Find the mean proportional between 81 & 100?
Options:
(A) A) 59
(B) B) 61
(C) C) 63
(D) D) 65
(E) E) 90

Formula = ab
A = 81 and B = 100
81100 = 9 10 = 90 So the answer is: (E).

Find a a prime number the options below.
Options:
(A) 78
(B) 80
(C) 81
(D) 82
(E) 83

Clearly, 83 is a prime number. So the answer is: (E).

How many 5 letter combinations can be made from the letters of the word VERMONT if the first letter has to be a vowel and the last letter has to be a consonant, and repetition is allowed?
Options:
(A) 3441
(B) 3433
(C) 3333
(D) 3430
(E) 3773

Vowel: E O
Consonants: V R M N T
First letter can be filled in 2 Ways ( As the question states that first letter has to be a vowel)
Fifth letter can be filled in 5 Ways ( As the question states that the last letter has to be a consonant)
Second letter can be filled in 7 Ways
Third letter can be filled in 7 Ways
Fourth letter can be filled in 7 Ways
So, Total number of ways = 2 Ways x 7 Ways x 7 ways x 7 ways x 5 ways = 3430 So the answer is: (D).

Seven different objects must be divided among three persons. In how many ways this can be done if at least one of them gets exactly one object.
Options:
(A) 223
(B) 776
(C) 287
(D) 196
(E) 211
Division of m+n+p objects into three groups is given by (m+n+p)!m!n!p!(m+n+p)!m!n!p!
But 7 = 1 + 3 + 3 or 1 + 2 + 4 or 1 + 1 + 5
So The number of ways are (7)!1!3!3!12!(7)!1!3!3!12! + (7)!1!2!4!(7)!1!2!4! + (7)!1!1!5!12!(7)!1!1!5!12! = 70 + 105 + 21 = 196 So the answer is: (D).