Mathematix for the Not so Insane Quiz

Quiz Answer Key and Fun Facts

1. If ten parallel lines were intersected by another nine parallel lines, then how many parallelograms are formed?

Answer: 1620

To form a parallelogram, you need two parallel lines intersecting with two parallel lines. Therefore the total number of parallel lines is 10C2 * 9C2. (i.e. 2 lines can be chosen from 10 parallel lines multiplied by 2 lines chosen from the other 9 parallel lines)
For those of you unfamiliar with Combinations: when you chose r things from n things, nCr = n!/(n-r)!r! and n! = n(n-1)(n-2)....1.

2. If a^b + b^a = 145 and a, b are positive integers, then how many such pairs are possible?

Answer: 2

The two values can be (144,1) and (4,3).

3. A and B are the roots of the equation x^2 - 6x + p = 0. We also know that 3A + 2B = 5. What are the roots of x^2 - px + 2070=0?

Answer: -45, -46

From the first equation, we know that A + B = 6 (for a quadratic equation ax^2 + bx + c = 0, the sum of the roots = -b/a). Solving A and B with the help of equation 2, we get A = -7 and B = 13. Now product of the roots is c/a, therefore 'p' = -7*13 = -91.
Solving the roots of the equation x^2 + 91p + 2070 = 0, we find that the roots are -45 and -46.

4. How many fractions of the form p/q do not form an integer, where p, q belong to the set {1,2,3,4,5,6,7,8,9}?

Answer: 58

If we count the total number fractions of the form p/q, p can be any of the 9 numbers and q can also be any of the 9 numbers. Thus, 9 * 9 = 81 is a possible form. The unfavourable form is that which gives an integer, which is possible if (1) p = q [9 cases] ;(2) p = 2q [4 cases(e.g p
=4, q =2 etc); (3) p = 3q [3 cases]; (4) p = 4q (2 cases) ; from p = 5q to p = 9q, there will be one case each.
So total number of unfavourable cases = 9 + 4 + 3 + 2 + 1 + 1 + 1 + 1 + 1 = 23.
Therefore answer required = total cases - total unfavourable cases = 81 - 23 = 58.

5. If 24th November of 2002 happens to be a Sunday, then on what date would the second Sunday of December 5002 fall?

Answer: 12th December

Between 2002 and 5002, there are 3000 years i.e. 7*400 + 1*200. The number of odd days in 400 years is 0 and in 200 years is 3. So the 24th November of 5002 is a Wednesday, and therefore the second Sunday of December will be 12th December.
(Odd days is the remainder when a number is divided by 7. This is used to calculate day and dates. For e.g. in a normal year, there are 365 days, therefore the number of odd days is 365/7 = 1. What this means is that if it is Tuesday today, then next year, it will be a Wednesday (unless there is a leap year in between).

6. You are given a wire of length 10 metres. You can make any shape by bending the wire (if you want). What shape should you make the wire into to get the maximum area?

Answer: Circle

For equal perimeter (or circumference), the polygon with maximum number of sides will have greatest area. A circle has an infinite number of sides, hence it will have maximum area. Using the same logic, a line has the least area (=0).

7. The population of a country doubles every 10 years. The population 80 years from now will be how many times bigger than the population this year?

Answer: 256

In year (x + 10), the population will be two times the population in year x. In year (x + 20), the population will be 4 times the population in year x. So, we see that the population in after 80 years (8*10) is 2^8 times the population in year x - i.e. 256 times.

8. Simple one : m and n (n > m) are positive integers for which the following operation is defined: m $ n = sum of all integers greater than m but less than n. What is the value of (107 $ 119)-(109 $ 120)?

Answer: 98

The value is (108 + 109 + 110+....+ 118) - (110 + 111 + 112 +...+ 119) which is equal to 108 + 109 - 119 = 98.

9. What is the acute angle between the lines x - y + 5 = 0 and x - 3^(1/2)y + 5*[3^(1/2)]? (Note: 3^(1/2) means square root of 3.)

Answer: 15

Slopes of the two equations are 1 (m1) and 1/ [3^(1/2)] (m2).
A line of the form y = mx + c has a slope of m.
If @ is the angle between any two lines, then tan@ = (m1 - m2)/(1 + m1*m2.
Solving the equation, we get @ = 15.

10. Ace goes to a furniture shop to buy a sofa set and a centre table. He bargains for a 10% discount on the centre table and a 25% discount on the sofa set. However, the shopkeeper, by mistake, interchanged the discount figures while making the bill and Ace paid accordingly. When compared to what he should have paid for his purchases, what percentage did Ace pay extra given that the centre table costs 40% as much as the sofa set?

Answer: 8.1%

Let the cost of sofa set be 100 and the cost of centre table be 40 (for example). Amount payable by Ace = 75 + 36 = 111. Amount calculated by shopkeeper= 90 + 30 = 120. Percentage excess paid by Ace=(9*100%)/111 = 8.1%.

Source: Author ace_sodium

This quiz was reviewed by FunTrivia editor CellarDoor before going online.
Any errors found in FunTrivia content are routinely corrected through our feedback system.

Continuing the series .....

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>

NEXT>