Saturday, March 16, 2013

problem solving - circumcentre,orthocentre


A straight line 4x + 3y = 24 forms a triangle with the coordinate axes. What is the distance between the orthocentre and circumcentre of the triangle so formed?

Explanation:

Consider the diagram below where the line 4x + 3y = 24 forms a triangle with the coordinate axes.
We see that it forms a right triangle with sides 6, 8 and 10. And in a right triangle, the orthocenter lies at the vertex where the right angle is formed. Thus, orthocenter lies at (0,0). The circumcenter in a right triangle is at the mid-point of the hypotenuse. Thus, mid-point of (6,0) and (0,8) is (3,4). Thus, the distance between (0,0) and (3,4) is 5 units.
If you think about it the other way, the line joining the orthocenter and the circumcenter is nothing but the median to the hypotenuse. And we also know that the median to the hypotenuse is the circumradius too. And in a right triangle, the circumradius is half of the hypotenuse. Hence 5 units.

Wednesday, March 6, 2013

problem solving - permutations difficult type


In how many rearrangements of the word SCINTILLATING will no two 'I' come together?

Explanation:

Let us consider the following arrangement without the three I’s.

_S_C_N_T_L_L_A_T_N_G_

No two I’s would come together if the I’s are placed in the blanks. So, the question boils down to first finding out the number of ways in which three blanks can be chosen out of 11 blanks. This can be done in 11C3 ways.

Having placed the I’s, now let us find out all possible arrangements of the remaining letters. i.e., all possible arrangements of

SCNTLLATNG

The number of ways in which SCNTLLATNG can be arranged is 10!/(2!*2!*2!)

Therefore, the number of rearrangements of the word SCINTILLATING where no two I’s would come together is 11C3*10!/(2!*2!*2!).

problem solving - same base, same height type


An equilateral triangle and a parallelogram lie on the same base and between the same parallel lines. The area of the parallelogram is 200*root(3) sq cm. What is the length of the side of the triangle?

Explanation: Two triangles with the same base and between the same parallel lines will have equal areas. Likewise, a triangle and a parallelogram with the same base and between the same parallel lines will have areas in the ratio 1:2 respectively.




Therefore,

2*Area of the equilateral triangle = 200*root(3)
2*root(3)*a2/4 = 200*root(3)
a = 20.


Friday, February 15, 2013

problem solving - planet circles

Consider a planet P1(like earth) that is spherical in shape. It has an atmospheric layer around it that also takes the same shape as that of the planet. An adventurist launches a rocket from a neighboring planet P2 to this planet P1, cuts the atmospheric layer at point A, grazes the planet P1 and goes out of the atmospheric layer at point B. If AB = 100kms, what is the area of cross section of the atmospheric layer?

Explanation:

Consider the following diagram.


The question essentially boils down to a case where a chord AB to the outer circle is a tangent to the inner circle. We have to find out the area between the two circles.

Therefore, Required Area = Area of outer circle - Area of inner circle
                                      = pi*(502 + r2) - pi*r2
                                                   = 2500*pi

Sunday, January 27, 2013

quantitative comparison - numberline, algebra


p, q and r are three points on the real number line where q = (2pr)/(p+r).
Quantity A: mod(1/p - 1/r)
Quantity B: mod(1/q - 1/p)

Explanation:

q = (2pr)/(p+r)

qp + qr = 2pr

(1/r) + (1/p) = (2/q)

[(1/r) + (1/p)]/2 = (1/q)

Thus we see that (1/q) is the arithmetic mean of (1/r) and (1/p).

Thus mod(1/p - 1/r) will be greater than mod(1/q - 1/p).
Quantity A is greater.

Friday, January 25, 2013

problem solving - chess board probability

What is the probability that two squares(smallest dimension) selected randomly from a chess board have only one common corner?

Explanation:



From the diagram you see that considering the top two rows, there are 14 ways of choosing two squares with just one common corner.

Like wise rows (2, 3), (3, 4)…….(7, 8) can be considered.
Therefore no of ways = 14 x 7 = 98

Total no of ways of choosing two squares from a chess board = 64C2

Probability = 

 =



problem solving - general algebra

When four is added to six times a number and the result squared, the result obtained is four times the square of the sum of the number and its next multiple. What is the number?

Explanation:


The question can be directly converted to an equation as follows.
Let ‘n’ be the number.
(4 + 6n)2 = 4(n + 2n)2
22(2 + 3n)2 = 4(3n)2
4 + 12n + 9n2 = 9n2
n =