Q30) Find the sum of all positive integral values of n such that 1001 + n^2 is a perfect square a) 140 b) 548 c) 640 d) 671

Let us number the vertices from 1 – 6, there are three different types of triangles that can be formed.Type 1 : (1, 2, 3) [ this is same as (2, 3, 4), (3, 4, 5) … etc)Type 2 : (1, 2, 4) [ this is same as (1, 2, 5) (2, 3, 5) (3, 4, 6)]Type 3 : (1, 3, 5) and (2, 4, 6) have same shape and area So there will be three triangles of different areas that can be formed from a regular hexagon.There are 6 triangles of Type 1, 12 of type 2 and 2 of type 3.Adding these we get 6 + 12 + = 20 triangles. Total number of triangles possible = 6C3 = 20

S = 1 + 4x + 7x^2 + 10x^3 + … Sx = x + 4x^2 + 7x^3 + 10x^4 + … S (1 – x) = 1 + 3x + 3x^2 + 3x^3 + … S (1 – x) = 1 + 3x/( 1 – x) 35/16(1 – x) = (1 – x + 3x) / ( 1 – x) = ( 1 + 2x ) / ( 1 – x) 35 (1 – x) ^2 = 16 (1 + 2x ) 35x^2 – 102x + 19 = 0 (7x – 19) (5x – 1) = 0 x = 19/7, 1/5 But x # 19/7 (because 19/7 > 1) So x = 1/5

(x+y)/xy = 1/x + 1/y Only (1,1) and (2,2) satisfy the equation Hence option C is correct.

Q100) There are 2n consecutive natural numbers. If (n + 1) numbers are randomly selected, then what is the probability that their HCF is 1?

Q30) Pipe A takes 3/4 of the times required by pipe B to fill the empty tank individually. When an outlet pipe C also opened simultaneously with pipe A and pipe B it takes 3/4 more time to fill the empty tank than it takes when only pipe A and pipe B are opened together. if it takes to fill 33hrs when all the 3 pipes are opened simultaneously, then in what time pipe C can empty the full tank by operating alone.

Q100) There is a 3 digit no. ABC. The no. Is a perfect square and the no. of factors of ABC is also a perfect square. If A + B + C is also a perfect square then what is the no of factors of the six digit number ABCABC.what is the no of factors of the 6 digit no ABCABC if the cube of the product of the digits of the no ABC is not divisible by 5?

OA : required probability is 1