Q: Let a=3 2 and b= 1 5^ 1 / 6 6 . If x, y R are such that gathered 3 x+2 y= _a(18)^ 5 4 \\2 x-y= _b( 1080 ) gathered and then 4 x+5 y is equal to . ________.

aligned & 3 x+2 y= _ 3 2 (3 2 )^ 5 2 = 5 2 \\ & 6 x+4 y=5 (1) aligned 2 x-y= _ 1 5^ 1 / 6 6 (5^ 1 6 6 )^3=-3 array ll & 2 x-y=-3 .....(2)\\ & equation (1) -(2) \\ & 4 x+5 y=8 array

Question 1

Let a=3 2 and b= 1 5^ 1 / 6 6 . If x, y R are such that gathered 3 x+2 y= _a(18)^ 5 4 \2 x-y= _b( 1080 ) gathered and then 4 x+5 y is equal to . ________.

Accepted Answer

aligned & 3 x+2 y= _ 3 2 (3 2 )^ 5 2 = 5 2 \ & 6 x+4 y=5 (1) aligned 2 x-y= _ 1 5^ 1 / 6 6 (5^ 1 6 6 )^3=-3 array ll & 2 x-y=-3 .....(2)\ & equation (1) -(2) \ & 4 x+5 y=8 array

Question 2

The product of all positive real values of x satisfying the equation x 16 log 5 x 3 - 68 log 5 x = 5 - 16 is

Accepted Answer

Taking log to the base 5  on both sides 16 log 5 x 3 - 68 log 5 x log 5 x = - 16 Let log 5 x = t 16 t 4 - 68 t 2 + 16 = 0 &#8658; 4 t 4 - 16 t 2 - t 2 + 4 = 0 &#8658; 4 t 2 - 1 t 2 - 4 = 0 &#8658; t = &#177; 1 2 , &#177; 2 So log 5 x = &#177; 1 2 or  &#177; 2 &#8658; x = 5 1 2 , &#160; 5 - 1 2 , &#160; 5 2 , &#160; 5 - 2 Hence, the product is  5 &#215; 1 5 &#215; 25 &#215; 1 25 = 1

Basic of Mathematics — Previous Year Questions & Solutions