#84. Prove that the logarithmic function is strictly increasing whenever it is defined.

#85. Find the dimensions and the perimter of a rectangle of area 96 square centimeters whose perimeter is the least.

#86. Find the general solution of the differential equation

.

#87. The bottom of a cuboidal swimming pool is 25 meters wide and 40 meters long. Water is pumped into the tank at the rate of 500 m³ per hour. Find the rate at which the water level is rising.

Answers to #84, #85, #86, and #87:-
All correct, JaneFairfax!!!

#91. If a, a+2, a+4 are prime numbers where

, what is the number of values a can be given?

#92. A survey on a sample of 25 new cars being sold at a local automobile dealer was conducted to study which of the three popular facilities - air-conditioning, radio and power windows - were already installed. The survey found

15 had air conditioning;
2 had air conditioning and power windows but no radio;
12 had radio;
6 had air conditioning and radio but no power windows;
11 had power windows;
4 had radio and power windows; and
3 had all the three facilities.

What is the number of cars that had none of these facilities?

Answer to #91:

Correct, JaneFairfax! Good!

Answer to #89:
Correct, JaneFairfax!!!

Answer to #90:-

Explanation to #90:
Please understand, I'd hate to be unfair and say someone's wrong without enough reason.
What makes you think that 11 or 88 read upside down would cause misinterpretation? They would still appear as 11 and 88 respectively. It was stated in the question 'the code, handwritten in a slip can however create misinterpretation when read upside down - for example, the code 91 may appear as 16'. Which means 1 is written as |. And

as

.

#93. The salary of an employee is $1,600. His expenditure on food, clothing, house rent and entermainment are represented by a pie chart making central angles 157.5°, 67.5°, 54°, and 36° respectively. What is his monthly saving?

#94. For any integer n, what is the argument of

?

#95. In a Geometric Progression, both the first term and the common ratio are

.
What is the magnitude of the

term of the Geometric Progression?

#96. If

, then what is the value of

?

Answers to #95 and #96.
Answer to #95 is correct.
Excellent!!!

Regarding #96, I guess you've made a minor mistake somewhere. Please try again.

Answer to #96:-
Correct, JaneFairfax!



#97. If

are the roots of the equation

, then what is the value of

?

#98. If

, then what is the value of

?

#99. If n is an integer other than a multiple of 3, then what is the value of

?

#100. If n is a multiple of 3, then what is the value of

?

Answer to #97:-
You are correct, JaneFairfax!


#101. If

,
then, what is the value of

?

#102. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

#103. Five digit numbers are formed using 0, 1, 2, 3, and 4. How many numbers would be there in which atleast one digit is repeated?