31). The LCM of two integers is 1237. What is their HCF?
A). 37 
B). 19 
C). 1 
D). Cannot be determined 

32). What is \( \Large \frac{1}{ab} \)\( \Large \frac{1}{a+b} \)\( \Large \frac{2b}{a^{2}+b^{2}} \)\( \Large \frac{4b^{3}}{a^{4}+b^{4}} \)\( \Large \frac{8b^{7}}{a^{8}b^{8}} \)
A). \( a+b\) 
B). \( ab\) 
C). \( 1\) 
D). \( 0\) 

33). A man rides one third of the distances from A to B at the rate of \( x\)km/hr and the remainder at the rate of \( 2y\) km/hr. If he had travelled at a uniform rate of \( 6z\) km/hr, he could have ridden from A to B and back again in he same time Which one of the following is correct?
A). \( z\) = \( x\) + \( y\) 
B). \(3 z\) = \( x\) + \( y\) 
C). \( \Large \frac{1}{z} \) = \( \Large \frac{1}{x} \)+\( \Large \frac{1}{y} \) 
D). \( \Large \frac{1}{2z} \) = \( \Large \frac{1}{x} \)+\( \Large \frac{1}{y} \) 

34). If \( ax\) + \( by\)  \( 2\) = 0 and \( axby\) = \( 1\), where \( a\) \( \ne\)\( 0\),\( b\) \( \ne\)\( 0\) then what is \( \left( a^{2} x + b^{2 y}\right)\) equal to?
A). \( a\) +\( b\) 
B). \( 2ab\) 
C). \( a^{3}\) +\( b^{3}\) 
D). \( a^{4}\) +\( b^{4}\) 

35). Consider the following statements : 1. \( \left( a  b  c \right)\) is one of the factors of \( 3abc\) + \( b^{3}\) + \( c^{3}\)  \( a^{3}\) 2. \( \left( b + c  1 \right)\) is one of the factors of \( 3abc\) + \( b^{3}\) + \( c^{3}\)  \( 1\) Which of the above statements is/are correct?
A). 1 only 
B). 2 only 
C). Both 1 and 2 
D). Neither 1 nor 2 

36). \( 7^{10}\)  \( 5^{10}\) is divisible by
A). 10 
B). 7 
C). 5 
D). 11 

37). Consider the following statements in respect of four spheres A, B, C and D having respective radii 6, 8, 10 and 12 cm : 1. The surface area of sphere C is equal to the sum of surface areas of spheres A and B. 2. The volume of sphere D is equal to the sum of volumes of spheres A, B and C. Which of the above statements is/are correct?
A). 1 only 
B). 2 only 
C). Both 1 and 2 
D). Neither 1 nor 2 

38). What is the number of divisors of 360?
A). 12 
B). 18 
C). 24 
D). None of these 

39). The multiplication of a threedigit number \( XY5\)with digit \( Z\) yields \( X215.\)What is \( X + Y + Z \)equal to?
A). 13 
B). 15 
C). 17 
D). 18 

40). If the equation \( x^{2}\) + \( 2\)\( \left( 1+k\right)\)\( x\) + \( k^{2}\) = \( 0\) has equal roots, then what is the value of k?
A). \( \Large \frac{1}{2} \) 
B). \( \Large \frac{1}{2} \) 
C). \( 1\) 
D). \( 1\) 
