Problem Solving

soni_pallavi wrote:Dear All

Plz help with the following questions

q1)Tanya prepared 4 different letters to be sent to 4 different addresses.For each letter,she prepared an envelope with its correct address.If the 4 letterd are to be put into the 4 envelopes at random what is the probability that only one letter will be put into the envelope with its correct address.

Let the envelopes be E1, E2, E3, E4 and the corresponding letters be L1, L2, L3, L4.
Suppose L1 goes to E1 and the other letters do not go in their corresponding envelopes.
So, E2 will have either L3 or L4
If E2 has L3, E3 will have L4, E4 will have L2.
If E2 has L4, E3 will have L2, E4 will have L3.
So, for L1 going to E1 we have 2 arrangements where other letters are not going into right envelopes.
Similarly for L2 going to E2 or L3 going to E3 or L4 going to E4, we have 2 arrangements each.

There are 2 * 2 * 2 * 2 = 8 ways in which only one letter goes to right envelope.

Total number of all possible arrangements of letters in any envelopes is 4! = 24.

Required probability is 8/24 = [spoiler]1/3[/spoiler].

soni_pallavi wrote:Dear All

q2) If n denotes a number to the left of 0 on the numberv line such that teh square of n is less than 1/100 then the reciprocal of n must be...

Ans..Less than 10 (plz explain how?)

The Question is:

if n denotes a number to the left of 0 on the number line such that the square of n is less than 1/100, then the reciprocal of n must be

a. less than -10
b. between -1 and -1/10
c. between -1/10 and 0
d. between 0 and 1/10
e. greater than 10

Explanation:

n is to the left of 0 on the number line, i.e. n < 0
As n² < 1/100, n lies between -1/10 and 1/10

Hence, -1/10 < n < 0
--> 1/n < -10

The correct answer is A.

soni_pallavi wrote:Dear All

q3)For every integer k from 1 to 10 inclusive kth term is given by (-1)^k+1(1/2^k).If T is the sum of the first 10 terms then T is

Ans Between 1/4 and 1/5 (Plz explain how)

Question is:

For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) x (1/2^k). If "T" is the sum of the first 10 terms in the sequence, then T is:

A. greater than 2
B. between 1 and 2
C. between 1/2 and 1
D. between 1/4 and 1/2
E. less than 1/2

Explanation:

We know that the sum of n terms of a geometric series is given by:
S(n) = a(1 - r�)(1 - r), where a is the first term, r is the common ratio of the geometric progression and n = number of terms.

Here, a = 1/2, r = -1/2, n = 10
T = 1/2[1 - (-1/2)^10]/[1 + 1/2]
= 1/2[1 - 1/1024]/[3/2]
= 1/2 * 1023/1024 * (2/3)
= (1023/1024) * (1/3)
Now (1023/1024) = 1 approx, so T = 1/3, which lies between 1/4 and 1/2.

The correct answer is D.