Match the following Column I Column IIAIf log45 75 - x and log135 375 - y then xy - 5x - y equalsp3BThe number of real solutions of the equation Xlog3X2-9-2log3X-5-33-2q1CThe number of real solutions x- y- z of the system of equations log2xy - logx logy- logyz - logy logz log2zx - logz logx isr4DIf log1227 -x and log616 -y then y3-x-3-x equalss2

(A) (log 75)log 45 =x and nbsp; log 375log 135=y ⇒ xy+5(x y)= ( log 3+2log 52log +log 5 ) ( log 5+3log 53log 3+log 5 )+5 [ log 3+3log 5 log 3+3log 53log 3+l ...

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Byju's Answer

Standard XII

Mathematics

Use of Monotonicity for Proving Inequalities

Match the fol...

Question

Match the following

Column I Column II

(A) If $l o g_{45} 75 = x$ and $l o g_{135} 375 = y$ then $x y + 5 (x - y)$ equals (p) 3

(B) The number of real solutions of the equation $X^{(l o g_{3} X)^{2} - \frac{9}{2} (l o g_{3} X - 5)} = 3^{3 / 2}$ (q) 1

(C) The number of real solutions (x, y, z) of the system of equations log(2xy) = logx logy, logyz = logy logz log(2zx) = logz logx is (r) 4

(D) If $l o g_{12} 27 = x$ and $l o g_{6} 16 = y t h e n \frac{y (3 + x)}{(3 - x)}$ equals (s) 2

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Solution

(A)
$\frac{(log 75)}{log 45} = x and \frac{log 375}{log 135} = y$
$\Rightarrow x y + 5 (x - y) = (\frac{log 3 + 2 log 5}{2 log + log 5}) (\frac{log 5 + 3 log 5}{3 log 3 + log 5}) + 5 [\frac{log 3 + 3 log 5}{} - \frac{log 3 + 3 log 5}{3 log 3 + log 5}] = 1$
(B)
$[({log}_{3} X)^{2} - \frac{9}{2} ({log}_{3} X)] {log}_{3} X = \frac{3}{2}$
Number of solutions $\to 3$
(C) 2
(D) 4

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	Column I		Column II
(A)	If $l o g_{45} 75 = x$ and $l o g_{135} 375 = y$ then $x y + 5 (x - y)$ equals	(p)	3
(B)	The number of real solutions of the equation $X^{(l o g_{3} X)^{2} - \frac{9}{2} (l o g_{3} X - 5)} = 3^{3 / 2}$	(q)	1
(C)	The number of real solutions (x, y, z) of the system of equations log(2xy) = logx logy, logyz = logy logz log(2zx) = logz logx is	(r)	4
(D)	If $l o g_{12} 27 = x$ and $l o g_{6} 16 = y t h e n \frac{y (3 + x)}{(3 - x)}$ equals	(s)	2

(A) (log75)log45=x and log375log135=y ⇒xy+5(x−y)=(log3+2log52log+log5)(log5+3log53log3+log5)+5[log3+3log5−log3+3log53log3+log5]=1 (B) [(log3X)2−92(log3X)]log3X=32 Number of solutions →3 (C) 2 (D) 4