online user counter

MATHMAA

KYPY2013solution

This page contains KYPY2013solution 2 and 3 in detail. We have three different cases which you can see in detail below.

KYPY2013solution 2

2) If a,b are natural numbers such that 2013+a² = b², then the minimum possible value of ab is
a) 671        b) 668        c) 658      d) 645

Solution:-
First we have to factor 2013+a² = b².
2013 + a² = b²
Subtracting a² on both sides, we get,
b² - a² = 2013
Applying the formula x² - y² = (x+y)(x-y), we get,
(b+a)(b-a) = 2013
Now we have to split 2013 as prime factors. We get,
2013 = 11 * 3 * 61.
We have to write these factors as the product of two numbers so we can write them as
61 times 33  or  183 times 11  or  671 times 3
Case I:
(b+a)(b-a) = 61*33
b+a = 61
b-a  = 33
Solving these, we get,
b = 47 and a = 14 so ab = 47*14 = 658
Case II:
(b+a)(b-a) = 183*11
b+a = 183
b-a  = 33
Solving these, we get,
b=108 and a=75 so ab = 108*75 = 8100
Case III:
(b+a)(b-a) = 671 times 3
b+a = 671
b-a  = 3
Solving these, we get,
b=337 and a=334 so ab = 337*334 = 112558
Now the minimum value of ab is 658 therefore the correct option is (C).

KVPY2013solution 3

3) The number of values of b for which there is an isosceles triangle with side lengths b+5, 3b-2 and 6-b is
a) 0            b) 1          c) 2           d)3

Solution:
The triangle in which we have lengths of any two sides equal is called an isosceles triangle. So here we are given the lengths of an isosceles triangle b+5, 3b-2, 6-b.
Here the possible equal sides are either b+5 and 3b-2 or 3b-2 and 6-b or b+5 and 6-b.
Case I:
The possible isosceles triangle with equal side lengths b+5 and 3b-2.
b+5 = 3b-2
2b = 7
b = $$\frac{7}{2}$$
b = 3.5

Now the length of sides of the triangle are 8.5, 8.5, 2.5. So triangle is possible with these measurements.

Case II:
The possible isosceles triangle with equal side lengths of 3b-2 and 6-b.
3b-2 = 6-b
4b = 8
b = 2

Now the length of sides of the triangle are 7, 4, 4. So triangle is possible with these measurements.
Case III:
The possible isosceles triangle with equal side length of b+5 and 6-b.
b+5 = 6-b
2b = 1
b = $$\frac{1}{2}$$
b = 0.5
Now the length of sides of the triangle are 5.5, -0.5, 5.5 which is not possible as the length cannot be negative.

Hence possible values of b are 3.5, 2. There are 2 possible values. So the correct option is (C).

To go the previous solution click Back and further solutions click Next. Share your views to update if necessary.