
solve

Solution




61.
A.
find n if

Solution



B.
calculate n if

Solution



62.
calculate if

Solution



63.
2n + 2n-1 = 48
Calculate n.

Solution

2.2n-1 + 2.n-1 = 48
3.2n-1 = 48
2n-1 = 16 = 24
n -1 = 4
n = 5

64.
x3 + x2 = 392.
Calculate x.

Solution

x2(x+1) = 23.72
x2(x+1) = (7+1).72
x = 7

65.
If calculate

Solution

Successively multiply by a, b, c and add :



66.
A.
n is a positive integer.
Prove that is divisible by 7

Solution

This is a proof based on induction.
Assume that the formula is correct for some value of n.
Then prove that the formula would also be true if n is replaced by (n+1):



The formula is OK for n=1.
So also for n=2,3,4,......

B.
N is a positive integer.
Prove that the fraction is not reducable for any N

Solution

Note: GCD(A,B) means the greatest common divisor of A and B.
We use the Euclides lemma stating that GCD(A,B) = GCD(A-B,B).

GCD(21N+4,14N+3)=GCD(7N+1,14N+3)=GCD(14N+3,7N+1)=
GCD(7N+2,7N+1)=GCD(1,7N+1)=1

There are no common factors, the fraction cannot be reduced.

C.
x and y are integers.
Prove that :
    if 2x + 3y is divisible by 17 than also 9x + 5y is divisible by 17.

Solution



67.
A.
x4 + x2 = 90.
Calculate x3

Solution

Let p=x2

p2 + p - 90 = 0
(p+10)(p-9)=0
p=9
x=3
x3 = 27.

B.
8n = 27.
Calculate 4n.

Solution



C.
32x+1 + 3 = 3x+2 + 3x.
Calculate x.

Solution

3.32x + 3 = 9.3x + 3x.
Let 3x = p.
3p2 + 3 = 10p.
3p2 - 10p + 3 = 0.
(3p-1)(p-3) = 0.
3p-1=0; p=1/3; x = -1.

p-3=0; p=3; x = 1.

D.
simplify

Solution



E.
2x-3y=4
calculate

Solution



F.
solve x op

Solution



68.
A.



x=? y=?

Solution



B.
x4 + x2 = 90.
x3 = ?

Solution

Let x2 = p.
p2 + p - 90 = 0.
(p-9)(p+10) = 0.
p=9.....x=3.....x3 = 27

C.



x=?

Solution



D.
2x = 216
3y = 216

calculate:


Solution



E.
x and y are positive integers.
x2 + 4xy - 12y2 = 148.

Calculate x2 + y2

Solution

148 = 4 * 37
148 = 2 * 74

x2 + 4xy - 12y2 = (x + 6y)(x - 2y) = 148.

x + 6y = 37
x - 2y = 4.............no solution in integers

x + 6y = 74
x - 2y = 2...........8y=72..........y=9; x=20

x2+ y2 = 481.

69.



Find the function of above third degree curve.

Solution.

General: y = ax3 + bx2 + cx + d.

To find a,b,c,d four equations are needed.
Two result from points (1,3) and (2,1).
These are extremes so, the derivative in these points is zero.

Derivative y'= 3ax2 + 2bx + c.
The four equations are: 
 a   +  b +  c + d = 3

 8a  + 4b + 2c + d = 1

 3a  + 2b +  c     = 0

 12a + 4b +  c     = 0
Coefficients a,b,c,d are supplied to the application "LIVE" to solve this system of equations.



the solution:



Note:
Appliation LIVE may be found [hier].

Solution:
y = 4x3 - 18x2 + 24x - 7