1. Which of the following is the solution set of the equation x2-2x-15 = 0

Think of factors of the last digit that when subtracted gives the coefficient of the middle term

In this case, -15 can only be (3, -5), (-3, 5), (15, -1) or (-15, 1). Among these factors, (3, -5) is the only one whose sum will be equal to -3. So the solution set of the equation will be

X-5=0; x1 = 5

X+3=0; x2 = -3

By trial and error (best when using your calculator), try each choices and see which among them when substituted to x will make the equation true, that is, x2-2x-15 equals zero.

2. The next term in the progression 2, 8, 32, 128 is …

The progression above is a Geometric progression since there is a common ratio (r). In this case, the common ratio is 4 and the next terms are obtained by multiplying the previous term by 4. So, next to 128 will be 128X 4 = 512

3. Solve for x in the equation 3 lnx = 8

Using properties of logarithm: nlna = lna^n

In this case, lnx^3=8

By making both sides an exponent of e (Euler’s number), we can cancel the ln.

e^(lnx^3)=e^8

x^ 3 = e^8

x= e^ (8/3)

4. If x + y = 1 and x2+y2=2. Find x4+y4

Squaring both sides of x+y =1 we get (x+y) ^2=1^2; x2+2xy+y2=1;

But x2+y2=2 so 2+ 2xy=1 or xy=-1/2;

Square both sides, 4x2y2=1; x2y2=1/4

Let’s square both sides of x2+y2=2; (X2+y2) ^2=2^2 then x4+2x2y2+y4 = 4;

But x2y2=1/4 so, x4+y4+2(1/4) =4

Solving the equation will give us X4 +y4 =7/2

5. If lnx=5 and lny=2, lnx3/lnx4

Using properties of logarithm; lnx3=3lnx and lny4=4lny

So lnx3/lny4 will be 3lnx/4lny = 3(5)/4(2) =15/8

6. Joan can type the whole document in 3 hrs. On the other hand, James, being new to the job can type the same document in 5 hours. If they work together, how long can they finish 2 documents?

rate of Joan 1/3; rate of James 1/5

Rate=no job/time time = no of job/rate

If they work together, their rate will be added

t= 2 jobs/ (1job/3hrs+1job/5hrs)

t=3.75 hours

7. Three numbers are in the ratio 1: 2: 4. Find the numbers if their sum is 42.

add the ratios 1+2+4 = 7

Divide the sum of the numbers by the sum of the ratio; 42/7=6

Multiply 6 by each ratio. The numbers will be 6, 12, and 24

8. Find the sum of the coefficients in the expansion of (5x-4)^5

To find the sum of the coefficients of an expansion, just replace the variable by 1 in the expression.

In this case, (5x-4) ^5; [(5(1)-3] ^5=32

9. Evaluate the expression log 3+ log 2+ 4log7

use your calculator

Note: When there’s no base mentioned, it is automatically considered to be 10.

10. The sum of the infinite geometric progression is 3 and the common ratio is 1/3. What is the first term?

For an infinite Geometric progression, the sum is given by the formula S=a1/r where S is the sum and r is the ratio.

In this case, the Sum and the ratio were given so using the formula we can get the first term.

a1=Sr; a1=9(1/3) =3

**Solution**:Think of factors of the last digit that when subtracted gives the coefficient of the middle term

In this case, -15 can only be (3, -5), (-3, 5), (15, -1) or (-15, 1). Among these factors, (3, -5) is the only one whose sum will be equal to -3. So the solution set of the equation will be

X-5=0; x1 = 5

X+3=0; x2 = -3

**Alternative Solution:**By trial and error (best when using your calculator), try each choices and see which among them when substituted to x will make the equation true, that is, x2-2x-15 equals zero.

**B. (5, -3)**

2. The next term in the progression 2, 8, 32, 128 is …

**Solution**:The progression above is a Geometric progression since there is a common ratio (r). In this case, the common ratio is 4 and the next terms are obtained by multiplying the previous term by 4. So, next to 128 will be 128X 4 = 512

**C. 512**

3. Solve for x in the equation 3 lnx = 8

**Solution**:Using properties of logarithm: nlna = lna^n

In this case, lnx^3=8

By making both sides an exponent of e (Euler’s number), we can cancel the ln.

e^(lnx^3)=e^8

x^ 3 = e^8

x= e^ (8/3)

**A. e^ (8/3)**

4. If x + y = 1 and x2+y2=2. Find x4+y4

**Solution**:Squaring both sides of x+y =1 we get (x+y) ^2=1^2; x2+2xy+y2=1;

But x2+y2=2 so 2+ 2xy=1 or xy=-1/2;

Square both sides, 4x2y2=1; x2y2=1/4

Let’s square both sides of x2+y2=2; (X2+y2) ^2=2^2 then x4+2x2y2+y4 = 4;

But x2y2=1/4 so, x4+y4+2(1/4) =4

Solving the equation will give us X4 +y4 =7/2

**D. 7/2**

5. If lnx=5 and lny=2, lnx3/lnx4

**Solution:**Using properties of logarithm; lnx3=3lnx and lny4=4lny

So lnx3/lny4 will be 3lnx/4lny = 3(5)/4(2) =15/8

**B. 15/8**

6. Joan can type the whole document in 3 hrs. On the other hand, James, being new to the job can type the same document in 5 hours. If they work together, how long can they finish 2 documents?

**Solution:**rate of Joan 1/3; rate of James 1/5

Rate=no job/time time = no of job/rate

If they work together, their rate will be added

t= 2 jobs/ (1job/3hrs+1job/5hrs)

t=3.75 hours

**D. 3.75 hrs**

7. Three numbers are in the ratio 1: 2: 4. Find the numbers if their sum is 42.

**Solution**:add the ratios 1+2+4 = 7

Divide the sum of the numbers by the sum of the ratio; 42/7=6

Multiply 6 by each ratio. The numbers will be 6, 12, and 24

**A. 6, 12, 24**

8. Find the sum of the coefficients in the expansion of (5x-4)^5

**Solution**:To find the sum of the coefficients of an expansion, just replace the variable by 1 in the expression.

In this case, (5x-4) ^5; [(5(1)-3] ^5=32

**D. 32**

9. Evaluate the expression log 3+ log 2+ 4log7

**Solution**:use your calculator

Note: When there’s no base mentioned, it is automatically considered to be 10.

**D. 4.16**

10. The sum of the infinite geometric progression is 3 and the common ratio is 1/3. What is the first term?

**Solution**:For an infinite Geometric progression, the sum is given by the formula S=a1/r where S is the sum and r is the ratio.

In this case, the Sum and the ratio were given so using the formula we can get the first term.

a1=Sr; a1=9(1/3) =3

**A. 3**

**Shortcut to questions.**Note: Writing solution to a math problem in a webpage like these is extremely difficult so I expect that some of you may be confused by the way I wrote the solution. If you have questions and clarifications, don't hesitate to send me a message at admin@tupmechanical.com or post a comment below.

Your answer to #2 is incorrect since 32 times 4 is not 64 so the geometric ratio cannot be 4

ReplyDelete@andrew: Thanks. Questions and answers are corrected accordingly.

ReplyDelete@admin

ReplyDeleteanswers to #8 and #10 is incorrect..

to #8:

since the coefficient has constant so the formula should be : (coef.X + K)^n - (K)^n

so the answer should be 1025

to #10:

the formula should be S=a1/1-r

so the answer should be 2..

tama ka dude..mali mali talaga toh,,,tsk2x

Deletepakireview po ng #5,, mali kasi ang given,,Iny4 dapat...

ReplyDeletemali mali naman.

ReplyDeleteFor the admin, with all due respect, please kindly check first your review material before being published here. I thought it would be really helpful. It just confuses your visitors. :(

ReplyDeleteThank you!

Dude hope you did some writing or explanation about this numbers. Thank you for sharing this very good math information.

ReplyDeleteIn fact, this is too complicated for me! I think that a simple calculator will ease these calculations!

ReplyDeleteThanks for the provided calculations! The problem was quite confused, but you were managed to resolve it finally!

ReplyDeleteCorrections

ReplyDelete# 8: ans = 1025; (first coefficient + 2nd coef.)^exp. - (2nd coef.)^exp.

# 10: ans = 2; Sum Geo. seq. infinity = (1st no.)/1-ratio