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If a 3-digit number (000 to 999) is chosen at random, find the probability that exactly one digit will be less than 2.
there are some things not clear ..a 3 digit number if you say ,it can be only 100 to 999 ,not 000 to 999..please check how/where you got the idea. any way i will give the method
total 3 digit numbers = 999-100+1 = 900
now less than 2 we can have 1 or 0 . but in the first digit we can not have zero.hence numbers with 1 in the first digit can be
1(1st.digit)*8(second digit)*8(3rd.digit)= 64..(second & 3rd. digits will have to be between 2 and 9 ..that is 8 possibilities)
Similarly with 1 or zero in the second digit we can have
8*2*8 = 128
Similarly with 1 or zero in the 3rd. digit we can have
8*8*2 = 128
so total possibilities =64+128+128=320
probability =320/900 =32/90 =16/45

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Given that, for all values of x:
Find the values of p,q and r
the given equation is a second degree polynomial in x and should have 2 roots or it should be satisfied by 2 values of x . but it is given that it is true for all values of x . so it is an identity . so coefficients of corresponding terms on either side of the identity shall be equal
l.h.s.==r.h.s.==
hence 3=p
12 = 2p*q or...... q=12/2*p = 12/2*3 = 12/6 =2
5= = 3*2^2+r=12+r...or.....r=5-12 = -7

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=
Let this be equal to
= =

Since denominators are same , equating the numerators we get
=x
since this is an identity , it is true for any value of x
hence take x=1,and -2 and 2 to get a ,b , c values
for x=1 ,we have
a(1+2)(1-2)=1
a = -1/3 similarly we get b=-2/12=-1/6 and c=2/4 = 1/2
hence
=

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|x+8|<0
absolute value of a function is always positive
it can never be less than zero
so there is no solution for this problem

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domain for f(x)=(2x+1)/(x+6).
certain operations in maths like division by zero ,square root of a negative numbers are not allowed .
so by domain of a function we mean such values of independent variable ( usually x ) which make the function exist with permissible operations.
we have to look at denominator here .it has x+6 ..it should not be zero
so the domain is all values of x in the real number set except x=-6

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the general expression for odd integer is 2n+1...since ..n is any integer, 2n is always even . if we add one to an even number we get an odd number
so let the first odd integer be 2n+1
the second point to be noted is that consecutive odd integers differ by 2 .
( like 1,3,5 etc..)..so now
1st. number = 2n+1
2nd.no.=2n+3
3rd.no. = 2n+5
4th.no. = 2n+7
5th. no. =2n+9
sum of 1st and 5th. nos .=2n+1+2n+9=4n+10.......(1)
three times the 4th. no. = 3*(2n+7)=6n+21
one less than above = 6n+21-1 = 6n+20.....(2)
(1) = (2) ....so 4n+10=6n+20
4n-6n=20-10=10
-2n=10
n=-5...so the five nos.are
1st. number = 2n+1 = 2(*-5)+1 = -9
2nd.no.=2n+3................=-7
3rd.no. = 2n+5................=-5
4th.no. = 2n+7................=-3
5th. no. =2n+9................=-1

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find the inverse of Y=4x and y=x-2
in these equations called functions , x is considered independent variable and y as dependent variable. generally you find Y on the l.h.s. expressed explicitly as equal to some function of x ....as 4x in the above example
to find inverse means we have to express x as function of y
hence we have y = 4x
x = y/4 ...is the inverse
for the second example y = x - 2
x = y+2 ...is the inverse

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if 70% equals 175,000,000 how much is 100% ?
if you know algebra ..the method is
let the number be x
70% of the number = x*70/100
this is equal to 175000000
x*70/100 = 175000000
x = 175000000 *100/70 =250000000

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There are only 2 out comes true or false and only one success so
probability of success in one trial = 1/2
it is same for every question and does not depend on other questions / answers haence for 10 questions
probability = (1/2)*(1/2)*(1/2)*...10 times =

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If 10 cards are drawn without replacement from an ordinary deck, find the probability of obtaining exactly 3 aces or exactly 3 kings (or both). I tried
using (52 choose 10) for the denominator and (10 choose 3)^4 for the numerator(4 for each suite). It seems as though I'm missing something. Please help asap.
Total number of possibilities. N = ...taking 10 out of 52 cards = 52C10
number of successes....M =...taking 3 aces and rest seven other than aces.
taking 3 aces out of 4 aces ..=4C3
taking 7 cards out of rest other than aces...=48C7
M = 4C3 * 48C7
probability = M/N = 4C3 * 48C7 /52C10 = (4*48*47*46*45*44*43*42)*(10*9*8*7*6*5*4*3*2*1)/(7*6*5*4*3*2*1)*(52*51*50*49*48*47*46*45*44*43)
FOr 3 kings only the answer is same
for both 3 kings and 3 aces you can try your self on the above basis

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Find a vector x with ||x||=2 that has the same direction as the vector defined by the points A(3,2,1) and B(1,-1,0).
The opposite direction?
direction of ab = i(1-3)+j(-1-2)+k(0-1) = -2i-3j-k
vector x has sime direction .Hence vector|x|= s*(-2i-3j-k) where s is a scalar to be found ||x|| = s*||(-2i-3j-k)||=s*
s*=s*= 2...given
Hence s = (2/*)
vector |x| = (2/*)*(-2i-3j-k) ....answer
if the direction is opposite then vector |x| = (2/*)*(2i+3j+k)

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The length of a rectangle is 2cm less than 7 times the width. The perimeter is 60cm. Find the width and length.
let length be =L
let width be = W
7 times width =7W
2 cm. less than above = 7W-2 = L
perimeter =2L+2W = 60
2*(7W-2)+2W = 60
14W+2W=60+2*2=64
16W = 64
W =64/16 = 4
L=7*4-2 = 26
SECOND PROBLEM
The first side of a triangle is 8m shorter than the second side. The third side is 4times as long as the first side. The perimeter is 26m. Find the length of each side.
Let the first side =x
let the second side =y
let the third side =z
8m shorte than second side = y-8 = x OR y = x+8
4 times the first side =4*x= z
perimeter =x+y+z = 26
x+(x+8)+4*x = 26
x+x+4x=26-8 = 18
6x = 18
x=18/6 = 3
y= 3+8 = 11
z=4*3 = 12

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let the number be =n
squareroot of the number =
number devided by 40 = n/40
hence n/40 =
n/=40
=40
n=40*40 = 1600

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The Kings won two less than three times as many games as they lost. They played 82 games. How many wins and losses did the team have?
let the games won by kings be ....W
let the games lost by kings be ...L
3 times the games lost = 3*L
2 less than above = 3*L-2 = W
total games played =L+W = 82
L+3*L -2 = 82
4*L= 82+2 = 84
L = 84/4 =21
W = 82 - 21 = 61

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let the length of rectangle be ...L
let the width of rectangle be ...W
three times width = 3w = L
perimeter =2*L+2*W= 128
2*3*W+2*W = 128
8*W = 128....
W = 128/8 = 16
L=3*W=3*16 = 48

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x+y=180
x=2y-30
2y-30+y=180
3y=180+30=210
y=70
x=110

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13.7b-6.5= -2.3b+8.3
13.7b+2.3b=8.3+6.5=14.8
b(13.7+2.3)=16b= 14.8
b=14.8/16 =3.7/4

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3n-2/5 = 7/10
3n = 7/10 + 2/5 = (7+4)/10 = 11 / 10
n= 11/(10*3) = 11/30

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3n-2/5 = 7/10
3n = 7/10 + 2/5 = (7+4)/10 = 11 / 10
n= 11/(10*3) = 11/30

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let the 2 angles be x and y .
they are complementary hence x+y = 90 .......(1)
it is given that one say x measures 30 degrees more than 3 times the other that is y.
three times y = 3y...
30 degree more makes it 3y+30..
this is equal to x.
hence we have x=3y+30.....(2)..
substituting this in eqn.(1), we get
3y+30+y = 90
4y = 90-30 = 60
y=60/4 =15
but x+y = 90 so x=90-y=90-15 = 75
hence x=75 and y = 15.

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It is quite simple procedure . Let us take one simple equation to illutrate the principle involved Say...Solve |x+4|=5
|x+4| is read as Absolute value or MOD of x+4 .this means that whether x+4 is negative or positive ,we should take its positive value .that is if x+4 turns out to be positive then take its value as such that is x+4 only; and if x+4 turns out to be negative ,then take its negative value so that we get a positive value only.that is in that case we should take -(x+4) as its value. so this means we have 2 cases and hence 2 equations and so 2 solutions . in these problems we should therefore write the 2 cases and 2 equations first.
case1...x+4 is positive ..then |x+4|=x+4 = 5 or x=5-4 =1 is the answer.
case2...x+4 is negative ..then |x+4|=-(x+4) = 5 or -x=5+4 =9 or x=-9 is the answer.
Hence we should write the solution as x=1 or -9 ........answer.
So now you see that if x+4=5 is the problem ,we will get only one answer namely x=1 ; but if the problem is |x+4|=5 , we get 2 answers namely x=1 or -9 .hope you understood .

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I need help: I have the formulas I just do not know how to apply them correctly. Here is the problem it is 9 ? out of : given points p(-1,2) and n(5,-7)...find (1) the midpoint. (2) the distance (3) the slope (4) slope of a line parallel to segment pn (5) slope of a line perpendicular to segment pn (6) slope-intercept form of the equation the line (7) point-slope form of the equeation of the line (8) standard form of the equation of the line (9) graph the points....Thanks
(1) midpoint --I can do...-1+5/2= 4/2 =2 2+-7/2= -5/2= -5/2 (2, -5/2)
(2) distance-
(3) the slope y2-y1/x2-x1= -7-2/5- (-1)= -9/6=-3/2
Good You know the formulae and did fairly well..let us standardise and recount all the formulae for the above questions
Let the 2 points as you called P and Q say be represented by their coordinates
(x1,y1) and (x2,y2) respectively . So in your example we have x1=-1 ; y1=2 ;
x2=5 ; and y2 = -7...you did substitutions in the formula correctly for the midpoint and slope . hence ,knowing the fomulae for the other questions , you should be able to do the problem your self by substituting for x1,y1,x2 and y2.
mid point is ...(x1+x2)/2 ; and (y1+y2)/2..you did this ok.=(2,-5/2)
distance is .... =square root of 117
slope is ........(y2-y1)/(x2-x1)...you did this ok .=-3/2
slope of a line parallel to the above segment ....same as above since parallel lines have identical slopes =-3/2
slope of a line perpendicular to the above line ....-(x2-x1)/(y2-y1) since the rule is that product of slopes of 2 perpendicular lines should be equal to -1. answer is 2/3
slope intercept form equation of the line
Y=X*((y2-y1)/(x2-x1))+[y1-x1*((y2-y1)/(x2-x1)))answer is Y=(-3/2)x+(1/2)
where ((y2-y1)/(x2-x1))is the slope and (y1-x1*((y2-y1)/(x2-x1)))is the intercept.
point slope form is Y-y1=(X-x1)*((y2-y1)/(x2-x1))answer is Y-2=(-3/2)*(X+1)
standard form of eqn. is X*((y2-y1)/(x2-x1))-Y+(y1-x1*((y2-y1)/(x2-x1)))=0
answer is (-3/2)*X - Y +(1/2)=0 ..or multiplying with -2 throughout to remove 2 in the denominator we get 3X+Y-1=0
to plot the graph take the last equation 3X+Y-1=0 , give different values for X , say 0,2 and 4 (2 or 3 would do as it is a straight line),find corresponding values of Y from this equation and plot to a suitable scale.

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The second of three numbers is 8 more that the first and the third number is 3 less than 3 times the first. If the tird number is 15 more than the second, find the three numbers.
Here's what I have for the variables:
First number = x
Second number = x + 8
third number = 3x - 3 and it is x + 8 + 15
Where do I go from here??
OH!! DEAR!! DEAR!! YOU GOT IT !! YOU ARE ALMOST AT THE FINISH LINE !!! JUMP AHEAD ! YOU WILL GET THE ANSWER!!!
You rightly concluded that the third number is 3x-3 and it is x+8+15...MEANS WHAT ???? THEY ARE EQUAL ...SO
3x-3=x+8+15
3x-x=8+15+3
2x=26...or x=13..... hence the 3 numbers are as per your own working
First number = x =13
Second number = x + 8=13+8 =21
third number = 3x - 3=3*13-3=36..check this if you want with
x + 8 + 15 = 13+8+15 =36

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x - 6y = 3....(1) ; 3x+ 2y = 1.....(2) Let me explain elimination to you.there are 2 unknowns. we can easily solve if there is only 1 unknown.so we try to eliminate 1 unknown. say x .
see coefficient of x in both eqns.,1 and 3. find their lcm .it is 3. so multiply eqn.1 with 3 to get 3x and eqn.2 with 1 to get the same 3x.having made coefficient of x same in the 2 eqns.,add or subtract them to eliminate x.
eqn.(1)*3....3x-18y=9......(3)
eqn.(2)*1....3x+2y=1........(4)
eqn(3)-eqn.(4)......-18y-2y=9-1...or -20y=8..or y=-8/20=-2/5.
now substitute this value of y in any eqn.say eqn.(1) to get x value
x-6*(-2/5)=3...
x=3-12/5=(15-12)/5 = 3/5.
to check your answer substitute these 2 values of x and y in the given equations and check for compliance.
Eqn.(1)....3/5 -6*(-2/5)= 3/5 +12/5 =15/5 = 3 =rhs ...ok.
Eqn.(2)....3*(3/5)+2*(-2/5)=9/5-4/5=5/5=1 = rhs ...ok
now i leave the second problem for you to do as practice

6x + -4y = 12
9x - 6y = 18

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in these equations or functions as we call them you know that x is called the independent variable and y the dependent variable. that is we give different values to x and find the corresponding values of y.that is how you draw the graphs of the functions .
now to start with domain is the set of all values x can take and range is the corresponding set of values y can take .
what do we mean by the set of values x can take .As you know certain operations are prohibited or meaning less in maths.like division by zero.so if we have a function y=1/x , we say x can take any value except zero.so domain of x is all real numbers except zero . similarly if y= square root of x ,then domain is ... x can take all real non negative numbers.x cannot be negative as square root of negative number does not exist.
after determining the domain ,we find the set of corresponding values of y and call it the range of y.clear ?
now y=3*x-4 .. as there are no problem areas like division by x or squareroot of x etc..we can conclude that domain of x is all real numbers and range of y is also all real numbers
now y = -2x^2 + 2...domain of x is all real numbers as above.but as far as range is concerned we see that ...since square of x is always positive,its minimum value can be 0.hence -(square of x) has a maximum value of 0.hence has a maximum value of 1. hence y has a maximum value of 2 . hence range of y is all real numbers less than or equal to 2.you can check this from your graph too. you will find that as x changes from -infinity to +infinity , y changes from +2 to -infinity.

y = x^3...you can see here domain and range are set of all real numbers.

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let length of rectangle be L and width be W.
area = L*W=13.5..........(1)
perimeter = 2*(L+W)
length of one side =L = 0.2 times perimeter=0.2*2*(L+W)
L=0.4*L+0.4*W
0.6*L=0.4*W
1.5*L = W....substituting in (1)we get
1.5*L*L=13.5
L*L= 13.5/1.5 =9
L=3
hence W = 13.5/3 = 4.5

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let the smaller number be x and greater number be y.
sum of numbers = x+y = 18.......(1)
three times the greater number = 3*y
four times the smaller number = 4*x
three times the greater number exceeds four times the smaller number by 5
hence 3*y-4*x=5
-4*x+3*y=5 .........(2).we have 2 unknowns.let us eliminate one unknown say x.for this multiply eqn.(1) with 4 and add to eqn.(2).
eqn.(1)*4 gives us ....4*x+4*y=4*18=72.......(3)adding to eqn.(2)
-4*x+3*y=5......(2) we get
4*y+3*y=7*y=72+5=77
y=77/7=11 substituting in 1 we get
x+11=18 or x=18-11=7

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Let x be the smaller number and y the bigger number.
we have x+y=133.......(1)
and 4*x=3*y ..........(2)
multiplying (1) with 3 throught we get 3*x+3*y=3*133...substituting 3*y=4*x in this we get 3*x+4*x=399
7*x=399
x=399/7=57
since 4*x=3*y=4*57
y=4*57/3=4*19=76
hence x=57 and y=76....

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Let us assume that vertical line means line perpendicular to X axis or parallel to Y axis.Its equation is given by ...X=K where K is a constant to be found .Since this vertical goes through the point (4,-5), it should satisfy the equation So substituting in X=K , we get 4=K...hence the equation of the vertical containing (4,-5) is X=4

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Similar problem was solved earlier as given below

lcm is 15. hence

Multiplying both sides with 15 , we get ....
-2x-10=1
-2x=1+10=11
x=11/2
You may see this and solve your problem.If still you have any difficulty , we shall clarify on hearing from you

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lcm is 15. hence

Multiplying both sides with 15 , we get ....
-2x-10=1
-2x=1+10=11
x=11/2