Algebra and probability

Question

1. If x is a negative integer, which of the following expression has the least value?
a. X^2 – 0.5
b. X^2 – 0.51
c. X^3 – 0.502
d. X^3 – 0.511
e. X^4 – 0.512

2. If n is a multiple of 5 and n = p^2q, where p and q are prime numbers, which of the following will be a multiple of 25?
a. P^2
b. Q^2
c. Pq
d. P^2q^2
e. P^3q

3. A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following is represents the total area, in square meters, of the circular and square regions in terms of r?
a. Pi r^2
b. Pi r^2 +10
c. Pi r^2 + ¼ Pi^2 r^2
d. Pi r^2 + (40- 2Pir)^2
e. Pi r^2 + (10- ½ Pi r)^2
4. For which of the following functions is F(a+b) = F(a) + F(b) for all positive value of a and b?
a. F(x) = x^2
b. F(x) = x + 1
c. F(x) = SQRT (x)
d. F(x) = 2/x
e. F(x) = -3x

5. From a bag containing 12 identical blue balls Y identical yellow balls, and no other balls, one ball will be removed at random. If the probability is less than 2/5 that the removed ball will be blue, what is the least number of yellow balls that must be in the bag?
a. 17
b. 18
c. 19
d. 20
e. 21

6. If each term in in the sum a1 + a2 +....+ an is either 7 or 77 and the sum equals 350, which of the following could be n?
a.38
b.39
c.40.
d41.
e.42

Night reader · Answer

x<0, find the least value?
x=-1
a. 1-0.5=0.5
b. 1-0.51=0.49
c. -1-0.502=-1,502
d. -1-0.511=-1,511
e. 1-0.512=0.abc

IOM d

Tagne wrote:1. If x is a negative integer, which of the following expression has the least value?
a. X^2 – 0.5
b. X^2 – 0.51
c. X^3 – 0.502
d. X^3 – 0.511
e. X^4 – 0.512

Night reader · Answer

n/5 =i (i is integer), n=p^2q {p,q are primes}. Find multiple (M) of 25 <> M/25-?
(p^2q)/5 offers the only condition when p=5, So [(5^2)^q]/5 <> (25^q)/5. Q can be anything here ...

a. p^2 Correct, since p=5
b. q^2 Not Correct, as Q can be anything here
c. PQ Not Correct as Q could be 3
d. ...
e ...

IOM A

Tagne wrote: 2. If n is a multiple of 5 and n = p^2q, where p and q are prime numbers, which of the following will be a multiple of 25?
a. P^2
b. Q^2
c. Pq
d. P^2q^2
e. P^3q

Night reader · Answer

W (wire) =40, C (circle) =2pi*r (r is radius), S (square) = 4a (a is square side). Find (square S) + (square C) ?
2pi*r+4a=40, a=(40-2pi*r)/4, a=10 -pi*r/2

Square of S = a^2
Square of C= pi*r^2

a^2 + pi*r^2 = (10 - pi*r/2)^2 + pi*r^2

IOM E

Tagne wrote: 3. A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following is represents the total area, in square meters, of the circular and square regions in terms of r?
a. Pi r^2
b. Pi r^2 +10
c. Pi r^2 + ¼ Pi^2 r^2
d. Pi r^2 + (40- 2Pir)^2
e. Pi r^2 + (10- ½ Pi r)^2

Night reader · Answer

a. (a+b)^2=a^2+b^2 Not Correct;
b. a+b+1=a+1 + b+1 Not Correct;
c. Sqrt (a+b) = Sqrt(a) +Sqrt(b) Not Correct;
d. 2/(a+b) =2/a +2/b Not Correct;
e. -3(a+b) = -3a + (-3b) Correct.

IOM E

Tagne wrote:4. For which of the following functions is F(a+b) = F(a) + F(b) for all positive value of a and b?
a. F(x) = x^2
b. F(x) = x + 1
c. F(x) = SQRT (x)
d. F(x) = 2/x
e. F(x) = -3x

Night reader · Answer

this problem is missing critical part i put in red

P(blue)<2/5. Find the least number of yellow balls?
The question says that less than (2/5) of (12 blue balls+ x yellow balls) in the bag are blue balls - OR, 12/(12+x) <2/5, 60<24+2x, 36<2x and x>13
Let's check the question again :"what is the least number of yellow balls that must be in the bag?"
answer choice A seems fitting

IOM A

Tagne wrote: 5. From a bag containing 12 identical blue balls, certain number of yellow balls, and no other balls, one ball will be removed at random. If the probability is less than 2/5 that the removed ball will be blue, what is the least number of yellow balls that must be in the bag?
a. 17
b. 18
c. 19
d. 20
e. 21

all numbers can be factored by 7 --> 350/7=50. Now we have 50 7-s and the 7-s (sevens) must be associated with 77.
we can have one 77 and (50-11) 7-s (sevens) OR total 40 numbers
two 77-s and (50-22) 7-s OR total 29 numbers
and so on with the numbers (n) decreasing, it's evident that n=40

IOM C

Tagne wrote: 6. If each term in in the sum a1 + a2 +....+ an is either 7 or 77 and the sum equals 350, which of the following could be n?
a.38
b.39
c.40.
d41.
e.42

srcc25anu · Answer

Q5. P(blue) < 2/5 12 / (12+y) < 2/5 or y > 18 hence minimum number of yellow balls for the probability of a blue ball to be drawn at random of less han 2/5 = 19 hence C

saurabh_maths · Answer

Night reader ,

for Q2 :

2. If n is a multiple of 5 and n = p^2q, where p and q are prime numbers, which of the following will be a multiple of 25?
a. P^2
b. Q^2
c. Pq
d. P^2q^2
e. P^3q

I think ans shd be D.

basically it means n/5 is an integer...

so p^2q is a multiple of 5. i.e atleast one of p or q is 5.

now in order to p^2q divisible by 25 ...look at cases :

p = 5 , q =2 (5 is prime and 2 is also prime)
or
P = 2 , q =5
in either case p^2q^2 will satisfy the condition.

option A will fail if we have p =2 and q = 5.

hence IMO it shd be D.

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